# Properties

 Label 2.2.56.1-5.1-b1 Base field $$\Q(\sqrt{14})$$ Conductor $$(-a + 3)$$ Conductor norm $$5$$ CM no Base change no Q-curve no Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{14})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 14$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 0, 1]))

gp: K = nfinit(Pol(Vecrev([-14, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);

## Weierstrass equation

$$y^2+xy=x^{3}-x^{2}+\left(-8a+30\right)x+27a-101$$
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([30,-8]),K([-101,27])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([30,-8])),Pol(Vecrev([-101,27]))], K);

magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![30,-8],K![-101,27]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-a + 3)$$ = $$\left(-a + 3\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$5$$ = $$5$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(387 a + 379)$$ = $$\left(-a + 3\right)^{9}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1953125$$ = $$5^{9}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{534684321}{1953125} a + \frac{3601913643}{1953125}$$ 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 $\left(-a + 4 : 4 a - 15 : 1\right)$ Height $$0.0531826671086175$$ 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: $$0.0531826671086175$$ Period: $$11.3567951855790$$ Tamagawa product: $$9$$ Torsion order: $$1$$ Leading coefficient: $$1.45279520759794$$ 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$$(j)_{-}$$
$$\left(-a + 3\right)$$ $$5$$ $$9$$ $$I_{9}$$ Split multiplicative $$-1$$ $$1$$ $$9$$ $$9$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Nn

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 5.1-b consists of this curve only.

## Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.