# Properties

 Base field $$\Q(\sqrt{-1})$$ Label 2.0.4.1-6400.2-e6 Conductor $$(80)$$ Conductor norm $$6400$$ CM no base-change yes: 320.f4,320.a4 Q-curve yes Torsion order $$4$$ 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

$$y^2 = x^{3} - i x^{2} - 4 x + 4 i$$
magma: E := ChangeRing(EllipticCurve([0, -i, 0, -4, 4*i]),K);

sage: E = EllipticCurve(K, [0, -i, 0, -4, 4*i])

gp: E = ellinit([0, -i, 0, -4, 4*i],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(80)$$ = $$\left(i + 1\right)^{8} \cdot \left(-i - 2\right) \cdot \left(2 i + 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$6400$$ = $$2^{8} \cdot 5^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(6400)$$ = $$\left(i + 1\right)^{16} \cdot \left(-i - 2\right)^{2} \cdot \left(2 i + 1\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$40960000$$ = $$2^{16} \cdot 5^{4}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{21296}{25}$$ 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: $\left(-i - 1 : i + 3 : 1\right)$

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

sage: gens = E.gens(); gens

Height: 0.4134369140091624

magma: [Height(P):P in gens];

sage: [P.height() for P in gens]

Regulator: 0.413436914009

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/2\Z\times\Z/2\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-2 : 0 : 1\right)$,$\left(i : 0 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[3]

## 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$$(j)_{-}$$
$$\left(i + 1\right)$$ $$2$$ $$4$$ $$I_{4}^*$$ Additive $$1$$ $$8$$ $$16$$ $$0$$
$$\left(-i - 2\right)$$ $$5$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(2 i + 1\right)$$ $$5$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## 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
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 6400.2-e consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is the base-change of elliptic curves 320.f4, 320.a4, defined over $$\Q$$, so it is also a $$\Q$$-curve.