Kodaira symbols

Kodaira symbols encode the type of reduction of an elliptic curve at a (finite) place.

The standard notation for Kodaira Symbols is as a string which is one of ${\mathrm{I}}_{\mathrm{m}}$, $\mathrm{I}\mathrm{I}$, $\mathrm{I}\mathrm{I}\mathrm{I}$, $\mathrm{I}\mathrm{V}$, ${\mathrm{I}}_{\mathrm{m}}^{\ast }$, ${\mathrm{I}\mathrm{I}}^{\ast }$, ${\mathrm{I}\mathrm{I}\mathrm{I}}^{\ast }$, ${\mathrm{I}\mathrm{V}}^{\ast }$, where $m$ denotes a nonnegative integer. These have been encoded by single integers by different people. For convenience we give here the conversion table between strings, the eclib coding and the PARI encoding.

Kodaira Symbol	Eclib coding	PARI Coding
${\mathrm{I}}_0$	$0$	$1$
${\mathrm{I}}_0^{\ast }$	$1$	$-1$
${\mathrm{I}}_{\mathrm{m}}$ $(m>0)$	$10m$	$m+4$
${\mathrm{I}}_{\mathrm{m}}^{\ast }$ $(m>0)$	$10m+1$	$-(m+4)$
$\mathrm{I}\mathrm{I}$	$2$	$2$
$\mathrm{I}\mathrm{I}\mathrm{I}$	$3$	$3$
$\mathrm{I}\mathrm{V}$	$4$	$4$
${\mathrm{I}\mathrm{I}}^{\ast }$	$7$	$-2$
${\mathrm{I}\mathrm{I}\mathrm{I}}^{\ast }$	$6$	$-3$
${\mathrm{I}\mathrm{V}}^{\ast }$	$5$	$-4$

AUTHORS:

• David Roe <roed@math.harvard.edu>

• John Cremona

sage.schemes.elliptic_curves.kodaira_symbol.KodairaSymbol(symbol)

Return the specified Kodaira symbol.

INPUT:

• symbol – string or integer; either a string of the form “I0”, “I1”, …, “In”, “II”, “III”, “IV”, “I0*”, “I1*”, …, “In*”, “II*”, “III*”, or “IV*”, or an integer encoding a Kodaira symbol using PARI’s conventions

OUTPUT:

(KodairaSymbol) The corresponding Kodaira symbol.

EXAMPLES:

Sage

sage: KS = KodairaSymbol
sage: [KS(n) for n in range(1,10)]
[I0, II, III, IV, I1, I2, I3, I4, I5]
sage: [KS(-n) for n in range(1,10)]
[I0*, II*, III*, IV*, I1*, I2*, I3*, I4*, I5*]
sage: all(KS(str(KS(n))) == KS(n) for n in range(-10,10) if n != 0)
True

Python

>>> from sage.all import *
>>> KS = KodairaSymbol
>>> [KS(n) for n in range(Integer(1),Integer(10))]
[I0, II, III, IV, I1, I2, I3, I4, I5]
>>> [KS(-n) for n in range(Integer(1),Integer(10))]
[I0*, II*, III*, IV*, I1*, I2*, I3*, I4*, I5*]
>>> all(KS(str(KS(n))) == KS(n) for n in range(-Integer(10),Integer(10)) if n != Integer(0))
True

Sage Live

KS = KodairaSymbol
[KS(n) for n in range(1,10)]
[KS(-n) for n in range(1,10)]
all(KS(str(KS(n))) == KS(n) for n in range(-10,10) if n != 0)

class sage.schemes.elliptic_curves.kodaira_symbol.KodairaSymbol_class(symbol)

Bases: SageObject

Class to hold a Kodaira symbol of an elliptic curve over a $p$-adic local field.

Users should use the KodairaSymbol() function to construct Kodaira Symbols rather than use the class constructor directly.

Make live