# Properties

 Label 2.0.7.1-28672.9-g1 Base field $$\Q(\sqrt{-7})$$ Conductor norm $$28672$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$1$$

# Related objects

Show commands: 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

$${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(20a-24\right){x}+84a-84$$
sage: E = EllipticCurve([K([0,0]),K([-1,1]),K([0,0]),K([-24,20]),K([-84,84])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-24,20])),Pol(Vecrev([-84,84]))], K);

magma: E := EllipticCurve([K![0,0],K![-1,1],K![0,0],K![-24,20],K![-84,84]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(16a-176)$$ = $$(a)^{4}\cdot(-a+1)^{8}\cdot(-2a+1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$28672$$ = $$2^{4}\cdot2^{8}\cdot7$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(2809856a+2809856)$$ = $$(a)^{13}\cdot(-a+1)^{15}\cdot(-2a+1)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$31581162962944$$ = $$2^{13}\cdot2^{15}\cdot7^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{31083}{98} a - \frac{338111}{686}$$ 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 $\left(-a - 2 : 3 a + 4 : 1\right)$ Height $$1.64840266806596$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-a - 3 : 0 : 1\right)$ 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: $$1.64840266806596$$ Period: $$1.03178171370466$$ Tamagawa product: $$8$$  =  $$2\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$5.14271079861415$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$2$$ $$I_{5}^{*}$$ Additive $$-1$$ $$4$$ $$13$$ $$1$$
$$(-a+1)$$ $$2$$ $$2$$ $$III^{*}$$ Additive $$-1$$ $$8$$ $$15$$ $$0$$
$$(-2a+1)$$ $$7$$ $$2$$ $$I_{6}$$ 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 28672.9-g consists of curves linked by isogenies of degree 2.

## Base change

This elliptic curve is not a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.