# Properties

 Label 2.2.56.1-10.2-d2 Base field $$\Q(\sqrt{14})$$ Conductor $$(-a + 2)$$ Conductor norm $$10$$ 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

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

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([3,0])),Pol(Vecrev([1,0]))], K);

magma: E := EllipticCurve([K![1,0],K![-1,1],K![0,0],K![3,0],K![1,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-a + 2)$$ = $$\left(-a + 4\right) \cdot \left(-a - 3\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$10$$ = $$2 \cdot 5$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-4 a - 12)$$ = $$\left(-a + 4\right)^{4} \cdot \left(-a - 3\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$80$$ = $$2^{4} \cdot 5$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{5689}{20} a + \frac{21627}{20}$$ 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 $\left(0 : 1 : 1\right)$ Height $$0.115049867334663$$ 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: $$0.115049867334663$$ Period: $$20.1954884442536$$ Tamagawa product: $$2$$  =  $$2\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$1.24195671922458$$ 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$$(j)_{-}$$
$$\left(-a + 4\right)$$ $$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$\left(-a - 3\right)$$ $$5$$ $$1$$ $$I_{1}$$ 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
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 10.2-d consists of curves linked by isogenies of degree 3.

## Base change

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