# Properties

 Base field $$\Q(\sqrt{-1})$$ Label 2.0.4.1-1250.3-a1 Conductor $$(25 i + 25)$$ Conductor norm $$1250$$ CM no base-change yes: 50.a1,400.d1 Q-curve yes Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{-1})$$

Generator $$i$$, with minimal polynomial $$x^{2} + 1$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<i> := NumberField(R![1, 0, 1]);

sage: x = polygen(QQ); K.<i> = NumberField(x^2 + 1)

gp: K = nfinit(i^2 + 1);

## Weierstrass equation

$$y^2 + i x y + i y = x^{3} - 125 x + 552$$
magma: E := ChangeRing(EllipticCurve([i, 0, i, -125, 552]),K);

sage: E = EllipticCurve(K, [i, 0, i, -125, 552])

gp: E = ellinit([i, 0, i, -125, 552],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(25 i + 25)$$ = $$\left(i + 1\right) \cdot \left(-i - 2\right)^{2} \cdot \left(2 i + 1\right)^{2}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$1250$$ = $$2 \cdot 5^{4}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(5000)$$ = $$\left(i + 1\right)^{6} \cdot \left(-i - 2\right)^{4} \cdot \left(2 i + 1\right)^{4}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$25000000$$ = $$2^{6} \cdot 5^{8}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{349938025}{8}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank: $$1$$

magma: Rank(E);

sage: E.rank()

Generator: $\left(7 : -4 i + 2 : 1\right)$

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: gens = E.gens(); gens

Height: 0.1946976295619427

magma: [Height(P):P in gens];

sage: [P.height() for P in gens]

Regulator: 0.194697629562

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: Trivial magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]

## Local data at primes of bad reduction

magma: LocalInformation(E);

sage: E.local_data()

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(i + 1\right)$$ $$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$\left(-i - 2\right)$$ $$5$$ $$1$$ $$IV$$ Additive $$-1$$ $$2$$ $$4$$ $$0$$
$$\left(2 i + 1\right)$$ $$5$$ $$1$$ $$IV$$ Additive $$-1$$ $$2$$ $$4$$ $$0$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ except those listed.

prime Image of Galois Representation
$$3$$ 3B.1.2
$$5$$ 5B.1.3

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3, 5 and 15.
Its isogeny class 1250.3-a consists of curves linked by isogenies of degrees dividing 15.

## Base change

This curve is the base-change of elliptic curves 50.a1, 400.d1, defined over $$\Q$$, so it is also a $$\Q$$-curve.