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 , , , , , , , , where denotes a non-negative 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

AUTHORS:

• David Roe <roed@math.harvard.edu>
• John Cremona

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

Returns 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: 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

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

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

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

__cmp__(other)

Standard comparison function for Kodaira Symbols.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.kodaira_symbol import KodairaSymbol_class
sage: KS1 = KodairaSymbol_class(15); KS1
I11
sage: KS2 = KodairaSymbol_class(-34); KS2
I30*
sage: KS1 < KS2
True
sage: KS2 < KS1
False

sage: Klist = [KodairaSymbol_class(i) for i in [-10..10] if i!=0]
sage: Klist.sort()
sage: Klist
[I0,
I0*,
I1,
I1*,
I2,
I2*,
I3,
I3*,
I4,
I4*,
I5,
I5*,
I6,
I6*,
II,
II*,
III,
III*,
IV,
IV*]

__init__(symbol)

Constructor for Kodaira Symbol class.

INPUT:

• symbol (string or integer) – The string should be a

  standard string representation (e.g. III*) of a Kodaira symbol, which will be parsed. Alternatively, use the Pari encoding of Kodaira symbols as integers.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.kodaira_symbol import KodairaSymbol_class
sage: KodairaSymbol_class(14)
I10
sage: KodairaSymbol_class('III*')
III*

__repr__()

Return the string representation of this Kodaira Symbol.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.kodaira_symbol import KodairaSymbol_class
sage: KS = KodairaSymbol_class(15)
sage: str(KS) # indirect doctest
'I11'

__weakref__

list of weak references to the object (if defined)

_latex_()

Return the string representation of this Kodaira Symbol.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.kodaira_symbol import KodairaSymbol_class
sage: KS = KodairaSymbol_class(15)
sage: latex(KS)
$I_{11}$

_pari_code()

Return the Pari encoding of this Kodaira Symbol.

EXAMPLES:

sage: KodairaSymbol('I0')._pari_code()
1
sage: KodairaSymbol('I10')._pari_code()
14
sage: KodairaSymbol('I10*')._pari_code()
-14
sage: [KodairaSymbol(s)._pari_code() for s in ['II','III','IV']]
[2, 3, 4]
sage: [KodairaSymbol(s)._pari_code() for s in ['II*','III*','IV*']]
[-2, -3, -4]