# Properties

 Label 2.0.8.1-4050.3-c3 Base field $$\Q(\sqrt{-2})$$ Conductor $$(45 a)$$ Conductor norm $$4050$$ CM no Base change yes: 2880.q2,90.c2 Q-curve yes Torsion order $$6$$ 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

$$y^2+xy+y=x^{3}-x^{2}-4082x+14681$$
sage: E = EllipticCurve(K, [1, -1, 1, -4082, 14681])

gp: E = ellinit([1, -1, 1, -4082, 14681],K)

magma: E := ChangeRing(EllipticCurve([1, -1, 1, -4082, 14681]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(45 a)$$ = $$\left(a\right) \cdot \left(-a - 1\right)^{2} \cdot \left(a - 1\right)^{2} \cdot \left(5\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$4050$$ = $$2 \cdot 3^{4} \cdot 25$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(4271484375000)$$ = $$\left(a\right)^{6} \cdot \left(-a - 1\right)^{7} \cdot \left(a - 1\right)^{7} \cdot \left(5\right)^{12}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$18245578765869140625000000$$ = $$2^{6} \cdot 3^{14} \cdot 25^{12}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{10316097499609}{5859375000}$$ 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 $\left(25 : -190 a - 13 : 1\right)$ Height $$3.76255927290324$$ Torsion structure: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-9 : 229 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$3.76255927290324$$ Period: $$0.107857511695509$$ Tamagawa product: $$288$$  =  $$( 2 \cdot 3 )\cdot2\cdot2\cdot( 2^{2} \cdot 3 )$$ Torsion order: $$6$$ Leading coefficient: $$4.59133235974603$$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(-a - 1\right)$$ $$3$$ $$2$$ $$I_{1}^*$$ Additive $$-1$$ $$2$$ $$7$$ $$1$$
$$\left(a - 1\right)$$ $$3$$ $$2$$ $$I_{1}^*$$ Additive $$-1$$ $$2$$ $$7$$ $$1$$
$$\left(5\right)$$ $$25$$ $$12$$ $$I_{12}$$ Split multiplicative $$-1$$ $$1$$ $$12$$ $$12$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 4050.3-c consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is the base change of elliptic curves 2880.q2, 90.c2, defined over $$\Q$$, so it is also a $$\Q$$-curve.