Properties

 Label 2.0.7.1-49.1-CMa1 Base field $$\Q(\sqrt{-7})$$ Conductor norm $$49$$ CM yes ($$-7$$) Base change yes Q-curve yes Torsion order $$4$$ Rank $$0$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Base field$$\Q(\sqrt{-7})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))

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

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

Weierstrass equation

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

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-2,0])),Pol(Vecrev([-1,0]))], K);

magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![-2,0],K![-1,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(7)$$ = $$(-2a+1)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$49$$ = $$7^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-343)$$ = $$(-2a+1)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$117649$$ = $$7^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-3375$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z[(1+\sqrt{-7})/2]$$ (complex multiplication) Geometric endomorphism ring: $$\Z[(1+\sqrt{-7})/2]$$ sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{U}(1)$

Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/2\Z\oplus\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-\frac{1}{4} a - \frac{1}{2} : \frac{1}{8} a + \frac{1}{4} : 1\right)$ $\left(\frac{1}{4} a - \frac{3}{4} : -\frac{1}{8} a + \frac{3}{8} : 1\right)$ 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: $$4.94450460028255$$ Tamagawa product: $$4$$ Torsion order: $$4$$ Leading coefficient: $$0.934423537768746$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$7$$ $$4$$ $$I_0^{*}$$ Additive $$-1$$ $$2$$ $$6$$ $$0$$

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$7$$ 7B.1.4[2]

For all other primes $$p$$, the image is a split Cartan subgroup if $$\left(\frac{ -7 }{p}\right)=+1$$ or a nonsplit Cartan subgroup if $$\left(\frac{ -7 }{p}\right)=-1$$.

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies (excluding endomorphisms) of degree $$d$$ for $$d=$$ 2.
Its isogeny class 49.1-CMa consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 49.a4
$$\Q$$ 49.a2