# Properties

 Base field $$\Q(\sqrt{-1})$$ Label 2.0.4.1-93925.7-d9 Conductor $$(-30 i + 305)$$ Conductor norm $$93925$$ CM no base-change no Q-curve no Torsion order $$4$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

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

gp: K = nfinit(i^2 + 1);

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

## Weierstrass equation

$$y^2 + i x y + \left(i + 1\right) y = x^{3} + x^{2} + \left(1268 i - 1414\right) x - 22882 i + 16717$$
sage: E = EllipticCurve(K, [i, 1, i + 1, 1268*i - 1414, -22882*i + 16717])

gp: E = ellinit([i, 1, i + 1, 1268*i - 1414, -22882*i + 16717],K)

magma: E := ChangeRing(EllipticCurve([i, 1, i + 1, 1268*i - 1414, -22882*i + 16717]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-30 i + 305)$$ = $$\left(-i - 2\right) \cdot \left(2 i + 1\right) \cdot \left(-3 i - 2\right) \cdot \left(i + 4\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$93925$$ = $$5^{2} \cdot 13 \cdot 17^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(14640380000 i + 149256084375)$$ = $$\left(-i - 2\right)^{4} \cdot \left(2 i + 1\right)^{4} \cdot \left(-3 i - 2\right)^{4} \cdot \left(i + 4\right)^{10}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$22491719449501519140625$$ = $$5^{8} \cdot 13^{4} \cdot 17^{10}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{226834389543384}{59636082025} i + \frac{4972600364093721}{1490902050625}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

sage: gens = E.gens(); gens

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

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(17 i - 47 : 23 i + 8 : 1\right)$,$\left(-2 i + \frac{69}{4} : -\frac{73}{8} i - \frac{3}{2} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-i - 2\right)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(2 i + 1\right)$$ $$5$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$\left(-3 i - 2\right)$$ $$13$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$\left(i + 4\right)$$ $$17$$ $$4$$ $$I_{4}^*$$ Additive $$1$$ $$2$$ $$10$$ $$4$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 93925.7-d consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is not the base-change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.