# Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-13122.5-d3 Conductor $$(81 a)$$ Conductor norm $$13122$$ CM no base-change yes: 5184.p2,162.b2 Q-curve yes Torsion order $$1$$ Rank not available

# 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

$$y^2 + x y = x^{3} - x^{2} - 852 x + 19664$$
magma: E := ChangeRing(EllipticCurve([1, -1, 0, -852, 19664]),K);
sage: E = EllipticCurve(K, [1, -1, 0, -852, 19664])
gp (2.8): E = ellinit([1, -1, 0, -852, 19664],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(81 a)$$ = $$\left(a\right) \cdot \left(-a - 1\right)^{4} \cdot \left(a - 1\right)^{4}$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$13122$$ = $$2 \cdot 3^{8}$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(123834728448)$$ = $$\left(a\right)^{42} \cdot \left(-a - 1\right)^{10} \cdot \left(a - 1\right)^{10}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$15335039969789900488704$$ = $$2^{42} \cdot 3^{20}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$-\frac{1159088625}{2097152}$$ 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 not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: Trivial 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]

## 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$2$$ $$I_{42}$$ Non-split multiplicative $$1$$ $$1$$ $$42$$ $$42$$
$$\left(-a - 1\right)$$ $$3$$ $$1$$ $$IV^*$$ Additive $$-1$$ $$4$$ $$10$$ $$0$$
$$\left(a - 1\right)$$ $$3$$ $$1$$ $$IV^*$$ Additive $$-1$$ $$4$$ $$10$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cn
$$3$$ 3B.1.2
$$7$$ 7B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3, 7 and 21.
Its isogeny class 13122.5-d consists of curves linked by isogenies of degrees dividing 21.

## Base change

This curve is the base-change of elliptic curves 5184.p2, 162.b2, defined over $$\Q$$, so it is also a $$\Q$$-curve.