# Properties

 Base field $$\Q(\sqrt{2})$$ Label 2.2.8.1-784.1-a10 Conductor $$(28)$$ Conductor norm $$784$$ CM no base-change yes: 112.c1,448.a1 Q-curve yes Torsion order $$4$$ Rank $$0$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

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

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

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

gp (2.8): K = nfinit(a^2 - 2);

## Weierstrass equation

$$y^2 + a x y = x^{3} - x^{2} - 10922 x + 441166$$
magma: E := ChangeRing(EllipticCurve([a, -1, 0, -10922, 441166]),K);

sage: E = EllipticCurve(K, [a, -1, 0, -10922, 441166])

gp (2.8): E = ellinit([a, -1, 0, -10922, 441166],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(28)$$ = $$\left(a\right)^{4} \cdot \left(-2 a + 1\right) \cdot \left(2 a + 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$784$$ = $$2^{4} \cdot 7^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(1605632)$$ = $$\left(a\right)^{30} \cdot \left(-2 a + 1\right)^{2} \cdot \left(2 a + 1\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$2578054119424$$ = $$2^{30} \cdot 7^{4}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$\frac{2251439055699625}{25088}$$ 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: $$0$$
magma: Rank(E);

sage: E.rank()

magma: Generators(E); // includes torsion

sage: E.gens()

magma: Regulator(Generators(E));

sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $$\Z/2\Z\times\Z/2\Z$$ 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] $\left(-64 a - 30 : 15 a + 64 : 1\right)$,$\left(\frac{121}{2} : -\frac{121}{4} a : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[3]

## 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$4$$ $$I_{22}^*$$ Additive $$1$$ $$4$$ $$30$$ $$18$$
$$\left(-2 a + 1\right)$$ $$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(2 a + 1\right)$$ $$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 6, 9 and 18.
Its isogeny class 784.1-a consists of curves linked by isogenies of degrees dividing 36.

## Base change

This curve is the base-change of elliptic curves 112.c1, 448.a1, defined over $$\Q$$, so it is also a $$\Q$$-curve.