# Properties

 Label 2.0.7.1-225.1-a7 Base field $$\Q(\sqrt{-7})$$ Conductor $$(15)$$ Conductor norm $$225$$ CM no Base change yes: 735.c4,15.a4 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

$${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$$
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-80,0]),K([242,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-80,0])),Pol(Vecrev([242,0]))], K);

magma: E := EllipticCurve([K![1,0],K![1,0],K![1,0],K![-80,0],K![242,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(15)$$ = $$(3)\cdot(5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$225$$ = $$9\cdot25$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(15)$$ = $$(3)\cdot(5)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$225$$ = $$9\cdot25$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{56667352321}{15}$$ 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 $\left(\frac{3}{64} a + \frac{287}{64} : -\frac{105}{512} a - \frac{877}{512} : 1\right)$ Height $$2.75671230729636$$ Torsion structure: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(5 : -4 : 1\right)$ 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: $$2.75671230729636$$ Period: $$2.23570171261753$$ Tamagawa product: $$1$$  =  $$1\cdot1$$ Torsion order: $$4$$ Leading coefficient: $$0.582366377448412$$ 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$$(j)_{-}$$
$$(3)$$ $$9$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$(5)$$ $$25$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4, 8 and 16.
Its isogeny class 225.1-a consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base change of 735.c4, 15.a4, defined over $$\Q$$, so it is also a $$\Q$$-curve.