# Properties

 Base field $$\Q(\sqrt{-7})$$ Label 2.0.7.1-158.4-a1 Conductor $$(a - 13)$$ Conductor norm $$158$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{-7})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x + 2$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 2)
gp (2.8): K = nfinit(a^2 - a + 2);

## Weierstrass equation

$$y^2 + a x y + y = x^{3} + \left(a - 1\right) x^{2} - a x$$
magma: E := ChangeRing(EllipticCurve([a, a - 1, 1, -a, 0]),K);
sage: E = EllipticCurve(K, [a, a - 1, 1, -a, 0])
gp (2.8): E = ellinit([a, a - 1, 1, -a, 0],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(a - 13)$$ = $$\left(-a + 1\right) \cdot \left(6 a + 1\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$158$$ = $$2 \cdot 79$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(15 a - 37)$$ = $$\left(-a + 1\right)^{4} \cdot \left(6 a + 1\right)$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$1264$$ = $$2^{4} \cdot 79$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{226873}{1264} a - \frac{264187}{632}$$ 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: $$1$$
magma: Rank(E);
sage: E.rank()

Generator: $\left(1 : -a - 1 : 1\right)$

Height: 0.02018433583243251

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

Regulator: 0.0201843358324

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

## Torsion subgroup

Structure: Trivial 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]

## 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$$(j)_{-}$$
$$\left(-a + 1\right)$$ $$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$\left(6 a + 1\right)$$ $$79$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

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

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 158.4-a consists of this curve only.

## Base change

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