Eta-products on modular curves \(X_0(N)\)¶

This package provides a class for representing eta-products, which are meromorphic functions on modular curves of the form

\[\prod_{d | N} \eta(q^d)^{r_d}\]

where \(\eta(q)\) is Dirichlet’s eta function

\[q^{1/24} \prod_{n = 1}^\infty(1-q^n) .\]

These are useful for obtaining explicit models of modular curves.

See Issue #3934 for background.

AUTHOR:

David Loeffler (2008-08-22): initial version

sage.modular.etaproducts.AllCusps(N)[source]¶

Return a list of CuspFamily objects corresponding to the cusps of \(X_0(N)\).

INPUT:

N – integer; the level

EXAMPLES:

Sage

sage: AllCusps(18)
[(Inf), (c_{2}), (c_{3,1}), (c_{3,2}), (c_{6,1}), (c_{6,2}), (c_{9}), (0)]
sage: AllCusps(0)
Traceback (most recent call last):
...
ValueError: N must be positive

Python

>>> from sage.all import *
>>> AllCusps(Integer(18))
[(Inf), (c_{2}), (c_{3,1}), (c_{3,2}), (c_{6,1}), (c_{6,2}), (c_{9}), (0)]
>>> AllCusps(Integer(0))
Traceback (most recent call last):
...
ValueError: N must be positive

Sage Live

AllCusps(18)
AllCusps(0)

class sage.modular.etaproducts.CuspFamily(N, width, label=None)[source]¶

Bases: SageObject

A family of elliptic curves parametrising a region of \(X_0(N)\).

level()[source]¶

Return the level of this cusp.

EXAMPLES:

Sage

sage: e = CuspFamily(10, 1)
sage: e.level()
10

Python

>>> from sage.all import *
>>> e = CuspFamily(Integer(10), Integer(1))
>>> e.level()
10

Sage Live

e = CuspFamily(10, 1)
e.level()

sage_cusp()[source]¶

Return the corresponding element of \(\mathbb{P}^1(\QQ)\).

EXAMPLES:

Sage

sage: CuspFamily(10, 1).sage_cusp() # not implemented
Infinity

Python

>>> from sage.all import *
>>> CuspFamily(Integer(10), Integer(1)).sage_cusp() # not implemented
Infinity

Sage Live

CuspFamily(10, 1).sage_cusp() # not implemented

width()[source]¶

Return the width of this cusp.

EXAMPLES:

Sage

sage: e = CuspFamily(10, 1)
sage: e.width()
1

Python

>>> from sage.all import *
>>> e = CuspFamily(Integer(10), Integer(1))
>>> e.width()
1

Sage Live

e = CuspFamily(10, 1)
e.width()

sage.modular.etaproducts.EtaGroup(level)[source]¶

Create the group of eta products of the given level.

EXAMPLES:

Sage

sage: EtaGroup(12)
Group of eta products on X_0(12)
sage: EtaGroup(1/2)
Traceback (most recent call last):
...
TypeError: Level (=1/2) must be a positive integer
sage: EtaGroup(0)
Traceback (most recent call last):
...
ValueError: Level (=0) must be a positive integer

Python

>>> from sage.all import *
>>> EtaGroup(Integer(12))
Group of eta products on X_0(12)
>>> EtaGroup(Integer(1)/Integer(2))
Traceback (most recent call last):
...
TypeError: Level (=1/2) must be a positive integer
>>> EtaGroup(Integer(0))
Traceback (most recent call last):
...
ValueError: Level (=0) must be a positive integer

Sage Live

EtaGroup(12)
EtaGroup(1/2)
EtaGroup(0)

class sage.modular.etaproducts.EtaGroupElement(parent, rdict)[source]¶

Bases: Element

Create an eta product object. Usually called implicitly via EtaGroup_class.__call__ or the EtaProduct factory function.

EXAMPLES:

Sage

sage: EtaProduct(8, {1:24, 2:-24})
Eta product of level 8 : (eta_1)^24 (eta_2)^-24
sage: g = _; g == loads(dumps(g))
True
sage: TestSuite(g).run()

Python

>>> from sage.all import *
>>> EtaProduct(Integer(8), {Integer(1):Integer(24), Integer(2):-Integer(24)})
Eta product of level 8 : (eta_1)^24 (eta_2)^-24
>>> g = _; g == loads(dumps(g))
True
>>> TestSuite(g).run()

Sage Live

EtaProduct(8, {1:24, 2:-24})
g = _; g == loads(dumps(g))
TestSuite(g).run()

degree()[source]¶

Return the degree of self as a map \(X_0(N) \to \mathbb{P}^1\).

This is the sum of all the positive coefficients in the divisor of self.

EXAMPLES:

Sage

sage: e = EtaProduct(12, {1:-336, 2:576, 3:696, 4:-216, 6:-576, 12:-144})
sage: e.degree()
230

Python

>>> from sage.all import *
>>> e = EtaProduct(Integer(12), {Integer(1):-Integer(336), Integer(2):Integer(576), Integer(3):Integer(696), Integer(4):-Integer(216), Integer(6):-Integer(576), Integer(12):-Integer(144)})
>>> e.degree()
230

Sage Live

e = EtaProduct(12, {1:-336, 2:576, 3:696, 4:-216, 6:-576, 12:-144})
e.degree()

divisor()[source]¶

Return the divisor of self, as a formal sum of CuspFamily objects.

EXAMPLES:

Sage

sage: e = EtaProduct(12, {1:-336, 2:576, 3:696, 4:-216, 6:-576, 12:-144})
sage: e.divisor() # FormalSum seems to print things in a random order?
-131*(Inf) - 50*(c_{2}) + 11*(0) + 50*(c_{6}) + 169*(c_{4}) - 49*(c_{3})
sage: e = EtaProduct(2^8, {8:1,32:-1})
sage: e.divisor() # random
-(c_{2}) - (Inf) - (c_{8,2}) - (c_{8,3}) - (c_{8,4}) - (c_{4,2})
 - (c_{8,1}) - (c_{4,1}) + (c_{32,4}) + (c_{32,3}) + (c_{64,1})
 + (0) + (c_{32,2}) + (c_{64,2}) + (c_{128}) + (c_{32,1})

Python

>>> from sage.all import *
>>> e = EtaProduct(Integer(12), {Integer(1):-Integer(336), Integer(2):Integer(576), Integer(3):Integer(696), Integer(4):-Integer(216), Integer(6):-Integer(576), Integer(12):-Integer(144)})
>>> e.divisor() # FormalSum seems to print things in a random order?
-131*(Inf) - 50*(c_{2}) + 11*(0) + 50*(c_{6}) + 169*(c_{4}) - 49*(c_{3})
>>> e = EtaProduct(Integer(2)**Integer(8), {Integer(8):Integer(1),Integer(32):-Integer(1)})
>>> e.divisor() # random
-(c_{2}) - (Inf) - (c_{8,2}) - (c_{8,3}) - (c_{8,4}) - (c_{4,2})
 - (c_{8,1}) - (c_{4,1}) + (c_{32,4}) + (c_{32,3}) + (c_{64,1})
 + (0) + (c_{32,2}) + (c_{64,2}) + (c_{128}) + (c_{32,1})

Sage Live

e = EtaProduct(12, {1:-336, 2:576, 3:696, 4:-216, 6:-576, 12:-144})
e.divisor() # FormalSum seems to print things in a random order?
e = EtaProduct(2^8, {8:1,32:-1})
e.divisor() # random

is_one()[source]¶

Return whether self is the one of the monoid.

EXAMPLES:

Sage

sage: e = EtaProduct(3, {3:12, 1:-12})
sage: e.is_one()
False
sage: e.parent().one().is_one()
True
sage: ep = EtaProduct(5, {})
sage: ep.is_one()
True
sage: ep.parent().one() == ep
True

Python

>>> from sage.all import *
>>> e = EtaProduct(Integer(3), {Integer(3):Integer(12), Integer(1):-Integer(12)})
>>> e.is_one()
False
>>> e.parent().one().is_one()
True
>>> ep = EtaProduct(Integer(5), {})
>>> ep.is_one()
True
>>> ep.parent().one() == ep
True

Sage Live

e = EtaProduct(3, {3:12, 1:-12})
e.is_one()
e.parent().one().is_one()
ep = EtaProduct(5, {})
ep.is_one()
ep.parent().one() == ep

level()[source]¶

Return the level of this eta product.

EXAMPLES:

Sage

sage: e = EtaProduct(3, {3:12, 1:-12})
sage: e.level()
3
sage: EtaProduct(12, {6:6, 2:-6}).level() # not the lcm of the d's
12
sage: EtaProduct(36, {6:6, 2:-6}).level() # not minimal
36

Python

>>> from sage.all import *
>>> e = EtaProduct(Integer(3), {Integer(3):Integer(12), Integer(1):-Integer(12)})
>>> e.level()
3
>>> EtaProduct(Integer(12), {Integer(6):Integer(6), Integer(2):-Integer(6)}).level() # not the lcm of the d's
12
>>> EtaProduct(Integer(36), {Integer(6):Integer(6), Integer(2):-Integer(6)}).level() # not minimal
36

Sage Live

e = EtaProduct(3, {3:12, 1:-12})
e.level()
EtaProduct(12, {6:6, 2:-6}).level() # not the lcm of the d's
EtaProduct(36, {6:6, 2:-6}).level() # not minimal

order_at_cusp(cusp)[source]¶

Return the order of vanishing of self at the given cusp.

INPUT:

cusp – a CuspFamily object

OUTPUT: integer

EXAMPLES:

Sage

sage: e = EtaProduct(2, {2:24, 1:-24})
sage: e.order_at_cusp(CuspFamily(2, 1)) # cusp at infinity
1
sage: e.order_at_cusp(CuspFamily(2, 2)) # cusp 0
-1

Python

>>> from sage.all import *
>>> e = EtaProduct(Integer(2), {Integer(2):Integer(24), Integer(1):-Integer(24)})
>>> e.order_at_cusp(CuspFamily(Integer(2), Integer(1))) # cusp at infinity
1
>>> e.order_at_cusp(CuspFamily(Integer(2), Integer(2))) # cusp 0
-1

Sage Live

e = EtaProduct(2, {2:24, 1:-24})
e.order_at_cusp(CuspFamily(2, 1)) # cusp at infinity
e.order_at_cusp(CuspFamily(2, 2)) # cusp 0

q_expansion(n)[source]¶

Return the \(q\)-expansion of self at the cusp at infinity.

INPUT:

n – integer; number of terms to calculate

OUTPUT:

A power series over \(\ZZ\) in the variable \(q\), with a relative precision of \(1 + O(q^n)\).

ALGORITHM: Calculates eta to (n/m) terms, where m is the smallest integer dividing self.level() such that self.r(m) != 0. Then multiplies.

EXAMPLES:

Sage

sage: EtaProduct(36, {6:6, 2:-6}).q_expansion(10)
q + 6*q^3 + 27*q^5 + 92*q^7 + 279*q^9 + O(q^11)
sage: R.<q> = ZZ[[]]
sage: EtaProduct(2,{2:24,1:-24}).q_expansion(100) == delta_qexp(101)(q^2)/delta_qexp(101)(q)
True

Python

>>> from sage.all import *
>>> EtaProduct(Integer(36), {Integer(6):Integer(6), Integer(2):-Integer(6)}).q_expansion(Integer(10))
q + 6*q^3 + 27*q^5 + 92*q^7 + 279*q^9 + O(q^11)
>>> R = ZZ[['q']]; (q,) = R._first_ngens(1)
>>> EtaProduct(Integer(2),{Integer(2):Integer(24),Integer(1):-Integer(24)}).q_expansion(Integer(100)) == delta_qexp(Integer(101))(q**Integer(2))/delta_qexp(Integer(101))(q)
True

Sage Live

EtaProduct(36, {6:6, 2:-6}).q_expansion(10)
R.<q> = ZZ[[]]
EtaProduct(2,{2:24,1:-24}).q_expansion(100) == delta_qexp(101)(q^2)/delta_qexp(101)(q)

qexp(n)[source]¶

Alias for self.q_expansion().

EXAMPLES:

Sage

sage: e = EtaProduct(36, {6:8, 3:-8})
sage: e.qexp(10)
q + 8*q^4 + 36*q^7 + O(q^10)
sage: e.qexp(30) == e.q_expansion(30)
True

Python

>>> from sage.all import *
>>> e = EtaProduct(Integer(36), {Integer(6):Integer(8), Integer(3):-Integer(8)})
>>> e.qexp(Integer(10))
q + 8*q^4 + 36*q^7 + O(q^10)
>>> e.qexp(Integer(30)) == e.q_expansion(Integer(30))
True

Sage Live

e = EtaProduct(36, {6:8, 3:-8})
e.qexp(10)
e.qexp(30) == e.q_expansion(30)

r(d)[source]¶

Return the exponent \(r_d\) of \(\eta(q^d)\) in self.

EXAMPLES:

Sage

sage: e = EtaProduct(12, {2:24, 3:-24})
sage: e.r(3)
-24
sage: e.r(4)
0

Python

>>> from sage.all import *
>>> e = EtaProduct(Integer(12), {Integer(2):Integer(24), Integer(3):-Integer(24)})
>>> e.r(Integer(3))
-24
>>> e.r(Integer(4))
0

Sage Live

e = EtaProduct(12, {2:24, 3:-24})
e.r(3)
e.r(4)

class sage.modular.etaproducts.EtaGroup_class(level)[source]¶

Bases: UniqueRepresentation, Parent

The group of eta products of a given level under multiplication.

Element[source]¶: alias of EtaGroupElement

basis(reduce=True)[source]¶

Produce a basis for the free abelian group of eta-products of level N (under multiplication), attempting to find basis vectors of the smallest possible degree.

INPUT:

reduce – boolean (default: True); whether or not to apply LLL-reduction to the calculated basis

EXAMPLES:

Sage

sage: EtaGroup(5).basis()
[Eta product of level 5 : (eta_1)^6 (eta_5)^-6]
sage: EtaGroup(12).basis()
[Eta product of level 12 : (eta_1)^-3 (eta_2)^2 (eta_3)^1 (eta_4)^-1 (eta_6)^-2 (eta_12)^3,
 Eta product of level 12 : (eta_1)^-4 (eta_2)^2 (eta_3)^4 (eta_6)^-2,
 Eta product of level 12 : (eta_1)^6 (eta_2)^-9 (eta_3)^-2 (eta_4)^3 (eta_6)^3 (eta_12)^-1,
 Eta product of level 12 : (eta_1)^-1 (eta_2)^3 (eta_3)^3 (eta_4)^-2 (eta_6)^-9 (eta_12)^6,
 Eta product of level 12 : (eta_1)^3 (eta_3)^-1 (eta_4)^-3 (eta_12)^1]
sage: EtaGroup(12).basis(reduce=False) # much bigger coefficients
[Eta product of level 12 : (eta_1)^384 (eta_2)^-576 (eta_3)^-696 (eta_4)^216 (eta_6)^576 (eta_12)^96,
 Eta product of level 12 : (eta_2)^24 (eta_12)^-24,
 Eta product of level 12 : (eta_1)^-40 (eta_2)^116 (eta_3)^96 (eta_4)^-30 (eta_6)^-80 (eta_12)^-62,
 Eta product of level 12 : (eta_1)^-4 (eta_2)^-33 (eta_3)^-4 (eta_4)^1 (eta_6)^3 (eta_12)^37,
 Eta product of level 12 : (eta_1)^15 (eta_2)^-24 (eta_3)^-29 (eta_4)^9 (eta_6)^24 (eta_12)^5]

Python

>>> from sage.all import *
>>> EtaGroup(Integer(5)).basis()
[Eta product of level 5 : (eta_1)^6 (eta_5)^-6]
>>> EtaGroup(Integer(12)).basis()
[Eta product of level 12 : (eta_1)^-3 (eta_2)^2 (eta_3)^1 (eta_4)^-1 (eta_6)^-2 (eta_12)^3,
 Eta product of level 12 : (eta_1)^-4 (eta_2)^2 (eta_3)^4 (eta_6)^-2,
 Eta product of level 12 : (eta_1)^6 (eta_2)^-9 (eta_3)^-2 (eta_4)^3 (eta_6)^3 (eta_12)^-1,
 Eta product of level 12 : (eta_1)^-1 (eta_2)^3 (eta_3)^3 (eta_4)^-2 (eta_6)^-9 (eta_12)^6,
 Eta product of level 12 : (eta_1)^3 (eta_3)^-1 (eta_4)^-3 (eta_12)^1]
>>> EtaGroup(Integer(12)).basis(reduce=False) # much bigger coefficients
[Eta product of level 12 : (eta_1)^384 (eta_2)^-576 (eta_3)^-696 (eta_4)^216 (eta_6)^576 (eta_12)^96,
 Eta product of level 12 : (eta_2)^24 (eta_12)^-24,
 Eta product of level 12 : (eta_1)^-40 (eta_2)^116 (eta_3)^96 (eta_4)^-30 (eta_6)^-80 (eta_12)^-62,
 Eta product of level 12 : (eta_1)^-4 (eta_2)^-33 (eta_3)^-4 (eta_4)^1 (eta_6)^3 (eta_12)^37,
 Eta product of level 12 : (eta_1)^15 (eta_2)^-24 (eta_3)^-29 (eta_4)^9 (eta_6)^24 (eta_12)^5]

Sage Live

EtaGroup(5).basis()
EtaGroup(12).basis()
EtaGroup(12).basis(reduce=False) # much bigger coefficients

ALGORITHM: An eta product of level \(N\) is uniquely determined by the integers \(r_d\) for \(d | N\) with \(d < N\), since \(\sum_{d | N} r_d = 0\). The valid \(r_d\) are those that satisfy two congruences modulo 24, and one congruence modulo 2 for every prime divisor of N. We beef up the congruences modulo 2 to congruences modulo 24 by multiplying by 12. To calculate the kernel of the ensuing map \(\ZZ^m \to (\ZZ/24\ZZ)^n\) we lift it arbitrarily to an integer matrix and calculate its Smith normal form. This gives a basis for the lattice.

This lattice typically contains “large” elements, so by default we pass it to the reduce_basis() function which performs LLL-reduction to give a more manageable basis.

level()[source]¶

Return the level of self.

EXAMPLES:

Sage

sage: EtaGroup(10).level()
10

Python

>>> from sage.all import *
>>> EtaGroup(Integer(10)).level()
10

Sage Live

EtaGroup(10).level()

one()[source]¶

Return the identity element of self.

EXAMPLES:

Sage

sage: EtaGroup(12).one()
Eta product of level 12 : 1

Python

>>> from sage.all import *
>>> EtaGroup(Integer(12)).one()
Eta product of level 12 : 1

Sage Live

EtaGroup(12).one()

reduce_basis(long_etas)[source]¶

Produce a more manageable basis via LLL-reduction.

INPUT:

long_etas – a list of EtaGroupElement objects (which should all be of the same level)

OUTPUT:

a new list of EtaGroupElement objects having hopefully smaller norm

ALGORITHM: We define the norm of an eta-product to be the \(L^2\) norm of its divisor (as an element of the free \(\ZZ\)-module with the cusps as basis and the standard inner product). Applying LLL-reduction to this gives a basis of hopefully more tractable elements. Of course we’d like to use the \(L^1\) norm as this is just twice the degree, which is a much more natural invariant, but \(L^2\) norm is easier to work with!

EXAMPLES:

Sage

sage: EtaGroup(4).reduce_basis([ EtaProduct(4, {1:8,2:24,4:-32}), EtaProduct(4, {1:8, 4:-8})])
[Eta product of level 4 : (eta_1)^8 (eta_4)^-8,
 Eta product of level 4 : (eta_1)^-8 (eta_2)^24 (eta_4)^-16]

Python

>>> from sage.all import *
>>> EtaGroup(Integer(4)).reduce_basis([ EtaProduct(Integer(4), {Integer(1):Integer(8),Integer(2):Integer(24),Integer(4):-Integer(32)}), EtaProduct(Integer(4), {Integer(1):Integer(8), Integer(4):-Integer(8)})])
[Eta product of level 4 : (eta_1)^8 (eta_4)^-8,
 Eta product of level 4 : (eta_1)^-8 (eta_2)^24 (eta_4)^-16]

Sage Live

EtaGroup(4).reduce_basis([ EtaProduct(4, {1:8,2:24,4:-32}), EtaProduct(4, {1:8, 4:-8})])

sage.modular.etaproducts.EtaProduct(level, dic)[source]¶

Create an EtaGroupElement object representing the function \(\prod_{d | N} \eta(q^d)^{r_d}\).

This checks the criteria of Ligozat to ensure that this product really is the \(q\)-expansion of a meromorphic function on \(X_0(N)\).

INPUT:

level – integer; the N such that this eta product is a function on \(X_0(N)\)
dic – a dictionary indexed by divisors of N such that the coefficient of \(\eta(q^d)\) is r[d]. Only nonzero coefficients need be specified. If Ligozat’s criteria are not satisfied, a ValueError will be raised.

OUTPUT:

An EtaGroupElement object, whose parent is the EtaGroup of level N and whose coefficients are the given dictionary.

Note

The dictionary dic does not uniquely specify N. It is possible for two EtaGroupElements with different \(N\)’s to be created with the same dictionary, and these represent different objects (although they will have the same \(q\)-expansion at the cusp \(\infty\)).

EXAMPLES:

Sage

sage: EtaProduct(3, {3:12, 1:-12})
Eta product of level 3 : (eta_1)^-12 (eta_3)^12
sage: EtaProduct(3, {3:6, 1:-6})
Traceback (most recent call last):
...
ValueError: sum d r_d (=12) is not 0 mod 24
sage: EtaProduct(3, {4:6, 1:-6})
Traceback (most recent call last):
...
ValueError: 4 does not divide 3

Python

>>> from sage.all import *
>>> EtaProduct(Integer(3), {Integer(3):Integer(12), Integer(1):-Integer(12)})
Eta product of level 3 : (eta_1)^-12 (eta_3)^12
>>> EtaProduct(Integer(3), {Integer(3):Integer(6), Integer(1):-Integer(6)})
Traceback (most recent call last):
...
ValueError: sum d r_d (=12) is not 0 mod 24
>>> EtaProduct(Integer(3), {Integer(4):Integer(6), Integer(1):-Integer(6)})
Traceback (most recent call last):
...
ValueError: 4 does not divide 3

Sage Live

EtaProduct(3, {3:12, 1:-12})
EtaProduct(3, {3:6, 1:-6})
EtaProduct(3, {4:6, 1:-6})

sage.modular.etaproducts.eta_poly_relations(eta_elements, degree, labels=['x1', 'x2'], verbose=False)[source]¶

Find polynomial relations between eta products.

INPUT:

eta_elements – list; a list of EtaGroupElement objects. Not implemented unless this list has precisely two elements. degree
degree – integer; the maximal degree of polynomial to look for
labels – list of strings; labels to use for the polynomial returned
verbose – boolean (default: False); if True, prints information as it goes

OUTPUT: list of polynomials which is a Groebner basis for the part of the ideal of relations between eta_elements which is generated by elements up to the given degree; or None, if no relations were found.

ALGORITHM: An expression of the form \(\sum_{0 \le i,j \le d} a_{ij} x^i y^j\) is zero if and only if it vanishes at the cusp infinity to degree at least \(v = d(deg(x) + deg(y))\). For all terms up to \(q^v\) in the \(q\)-expansion of this expression to be zero is a system of \(v + k\) linear equations in \(d^2\) coefficients, where \(k\) is the number of nonzero negative coefficients that can appear.

Solving these equations and calculating a basis for the solution space gives us a set of polynomial relations, but this is generally far from a minimal generating set for the ideal, so we calculate a Groebner basis.

As a test, we calculate five extra terms of \(q\)-expansion and check that this doesn’t change the answer.

EXAMPLES:

Sage

sage: from sage.modular.etaproducts import eta_poly_relations
sage: t = EtaProduct(26, {2:2,13:2,26:-2,1:-2})
sage: u = EtaProduct(26, {2:4,13:2,26:-4,1:-2})
sage: eta_poly_relations([t, u], 3)
sage: eta_poly_relations([t, u], 4)
[x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]

Python

>>> from sage.all import *
>>> from sage.modular.etaproducts import eta_poly_relations
>>> t = EtaProduct(Integer(26), {Integer(2):Integer(2),Integer(13):Integer(2),Integer(26):-Integer(2),Integer(1):-Integer(2)})
>>> u = EtaProduct(Integer(26), {Integer(2):Integer(4),Integer(13):Integer(2),Integer(26):-Integer(4),Integer(1):-Integer(2)})
>>> eta_poly_relations([t, u], Integer(3))
>>> eta_poly_relations([t, u], Integer(4))
[x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]

Sage Live

from sage.modular.etaproducts import eta_poly_relations
t = EtaProduct(26, {2:2,13:2,26:-2,1:-2})
u = EtaProduct(26, {2:4,13:2,26:-4,1:-2})
eta_poly_relations([t, u], 3)
eta_poly_relations([t, u], 4)

Use verbose=True to see the details of the computation:

Sage

sage: eta_poly_relations([t, u], 3, verbose=True)
Trying to find a relation of degree 3
Lowest order of a term at infinity = -12
Highest possible degree of a term = 15
Trying all coefficients from q^-12 to q^15 inclusive
No polynomial relation of order 3 valid for 28 terms
Check:
Trying all coefficients from q^-12 to q^20 inclusive
No polynomial relation of order 3 valid for 33 terms

Python

>>> from sage.all import *
>>> eta_poly_relations([t, u], Integer(3), verbose=True)
Trying to find a relation of degree 3
Lowest order of a term at infinity = -12
Highest possible degree of a term = 15
Trying all coefficients from q^-12 to q^15 inclusive
No polynomial relation of order 3 valid for 28 terms
Check:
Trying all coefficients from q^-12 to q^20 inclusive
No polynomial relation of order 3 valid for 33 terms

Sage Live

eta_poly_relations([t, u], 3, verbose=True)

Sage

sage: eta_poly_relations([t, u], 4, verbose=True)
Trying to find a relation of degree 4
Lowest order of a term at infinity = -16
Highest possible degree of a term = 20
Trying all coefficients from q^-16 to q^20 inclusive
Check:
Trying all coefficients from q^-16 to q^25 inclusive
[x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]

Python

>>> from sage.all import *
>>> eta_poly_relations([t, u], Integer(4), verbose=True)
Trying to find a relation of degree 4
Lowest order of a term at infinity = -16
Highest possible degree of a term = 20
Trying all coefficients from q^-16 to q^20 inclusive
Check:
Trying all coefficients from q^-16 to q^25 inclusive
[x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]

Sage Live

eta_poly_relations([t, u], 4, verbose=True)

Python

>>> from sage.all import *
>>> eta_poly_relations([t, u], Integer(4), verbose=True)
Trying to find a relation of degree 4
Lowest order of a term at infinity = -16
Highest possible degree of a term = 20
Trying all coefficients from q^-16 to q^20 inclusive
Check:
Trying all coefficients from q^-16 to q^25 inclusive
[x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]

Sage Live

eta_poly_relations([t, u], 4, verbose=True)

sage.modular.etaproducts.num_cusps_of_width(N, d)[source]¶

Return the number of cusps on \(X_0(N)\) of width d.

INPUT:

N – integer; the level
d – integer; an integer dividing \(N\), the cusp width

EXAMPLES:

Sage

sage: from sage.modular.etaproducts import num_cusps_of_width
sage: [num_cusps_of_width(18,d) for d in divisors(18)]
[1, 1, 2, 2, 1, 1]
sage: num_cusps_of_width(4,8)
Traceback (most recent call last):
...
ValueError: N and d must be positive integers with d|N

Python

>>> from sage.all import *
>>> from sage.modular.etaproducts import num_cusps_of_width
>>> [num_cusps_of_width(Integer(18),d) for d in divisors(Integer(18))]
[1, 1, 2, 2, 1, 1]
>>> num_cusps_of_width(Integer(4),Integer(8))
Traceback (most recent call last):
...
ValueError: N and d must be positive integers with d|N

Sage Live

from sage.modular.etaproducts import num_cusps_of_width
[num_cusps_of_width(18,d) for d in divisors(18)]
num_cusps_of_width(4,8)

sage.modular.etaproducts.qexp_eta(ps_ring, prec)[source]¶

Return the \(q\)-expansion of \(\eta(q) / q^{1/24}\).

Here \(\eta(q)\) is Dedekind’s function

\[\eta(q) = q^{1/24}\prod_{n=1}^\infty (1-q^n).\]

The result is an element of ps_ring, with precision prec.

INPUT:

ps_ring – PowerSeriesRing; a power series ring
prec – integer; the number of terms to compute

OUTPUT: an element of ps_ring which is the \(q\)-expansion of \(\eta(q)/q^{1/24}\) truncated to prec terms.

ALGORITHM: We use the Euler identity

\[\eta(q) = q^{1/24}( 1 + \sum_{n \ge 1} (-1)^n (q^{n(3n+1)/2} + q^{n(3n-1)/2})\]

to compute the expansion.

EXAMPLES:

Sage

sage: from sage.modular.etaproducts import qexp_eta
sage: qexp_eta(ZZ[['q']], 100)
1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + q^22 + q^26 - q^35 - q^40 + q^51 + q^57 - q^70 - q^77 + q^92 + O(q^100)

Python

>>> from sage.all import *
>>> from sage.modular.etaproducts import qexp_eta
>>> qexp_eta(ZZ[['q']], Integer(100))
1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + q^22 + q^26 - q^35 - q^40 + q^51 + q^57 - q^70 - q^77 + q^92 + O(q^100)

Sage Live

from sage.modular.etaproducts import qexp_eta
qexp_eta(ZZ[['q']], 100)