# Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-13122.5-d4 Conductor $$(81 a)$$ Conductor norm $$13122$$ CM no base-change yes: 5184.p4,162.b4 Q-curve yes Torsion order $$3$$ Rank not available

# 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

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

gp: E = ellinit([1, -1, 0, 3, -1],K)

magma: E := ChangeRing(EllipticCurve([1, -1, 0, 3, -1]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(81 a)$$ = $$\left(a\right) \cdot \left(-a - 1\right)^{4} \cdot \left(a - 1\right)^{4}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$13122$$ = $$2 \cdot 3^{8}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(1458)$$ = $$\left(a\right)^{2} \cdot \left(-a - 1\right)^{6} \cdot \left(a - 1\right)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$2125764$$ = $$2^{2} \cdot 3^{12}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{3375}{2}$$ 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 not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

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: $$\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(1 : -2 : 1\right)$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(-a - 1\right)$$ $$3$$ $$3$$ $$IV$$ Additive $$-1$$ $$4$$ $$6$$ $$0$$
$$\left(a - 1\right)$$ $$3$$ $$3$$ $$IV$$ Additive $$-1$$ $$4$$ $$6$$ $$0$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$2$$ 2Cn
$$3$$ 3B.1.1
$$7$$ 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.p4, 162.b4, defined over $$\Q$$, so it is also a $$\Q$$-curve.