# Properties

 Base field $$\Q(\sqrt{-1})$$ Label 2.0.4.1-75466.2-a1 Conductor $$(215 i + 171)$$ Conductor norm $$75466$$ CM no base-change no Q-curve no Torsion order $$1$$ 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: K = nfinit(i^2 + 1);

## Weierstrass equation

$$y^2 + i x y + y = x^{3} + \left(i + 1\right) x^{2} + \left(-195 i - 233\right) x + 1273 i + 1841$$
magma: E := ChangeRing(EllipticCurve([i, i + 1, 1, -195*i - 233, 1273*i + 1841]),K);

sage: E = EllipticCurve(K, [i, i + 1, 1, -195*i - 233, 1273*i + 1841])

gp: E = ellinit([i, i + 1, 1, -195*i - 233, 1273*i + 1841],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(215 i + 171)$$ = $$\left(i + 1\right) \cdot \left(-4 i + 9\right) \cdot \left(10 i + 17\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$75466$$ = $$2 \cdot 97 \cdot 389$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(662522390 i + 1499956406)$$ = $$\left(i + 1\right)^{3} \cdot \left(-4 i + 9\right) \cdot \left(10 i + 17\right)^{6}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$2688805137151748936$$ = $$2^{3} \cdot 97 \cdot 389^{6}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{2526437929797748954525}{1344402568575874468} i + \frac{265587251101817213101}{1344402568575874468}$$ 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 not available.

magma: Rank(E);

sage: E.rank()

Regulator: not available

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

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: Trivial magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]

## 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$$(j)_{-}$$
$$\left(i + 1\right)$$ $$2$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$\left(-4 i + 9\right)$$ $$97$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(10 i + 17\right)$$ $$389$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 75466.2-a consists of curves linked by isogenies of degree 3.

## Base change

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