# Properties

 Label 2.0.8.1-5625.2-c2 Base field $$\Q(\sqrt{-2})$$ Conductor norm $$5625$$ CM no Base change yes Q-curve yes Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands: Magma / PariGP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))

gp: K = nfinit(Polrev([2, 0, 1]));

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

## Weierstrass equation

$${y}^2+{y}={x}^{3}-{x}^{2}+42{x}+443$$
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([42,0]),K([443,0])])

gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([42,0]),Polrev([443,0])], K);

magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![42,0],K![443,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(75)$$ = $$(-a-1)\cdot(a-1)\cdot(5)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$5625$$ = $$3\cdot3\cdot25^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-94921875)$$ = $$(-a-1)^{5}\cdot(a-1)^{5}\cdot(5)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$9010162353515625$$ = $$3^{5}\cdot3^{5}\cdot25^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{20480}{243}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$1$$ Generator $\left(-\frac{41}{2} : \frac{275}{4} a - \frac{1}{2} : 1\right)$ Height $$1.8091757775316562435795367594540824650$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$1.8091757775316562435795367594540824650$$ Period: $$0.65492061822256392831800259490154526705$$ Tamagawa product: $$3$$  =  $$1\cdot1\cdot3$$ Torsion order: $$1$$ Leading coefficient: $$5.0269629010179055044048511259925495446$$ Analytic order of Ш: $$1$$ (rounded)

## 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$$(j)_{-}$$
$$(-a-1)$$ $$3$$ $$1$$ $$I_{5}$$ Non-split multiplicative $$1$$ $$1$$ $$5$$ $$5$$
$$(a-1)$$ $$3$$ $$1$$ $$I_{5}$$ Non-split multiplicative $$1$$ $$1$$ $$5$$ $$5$$
$$(5)$$ $$25$$ $$3$$ $$IV^{*}$$ Additive $$1$$ $$2$$ $$8$$ $$0$$

## 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.1.4

## 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 elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 75.c2
$$\Q$$ 4800.be2