# Properties

 Base field $$\Q(\sqrt{-1})$$ Label 2.0.4.1-340.1-a1 Conductor $$(12 i + 14)$$ Conductor norm $$340$$ CM no base-change no Q-curve no Torsion order $$2$$ Rank $$0$$

# 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

$$y^2 + \left(i + 1\right) x y + \left(i + 1\right) y = x^{3} + \left(i + 1\right) x^{2} + \left(7 i - 19\right) x - 17 i + 18$$
magma: E := ChangeRing(EllipticCurve([i + 1, i + 1, i + 1, 7*i - 19, -17*i + 18]),K);

sage: E = EllipticCurve(K, [i + 1, i + 1, i + 1, 7*i - 19, -17*i + 18])

gp: E = ellinit([i + 1, i + 1, i + 1, 7*i - 19, -17*i + 18],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(12 i + 14)$$ = $$\left(i + 1\right)^{2} \cdot \left(-i - 2\right) \cdot \left(i + 4\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$340$$ = $$2^{2} \cdot 5 \cdot 17$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(-60884 i + 211112)$$ = $$\left(i + 1\right)^{4} \cdot \left(-i - 2\right)^{3} \cdot \left(i + 4\right)^{6}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$48275138000$$ = $$2^{4} \cdot 5^{3} \cdot 17^{6}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{8285953568288}{3017196125} i + \frac{8530762181584}{3017196125}$$ 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: $$0$$

magma: Rank(E);

sage: E.rank()

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: $$\Z/2\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-\frac{3}{2} i + 3 : -\frac{5}{4} i - \frac{11}{4} : 1\right)$ 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$$(j)_{-}$$
$$\left(i + 1\right)$$ $$2$$ $$1$$ $$IV$$ Additive $$-1$$ $$2$$ $$4$$ $$0$$
$$\left(-i - 2\right)$$ $$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$\left(i + 4\right)$$ $$17$$ $$2$$ $$I_{6}$$ 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
$$2$$ 2B
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 340.1-a consists of curves linked by isogenies of degrees dividing 6.

## Base change

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