# Properties

 Base field $\Q(\sqrt{-1})$ Label 2.0.4.1-5525.5-b9 Conductor $\left(25 i - 70\right)$ 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: K<i> := NumberField(x^2 + 1);
sage: K.<i> = NumberField(x^2 + 1)
gp (2.8): 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 (2.8): 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

 $\mathfrak{N}$ = $\left(25 i - 70\right)$ = $\left(-i - 2\right) \cdot \left(2 i + 1\right) \cdot \left(-3 i - 2\right) \cdot \left(i + 4\right)$ magma: Conductor(E); sage: E.conductor() $N(\mathfrak{N})$ = $5525$ = $5^{2} \cdot 13 \cdot 17$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $\mathfrak{D}$ = $\left(6025625 i + 29925000\right)$ = $\left(-i - 2\right)^{4} \cdot \left(2 i + 1\right)^{4} \cdot \left(-3 i - 2\right)^{4} \cdot \left(i + 4\right)^{4}$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $N(\mathfrak{D})$ = $931813781640625$ = $5^{8} \cdot 13^{4} \cdot 17^{4}$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $j$ = $\frac{226834389543384}{59636082025} i + \frac{4972600364093721}{1490902050625}$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): 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 rank and generators

Rank: $1$
magma: Rank(E);
sage: E.rank()

Generator: $\left(-5 i + \frac{41}{2} : \frac{137}{4} i - \frac{395}{4} : 1\right)$

Height: 1.12557182521

magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: 1.12557182521

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $\Z/4\Z\times\Z/4\Z$ magma: TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp (2.8): elltors(E)[1] $\left(-25 i - 17 : 23 i - 165 : 1\right)$,$\left(10 i - \frac{9}{2} : -\frac{13}{4} i - \frac{145}{4} : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[3]

## Local data at primes of bad reduction

magma: LocalInformation(E)
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type ord($\mathfrak{N}$) ord($\mathfrak{D}$) ord$(j)_{-}$
$\left(-i - 2\right)$ 5 $4$ $I_{4}$ Split multiplicative 1 4 4
$\left(2 i + 1\right)$ 5 $4$ $I_{4}$ Split multiplicative 1 4 4
$\left(-3 i - 2\right)$ 13 $4$ $I_{4}$ Split multiplicative 1 4 4
$\left(i + 4\right)$ 17 $4$ $I_{4}$ Split multiplicative 1 4 4

## Galois Representations

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