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 \(\rm{I}_m\), \(\rm{II}\), \(\rm{III}\), \(\rm{IV}\), \(\rm{I}^*_m\), \(\rm{II}^*\), \(\rm{III}^*\), \(\rm{IV}^*\), where \(m\) 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 |
|---|---|---|
| \(\rm{I}_0\) | \(0\) | \(1\) |
| \(\rm{I}^*_0\) | \(1\) | \(-1\) |
| \(\rm{I}_m\) \((m>0)\) | \(10m\) | \(m+4\) |
| \(\rm{I}^*_m\) \((m>0)\) | \(10m+1\) | \(-(m+4)\) |
| \(\rm{II}\) | \(2\) | \(2\) |
| \(\rm{III}\) | \(3\) | \(3\) |
| \(\rm{IV}\) | \(4\) | \(4\) |
| \(\rm{II}^*\) | \(7\) | \(-2\) |
| \(\rm{III}^*\) | \(6\) | \(-3\) |
| \(\rm{IV}^*\) | \(5\) | \(-4\) |
AUTHORS:
- David Roe <roed@math.harvard.edu>
- John Cremona
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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: 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
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class
sage.schemes.elliptic_curves.kodaira_symbol.KodairaSymbol_class(symbol)¶ Bases:
sage.structure.sage_object.SageObjectClass 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.