Properties

 Label 2.0.7.1-16128.5-n3 Base field $$\Q(\sqrt{-7})$$ Conductor $$(-96 a + 48)$$ Conductor norm $$16128$$ CM no Base change no Q-curve yes Torsion order $$2$$ Rank $$1$$

Related objects

Show commands for: 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

$$y^2=x^{3}+\left(a+1\right)x^{2}+\left(-31a-134\right)x-266a-644$$
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([0,0]),K([-134,-31]),K([-644,-266])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-134,-31])),Pol(Vecrev([-644,-266]))], K);

magma: E := EllipticCurve([K![0,0],K![1,1],K![0,0],K![-134,-31],K![-644,-266]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(-96 a + 48)$$ = $$\left(a\right)^{4} \cdot \left(-a + 1\right)^{4} \cdot \left(3\right) \cdot \left(-2 a + 1\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$16128$$ = $$2^{8} \cdot 7 \cdot 9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(9240576 a - 35586048)$$ = $$\left(a\right)^{16} \cdot \left(-a + 1\right)^{28} \cdot \left(3\right) \cdot \left(-2 a + 1\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1108307720798208$$ = $$2^{44} \cdot 7 \cdot 9$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{12134104351}{1376256} a - \frac{59946611}{688128}$$ 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 $\left(-14 : -26 a + 6 : 1\right)$ Height $$2.31081181470692$$ 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: $\left(-7 : 0 : 1\right)$ 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: $$2.31081181470692$$ Period: $$0.685091833066014$$ Tamagawa product: $$8$$  =  $$2\cdot2^{2}\cdot1\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$4.78689979783825$$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$2$$ $$I_{8}^*$$ Additive $$-1$$ $$4$$ $$16$$ $$4$$
$$\left(-a + 1\right)$$ $$2$$ $$4$$ $$I_{20}^*$$ Additive $$-1$$ $$4$$ $$28$$ $$16$$
$$\left(-2 a + 1\right)$$ $$7$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(3\right)$$ $$9$$ $$1$$ $$I_{1}$$ 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4, 8 and 16.
Its isogeny class 16128.5-n 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 a $$\Q$$-curve.