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KK<t> := FunctionField(GF(5^3));

E := EllipticCurve([t^2, t^3 + t]);

&*BadPlaces(E);

LocalInformation(E);

MordellWeilGroup(E);

KK<t> := FunctionField(GF(5^12));

E := EllipticCurve([t^2, t^3 + t]);

&*BadPlaces(E);

LocalInformation(E);

MordellWeilGroup(E);

Non-constant Non-Isotrivial Rational Elliptic Fibration

The Mordell-Weil group structure for a 'same' E/Fq(t) depends on q = p^r

Even for a fixed p = 5, we have |E(Fq(t))| = 1 for q = 5^3 and E(Fq(t)) = Z^4 for q = 5^12

KK<t> := FunctionField(GF(4007));

E1 := EllipticCurve([0, t]);

E2 := EllipticCurve([0, t^2]);

E3 := EllipticCurve([0, t^3]);

E4 := EllipticCurve([0, t^4]);

E5 := EllipticCurve([0, t^5]);

E6 := EllipticCurve([0, t^6]);

E7 := EllipticCurve([0, t^7]);

E1;

&*BadPlaces(E1);

LocalInformation(E1);

E2;

&*BadPlaces(E2);

LocalInformation(E2);

E3;

&*BadPlaces(E3);

LocalInformation(E3);

E4;

&*BadPlaces(E4);

LocalInformation(E4);

E5;

&*BadPlaces(E5);

LocalInformation(E5);

E6;

&*BadPlaces(E6);

LocalInformation(E6);

E7;

&*BadPlaces(E7);

LocalInformation(E7);

Non-constant Isotrivial with j = 1728 Rational Elliptic Surface

KK<t> := FunctionField(GF(4007));

E1 := EllipticCurve([0, t^3*(t-1)*(t-2)^5]);

E2 := EllipticCurve([0, t^3*(t-1)^2*(t-2)^5]);

E3 := EllipticCurve([0, t^3*(t-1)^3*(t-2)^5]);

E1;

&*BadPlaces(E1);

LocalInformation(E1);

E2;

&*BadPlaces(E2);

LocalInformation(E2);

E3;

&*BadPlaces(E3);

LocalInformation(E3);

Non-constant Isotrivial with j = 1728 Projective Elliptic K3 Surface

KK<t> := FunctionField(GF(4007));

E := EllipticCurve([-27, 54 - 2^6*3^6*t^1]);

&*BadPlaces(E);

LocalInformation(E);

Non-constant Non-Isotrivial Ulmer's Curve of Infinite Rank