# Properties

 Label 2.2.8.1-784.1-c3 Base field $$\Q(\sqrt{2})$$ Conductor norm $$784$$ CM no Base change yes Q-curve yes Torsion order $$4$$ Rank $$1$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{2})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 0, 1]))

gp: K = nfinit(Pol(Vecrev([-2, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);

## Weierstrass equation

$${y}^2+a{x}{y}+a{y}={x}^{3}-11{x}+10$$
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,1]),K([-11,0]),K([10,0])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-11,0])),Pol(Vecrev([10,0]))], K);

magma: E := EllipticCurve([K![0,1],K![0,0],K![0,1],K![-11,0],K![10,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(28)$$ = $$(a)^{4}\cdot(-2a+1)\cdot(2a+1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$784$$ = $$2^{4}\cdot7\cdot7$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(1568)$$ = $$(a)^{10}\cdot(-2a+1)^{2}\cdot(2a+1)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$2458624$$ = $$2^{10}\cdot7^{2}\cdot7^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{3543122}{49}$$ 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 $\left(-a + 1 : -3 a : 1\right)$ Height $$0.119959949362601$$ Torsion structure: $$\Z/2\Z\oplus\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-2 a - 1 : 2 : 1\right)$ $\left(\frac{3}{2} : -\frac{5}{4} a : 1\right)$ 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: $$0.119959949362601$$ Period: $$22.7571210471908$$ Tamagawa product: $$16$$  =  $$2^{2}\cdot2\cdot2$$ Torsion order: $$4$$ Leading coefficient: $$1.93036127010313$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$4$$ $$I_{2}^{*}$$ Additive $$-1$$ $$4$$ $$10$$ $$0$$
$$(-2a+1)$$ $$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$(2a+1)$$ $$7$$ $$2$$ $$I_{2}$$ Non-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.
Its isogeny class 784.1-c consists of curves linked by isogenies of degrees dividing 4.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 112.a1
$$\Q$$ 448.h1