# Properties

 Label 2.2.56.1-126.1-a4 Base field $$\Q(\sqrt{14})$$ Conductor $$(3 a)$$ Conductor norm $$126$$ CM no Base change yes: 294.g2,1344.i2 Q-curve yes Torsion order $$4$$ 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

$$y^2+xy+y=x^{3}+\left(-a-1\right)x^{2}+\left(109683a-410391\right)x-37770816a+141325451$$
sage: E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,0]),K([-410391,109683]),K([141325451,-37770816])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,-1])),Pol(Vecrev([1,0])),Pol(Vecrev([-410391,109683])),Pol(Vecrev([141325451,-37770816]))], K);

magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,0],K![-410391,109683],K![141325451,-37770816]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(3 a)$$ = $$\left(-a + 4\right) \cdot \left(3\right) \cdot \left(-2 a + 7\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$126$$ = $$2 \cdot 7 \cdot 9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(933897762)$$ = $$\left(-a + 4\right)^{2} \cdot \left(3\right)^{4} \cdot \left(-2 a + 7\right)^{16}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$872165029868608644$$ = $$2^{2} \cdot 7^{16} \cdot 9^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{84448510979617}{933897762}$$ 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 $\left(-250 a + 939 : -9640 a + 36070 : 1\right)$ Height $$1.14151965867083$$ 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: $\left(-26 a + 99 : 1532 a - 5734 : 1\right)$ 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: $$1.14151965867083$$ Period: $$2.48688727658968$$ Tamagawa product: $$128$$  =  $$2\cdot2^{4}\cdot2^{2}$$ Torsion order: $$4$$ Leading coefficient: $$6.06967538002856$$ 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$$(j)_{-}$$
$$\left(-a + 4\right)$$ $$2$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(-2 a + 7\right)$$ $$7$$ $$16$$ $$I_{16}$$ Split multiplicative $$-1$$ $$1$$ $$16$$ $$16$$
$$\left(3\right)$$ $$9$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## 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 and 8.
Its isogeny class 126.1-a consists of curves linked by isogenies of degrees dividing 8.

## Base change

This curve is the base change of elliptic curves 294.g2, 1344.i2, defined over $$\Q$$, so it is also a $$\Q$$-curve.