# Properties

 Base field $$\Q(\sqrt{2})$$ Label 2.2.8.1-392.1-c3 Conductor $$(14 a)$$ Conductor norm $$392$$ CM no base-change yes: 56.b1,448.c1 Q-curve yes Torsion order $$4$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 2)

gp: K = nfinit(a^2 - 2);

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

## Weierstrass equation

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

gp: E = ellinit([a, -1, a, -11, -11],K)

magma: E := ChangeRing(EllipticCurve([a, -1, a, -11, -11]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(14 a)$$ = $$\left(a\right)^{3} \cdot \left(-2 a + 1\right) \cdot \left(2 a + 1\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$392$$ = $$2^{3} \cdot 7^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(1568)$$ = $$\left(a\right)^{10} \cdot \left(-2 a + 1\right)^{2} \cdot \left(2 a + 1\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$2458624$$ = $$2^{10} \cdot 7^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{3543122}{49}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank: $$0$$

sage: E.rank()

magma: Rank(E);

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

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); $\left(-2 a + 1 : -a + 2 : 1\right)$,$\left(-\frac{3}{2} : \frac{1}{4} a : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$2$$ $$III^*$$ Additive $$1$$ $$3$$ $$10$$ $$0$$
$$\left(-2 a + 1\right)$$ $$7$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(2 a + 1\right)$$ $$7$$ $$2$$ $$I_{2}$$ 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.
Its isogeny class 392.1-c consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base-change of elliptic curves 56.b1, 448.c1, defined over $$\Q$$, so it is also a $$\Q$$-curve.