# Properties

 Label 2.0.7.1-16128.5-n2 Base field $$\Q(\sqrt{-7})$$ Conductor $$(-96 a + 48)$$ Conductor norm $$16128$$ CM no Base change yes: 336.d5,2352.l5 Q-curve yes Torsion order $$4$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))

gp: K = nfinit(Pol(Vecrev([2, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);

## Weierstrass equation

$$y^2=x^{3}+x^{2}-64x-460$$
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-64,0]),K([-460,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-64,0])),Pol(Vecrev([-460,0]))], K);

magma: E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![-64,0],K![-460,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-96 a + 48)$$ = $$\left(a\right)^{4} \cdot \left(-a + 1\right)^{4} \cdot \left(3\right) \cdot \left(-2 a + 1\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$16128$$ = $$2^{8} \cdot 7 \cdot 9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(66060288)$$ = $$\left(a\right)^{20} \cdot \left(-a + 1\right)^{20} \cdot \left(3\right)^{2} \cdot \left(-2 a + 1\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4363961650642944$$ = $$2^{40} \cdot 7^{2} \cdot 9^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{7189057}{16128}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$1$$ Generator $\left(-4 : -12 a + 6 : 1\right)$ Height $$1.15540590735346$$ Torsion structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-3 a - 4 : 0 : 1\right)$ $\left(3 a - 7 : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$1.15540590735346$$ Period: $$0.685091833066015$$ Tamagawa product: $$64$$  =  $$2^{2}\cdot2^{2}\cdot2\cdot2$$ Torsion order: $$4$$ Leading coefficient: $$4.78689979783825$$ Analytic order of Ш: $$1$$ (rounded)

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$4$$ $$I_{12}^*$$ Additive $$-1$$ $$4$$ $$20$$ $$8$$
$$\left(-a + 1\right)$$ $$2$$ $$4$$ $$I_{12}^*$$ Additive $$-1$$ $$4$$ $$20$$ $$8$$
$$\left(-2 a + 1\right)$$ $$7$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(3\right)$$ $$9$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ 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, 4 and 8.
Its isogeny class 16128.5-n consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base change of elliptic curves 336.d5, 2352.l5, defined over $$\Q$$, so it is also a $$\Q$$-curve.