# Properties

 Label 3.3.49.1-91.1-a3 Base field $$\Q(\zeta_{7})^+$$ Conductor $$(4 a + 1)$$ Conductor norm $$91$$ CM no Base change no Q-curve no Torsion order $$16$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\zeta_{7})^+$$

Generator $$a$$, with minimal polynomial $$x^{3} - x^{2} - 2 x + 1$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 2*x + 1)

gp: K = nfinit(a^3 - a^2 - 2*a + 1);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);

## Weierstrass equation

$$y^2+\left(a^{2}-2\right)xy=x^{3}+\left(a^{2}-3\right)x^{2}-4x-3a^{2}-a+9$$
sage: E = EllipticCurve(K, [a^2 - 2, a^2 - 3, 0, -4, -3*a^2 - a + 9])

gp: E = ellinit([a^2 - 2, a^2 - 3, 0, -4, -3*a^2 - a + 9],K)

magma: E := ChangeRing(EllipticCurve([a^2 - 2, a^2 - 3, 0, -4, -3*a^2 - a + 9]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(4 a + 1)$$ = $$\left(-a^{2} - a + 2\right) \cdot \left(-2 a^{2} + a + 2\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$91$$ = $$7 \cdot 13$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(44 a^{2} + 57 a - 125)$$ = $$\left(-a^{2} - a + 2\right)^{2} \cdot \left(-2 a^{2} + a + 2\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1399489$$ = $$7^{2} \cdot 13^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{3698907677516}{199927} a^{2} + \frac{293174005427}{28561} a + \frac{8312780816110}{199927}$$ 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: $$0$$ Torsion structure: $$\Z/2\Z\times\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-a + 1 : a^{2} - 2 : 1\right)$ $\left(-2 : a^{2} - 2 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$145.801004702548$$ Tamagawa product: $$8$$  =  $$2\cdot2^{2}$$ Torsion order: $$16$$ Leading coefficient: $$0.650897342422091$$ 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$$(j)_{-}$$
$$\left(-a^{2} - a + 2\right)$$ $$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(-2 a^{2} + a + 2\right)$$ $$13$$ $$4$$ $$I_{4}$$ 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
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 91.1-a consists of curves linked by isogenies of degrees dividing 16.

## Base change

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