# Properties

 Base field $$\Q(\sqrt{-1})$$ Label 2.0.4.1-5525.5-b9 Conductor $$(25 i - 70)$$ Conductor norm $$5525$$ CM no base-change no Q-curve no Torsion order $$16$$ 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

$$y^2 + i x y + i y = x^{3} + x^{2} + \left(-105 i + 39\right) x + 15 i - 399$$
magma: E := ChangeRing(EllipticCurve([i, 1, i, -105*i + 39, 15*i - 399]),K);

sage: E = EllipticCurve(K, [i, 1, i, -105*i + 39, 15*i - 399])

gp: E = ellinit([i, 1, i, -105*i + 39, 15*i - 399],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(25 i - 70)$$ = $$\left(-i - 2\right) \cdot \left(2 i + 1\right) \cdot \left(-3 i - 2\right) \cdot \left(i + 4\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$5525$$ = $$5^{2} \cdot 13 \cdot 17$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(6025625 i + 29925000)$$ = $$\left(-i - 2\right)^{4} \cdot \left(2 i + 1\right)^{4} \cdot \left(-3 i - 2\right)^{4} \cdot \left(i + 4\right)^{4}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$931813781640625$$ = $$5^{8} \cdot 13^{4} \cdot 17^{4}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{226834389543384}{59636082025} i + \frac{4972600364093721}{1490902050625}$$ 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: $\left(-5 i + \frac{41}{2} : \frac{137}{4} i - \frac{395}{4} : 1\right)$

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

sage: gens = E.gens(); gens

Height: 1.1255718252065914

magma: [Height(P):P in gens];

sage: [P.height() for P in gens]

Regulator: 1.12557182521

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/4\Z\times\Z/4\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-25 i - 17 : 23 i - 165 : 1\right)$,$\left(10 i - \frac{9}{2} : -\frac{13}{4} i - \frac{145}{4} : 1\right)$ 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$$(j)_{-}$$
$$\left(-i - 2\right)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(2 i + 1\right)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(-3 i - 2\right)$$ $$13$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(i + 4\right)$$ $$17$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## 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 5525.5-b 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.