# Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-5625.2-c1 Conductor $$(75)$$ Conductor norm $$5625$$ CM no base-change yes: 75.c1,4800.be1 Q-curve yes Torsion order $$1$$ Rank $$1$$

# 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

$$y^2 + y = x^{3} - x^{2} - 8 x - 7$$
sage: E = EllipticCurve(K, [0, -1, 1, -8, -7])

gp: E = ellinit([0, -1, 1, -8, -7],K)

magma: E := ChangeRing(EllipticCurve([0, -1, 1, -8, -7]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(75)$$ = $$\left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(5\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$5625$$ = $$3^{2} \cdot 25^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(1875)$$ = $$\left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(5\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$3515625$$ = $$3^{2} \cdot 25^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{102400}{3}$$ 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: $$1$$

sage: E.rank()

magma: Rank(E);

Generator: $\left(-\frac{1}{2} : \frac{5}{4} a - \frac{1}{2} : 1\right)$

sage: gens = E.gens(); gens

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

Height: 0.3618351555063314

sage: [P.height() for P in gens]

magma: [Height(P):P in gens];

Regulator: 0.361835155506

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: Trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(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$$(j)_{-}$$
$$\left(-a - 1\right)$$ $$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$\left(a - 1\right)$$ $$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$\left(5\right)$$ $$25$$ $$3$$ $$IV$$ Additive $$1$$ $$2$$ $$4$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$5$$ 5B.1.3

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 5625.2-c consists of curves linked by isogenies of degree 5.

## Base change

This curve is the base-change of elliptic curves 75.c1, 4800.be1, defined over $$\Q$$, so it is also a $$\Q$$-curve.