# Properties

 Label 3.3.49.1-71.1-a4 Base field $$\Q(\zeta_{7})^+$$ Conductor $$(4a^2-3a-5)$$ Conductor norm $$71$$ CM no Base change no Q-curve no Torsion order $$10$$ 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: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -2, -1, 1]))

gp: K = nfinit(Pol(Vecrev([1, -2, -1, 1])));

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

## Weierstrass equation

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

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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(4a^2-3a-5)$$ = $$(4a^2-3a-5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$71$$ = $$71$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(a+4)$$ = $$(4a^2-3a-5)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-71$$ = $$71$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{713073}{71} a^{2} + \frac{710952}{71} a + \frac{1959524}{71}$$ 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: $$\Z/10\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(0 : -a : 1\right)$ 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: $$411.200909713333$$ Tamagawa product: $$1$$ Torsion order: $$10$$ Leading coefficient: $$0.587429871019048$$ 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$$(j)_{-}$$
$$(4a^2-3a-5)$$ $$71$$ $$1$$ $$I_{1}$$ 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
$$5$$ 5B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 5 and 10.
Its isogeny class 71.1-a consists of curves linked by isogenies of degrees dividing 10.

## Base change

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