# Properties

 Base field $$\Q(\sqrt{15})$$ Label 2.2.60.1-8.1-b3 Conductor $$(4,2 a + 2)$$ Conductor norm $$8$$ CM no base-change no Q-curve yes Torsion order $$4$$ Rank not available

# Learn more about

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{15})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 15$$; class number $$2$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 15)

gp: K = nfinit(a^2 - 15);

## Weierstrass equation

$$y^2 + \left(a + 1\right) x y = x^{3} + \left(-a - 1\right) x^{2} + \left(-2605 a - 10081\right) x + 141300 a + 547257$$
magma: E := ChangeRing(EllipticCurve([a + 1, -a - 1, 0, -2605*a - 10081, 141300*a + 547257]),K);

sage: E = EllipticCurve(K, [a + 1, -a - 1, 0, -2605*a - 10081, 141300*a + 547257])

gp: E = ellinit([a + 1, -a - 1, 0, -2605*a - 10081, 141300*a + 547257],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(4,2 a + 2)$$ = $$\left(2, a + 1\right)^{3}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$8$$ = $$2^{3}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(16)$$ = $$\left(2, a + 1\right)^{8}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$256$$ = $$2^{8}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$1000188$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: 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()

Regulator: not available

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

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/2\Z\times\Z/2\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-21 a - 83 : 52 a + 199 : 1\right)$,$\left(\frac{21}{2} a + 39 : -\frac{99}{4} a - \frac{393}{4} : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[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$$(j)_{-}$$
$$\left(2, a + 1\right)$$ $$2$$ $$2$$ $$I_{1}^*$$ Additive $$1$$ $$3$$ $$8$$ $$0$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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 8.1-b consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is not the base-change of an elliptic curve defined over $$\Q$$. It is a $$\Q$$-curve.