# Properties

 Label 2.0.7.1-20412.2-d2 Base field $$\Q(\sqrt{-7})$$ Conductor $$(-108a+54)$$ Conductor norm $$20412$$ CM no Base change yes: 2646.f2,378.d2 Q-curve yes Torsion order $$3$$ 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

$${y}^2+{x}{y}={x}^{3}-{x}^{2}-12{x}+24$$
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([-12,0]),K([24,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-12,0])),Pol(Vecrev([24,0]))], K);

magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![-12,0],K![24,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-108a+54)$$ = $$(a)\cdot(-a+1)\cdot(-2a+1)\cdot(3)^{3}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$20412$$ = $$2\cdot2\cdot7\cdot9^{3}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-74088)$$ = $$(a)^{3}\cdot(-a+1)^{3}\cdot(-2a+1)^{6}\cdot(3)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$5489031744$$ = $$2^{3}\cdot2^{3}\cdot7^{6}\cdot9^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{7414875}{2744}$$ 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 $\left(\frac{5}{4} a + \frac{5}{4} : -\frac{25}{8} a + \frac{19}{8} : 1\right)$ Height $$1.27282881931807$$ Torsion structure: $$\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(1 : -4 : 1\right)$ 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: $$1.27282881931807$$ Period: $$2.03570983893765$$ Tamagawa product: $$6$$  =  $$1\cdot1\cdot( 2 \cdot 3 )\cdot1$$ Torsion order: $$3$$ Leading coefficient: $$2.61159355371822$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$(-a+1)$$ $$2$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$(-2a+1)$$ $$7$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$(3)$$ $$9$$ $$1$$ $$II$$ Additive $$-1$$ $$3$$ $$3$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Cs.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 20412.2-d consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base change of 2646.f2, 378.d2, defined over $$\Q$$, so it is also a $$\Q$$-curve.