Properties

 Label 3.3.49.1-56.1-a2 Base field $$\Q(\zeta_{7})^+$$ Conductor $$(-2 a^{2} - 2 a + 4)$$ Conductor norm $$56$$ CM no Base change yes: 14.a2 Q-curve yes Torsion order $$2$$ 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

$$y^2+xy+y=x^{3}-171x-874$$
sage: E = EllipticCurve(K, [1, 0, 1, -171, -874])

gp: E = ellinit([1, 0, 1, -171, -874],K)

magma: E := ChangeRing(EllipticCurve([1, 0, 1, -171, -874]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(-2 a^{2} - 2 a + 4)$$ = $$\left(2\right) \cdot \left(-a^{2} - a + 2\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$56$$ = $$7 \cdot 8$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(1835008)$$ = $$\left(2\right)^{18} \cdot \left(-a^{2} - a + 2\right)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$6178938688752320512$$ = $$7^{3} \cdot 8^{18}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{548347731625}{1835008}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(15 : -8 : 1\right)$ 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: $$0.288080952338379$$ Tamagawa product: $$6$$  =  $$3\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$0.555584693795445$$ Analytic order of Ш: $$9$$ (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$$(j)_{-}$$
$$\left(-a^{2} - a + 2\right)$$ $$7$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$\left(2\right)$$ $$8$$ $$2$$ $$I_{18}$$ Non-split multiplicative $$1$$ $$1$$ $$18$$ $$18$$

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
$$3$$ 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 6, 9 and 18.
Its isogeny class 56.1-a consists of curves linked by isogenies of degrees dividing 18.

Base change

This curve is the base change of elliptic curves 14.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.