# Properties

 Label 2.0.7.1-30492.5-r3 Base field $$\Q(\sqrt{-7})$$ Conductor $$(-132a+66)$$ Conductor norm $$30492$$ CM no Base change yes: 3234.v2,462.e2 Q-curve yes Torsion order $$4$$ Rank $$1$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{-7})$$

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

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

gp: K = nfinit(Pol(Vecrev([2, -1, 1])));

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

## Weierstrass equation

$${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-405{x}+4731$$
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-405,0]),K([4731,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-405,0])),Pol(Vecrev([4731,0]))], K);

magma: E := EllipticCurve([K![1,0],K![1,0],K![1,0],K![-405,0],K![4731,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-132a+66)$$ = $$(a)\cdot(-a+1)\cdot(-2a+1)\cdot(3)\cdot(-2a+3)\cdot(2a+1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$30492$$ = $$2\cdot2\cdot7\cdot9\cdot11\cdot11$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-6119866368)$$ = $$(a)^{14}\cdot(-a+1)^{14}\cdot(-2a+1)^{6}\cdot(3)^{2}\cdot(-2a+3)^{2}\cdot(2a+1)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$37452764362177511424$$ = $$2^{14}\cdot2^{14}\cdot7^{6}\cdot9^{2}\cdot11^{2}\cdot11^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{7347774183121}{6119866368}$$ 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 $\left(8 a + 7 : 32 a - 36 : 1\right)$ Height $$0.0394110598819251$$ Torsion structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-25 : 12 : 1\right)$ $\left(-\frac{21}{4} a + \frac{29}{2} : \frac{21}{8} a - \frac{31}{4} : 1\right)$ 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: $$0.0394110598819251$$ Period: $$0.316157574968820$$ Tamagawa product: $$9408$$  =  $$( 2 \cdot 7 )\cdot( 2 \cdot 7 )\cdot( 2 \cdot 3 )\cdot2\cdot2\cdot2$$ Torsion order: $$4$$ Leading coefficient: $$11.0766900569404$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$14$$ $$I_{14}$$ Split multiplicative $$-1$$ $$1$$ $$14$$ $$14$$
$$(-a+1)$$ $$2$$ $$14$$ $$I_{14}$$ Split multiplicative $$-1$$ $$1$$ $$14$$ $$14$$
$$(-2a+1)$$ $$7$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$(3)$$ $$9$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(-2a+3)$$ $$11$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$(2a+1)$$ $$11$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ 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 30492.5-r consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base change of 3234.v2, 462.e2, defined over $$\Q$$, so it is also a $$\Q$$-curve.