Kodaira symbols have their own type KodSym. Apart from the two functions that determine symbols for elliptic curves, there is a special creation function and a comparison operator, to test Kodaira symbols.

KodairaSymbol(E, p) : GeomEC -> MonStgElt

Given an elliptic curve E defined over Q and a prime number p, this function returns the reduction type of E modulo p in the form of a Kodaira symbol.

Given an elliptic curve E defined over Q, this function returns the reduction types of E modulo the bad primes in the form of a sequence of Kodaira symbols.

Given a string s, return a Kodaira symbol it represents. The values of s that are allowed are: "I0", "I1", "I2", ..., "In", "II", "III", "IV", and "I0*", "I1*", "I2*", ..., "In*", "II*", "III*", "IV*". The dots stand for "Ik" with k a positive integer. The `generic' type "In" is provided to be able to match types "I"n for any integer n>0 (and similarly for "In*"). This creation function is primarily useful for comparing values returned by the functions determining the Kodaira symbol for a curve.

Given two Kodaira symbols h and k, this function returns true if and only if either both are identical, or one is generic (of the form "In", or "In*") and the other is specific of the same type: "In" will compare equal with any of "I1", "I2", "I3", etc., and "In*" will compare equal with any of "I1*", "I2*", "I3*", etc. Note however that "In" and "I3" are different from the point of view of set creation.

> for n := 2 to 100 do                              
>    E := EllipticCurve([n,0]);
>    for p in BadPrimes(E) do
>       if KodairaSymbol(E, p) eq KodairaSymbol("I0*") then
>          print p, n;
>       end if;
>    end for;
> end for;                          
3 9
3 18
5 25
3 36
3 45
7 49
5 50
3 63
3 72
5 75
3 90
7 98
3 99
5 100