# Properties

 Base field $$\Q(\sqrt{-1})$$ Label 2.0.4.1-11025.2-c4 Conductor $$(105)$$ Conductor norm $$11025$$ CM no base-change yes: 105.a1,1680.f1 Q-curve yes Torsion order $$4$$ Rank not available

# 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 (2.8): K = nfinit(i^2 + 1);

## Weierstrass equation

$$y^2 + i x y + i y = x^{3} - 112 x + 469$$
magma: E := ChangeRing(EllipticCurve([i, 0, i, -112, 469]),K);
sage: E = EllipticCurve(K, [i, 0, i, -112, 469])
gp (2.8): E = ellinit([i, 0, i, -112, 469],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(105)$$ = $$\left(3\right) \cdot \left(-i - 2\right) \cdot \left(2 i + 1\right) \cdot \left(7\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$11025$$ = $$5^{2} \cdot 9 \cdot 49$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(13125)$$ = $$\left(3\right) \cdot \left(-i - 2\right)^{4} \cdot \left(2 i + 1\right)^{4} \cdot \left(7\right)$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$172265625$$ = $$5^{8} \cdot 9 \cdot 49$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{157551496201}{13125}$$ 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 group

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

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

## Torsion subgroup

Structure: $$\Z/4\Z$$ 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] $\left(5 : -3 i - 5 : 1\right)$ 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 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 $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(3\right)$$ $$9$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(7\right)$$ $$49$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 11025.2-c consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base-change of elliptic curves 105.a1, 1680.f1, defined over $$\Q$$, so it is also a $$\Q$$-curve.