kash% k := GF(5);
Finite field of size 5
kash% kx := PolynomialRing(k);
Univariate Polynomial Ring over GF(5)
kash% AssignNames_(kx, ["x"]);
kash% x := kx.1;
x
kash% kxy := PolynomialRing(kx);
Univariate Polynomial Ring over Univariate Polynomial Ring over GF(5)
kash% AssignNames_(kxy, ["y"]);
kash% y := kxy.1;
y
kash% f := y^2 - x^3 + 1;
y^2 + 4*x^3 + 1
kash% F := FunctionField(f);
Algebraic function field defined over Univariate rational function field 
over GF(5) by y^2 + 4*x^3 + 1
kash% Genus(F);
1
kash% ExactConstantField(F);
Finite field of size 5, extended by:
  ext1 := Mapping from: fld^fin: k to fld^fun: F
kash% x := Coerce(F, x);
x
kash% AssignNames_(F, ["y"]);
kash% y := F.1;
y
kash% y^2 - x^3 + 1;
0
kash% Divisor(x);
-2*(1/x, 1/x^2*y) + (x, y + x + 2) + (x, y + x + 3)
kash% Poles(x);
[ (1/x, 1/x^2*y) ]
kash% infty := Poles(x)[1];
(1/x, 1/x^2*y)
kash% Degree(2*infty);
2
kash% Basis(2*infty);
[
x,
1
]
kash% Degree(3*infty);
3
kash% Basis(3*infty);
[
y,
x,
1
]
kash% L := Places(F, 1);
[ (1/x, 1/x^2*y), (x, y + x + 2), (x, y + x + 3), (x + 2, y + 1), (x + 2, y + \
4), (x + 4, y) ]
kash% Evaluate(x, L[1]);
INFTY
kash% Evaluate(x, L[2]);
0
kash% Evaluate(x, L[4]);
3
kash% W := CanonicalDivisor(F);
(x + 4, y) + (x^2 + x + 1, y) - 3*(1/x, 1/x^2*y)
kash% GapNumbers(F, infty);
[ 1 ]
kash% o := MaximalOrderFinite(F);
Maximal Equation Order of F over Univariate Polynomial Ring in x over GF(5)
kash% Basis(o);
[ 1, y ]
kash% CoefficientRing(o);
Univariate Polynomial Ring in x over GF(5)
kash% Discriminant(o);
4*x^3 + 1
kash% Factorisation(x * o);
[ <Prime Ideal of o
Generators:
x
y + x + 2, 1>, <Prime Ideal of o
Generators:
x
y + x + 3, 1> ]