Properties

 Label 2.0.3.1-61731.2-g2 Base field $$\Q(\sqrt{-3})$$ Conductor norm $$61731$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$1$$

Related objects

Show commands: Magma / PariGP / SageMath

Base field$$\Q(\sqrt{-3})$$

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

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

gp: K = nfinit(Polrev([1, -1, 1]));

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

Weierstrass equation

$${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-238a-579\right){x}-3795a-5177$$
sage: E = EllipticCurve([K([1,0]),K([0,1]),K([1,0]),K([-579,-238]),K([-5177,-3795])])

gp: E = ellinit([Polrev([1,0]),Polrev([0,1]),Polrev([1,0]),Polrev([-579,-238]),Polrev([-5177,-3795])], K);

magma: E := EllipticCurve([K![1,0],K![0,1],K![1,0],K![-579,-238],K![-5177,-3795]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(-285a+171)$$ = $$(-2a+1)^{2}\cdot(-5a+3)^{2}\cdot(-5a+2)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$61731$$ = $$3^{2}\cdot19^{2}\cdot19$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(559859016a-1228796709)$$ = $$(-2a+1)^{3}\cdot(-5a+3)^{9}\cdot(-5a+2)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1135430553480940593$$ = $$3^{3}\cdot19^{9}\cdot19^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{5845799184}{130321} a + \frac{1712734635}{130321}$$ 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 $\left(\frac{1}{4} a - \frac{39}{2} : \frac{135}{4} a + \frac{91}{8} : 1\right)$ Height $$2.7690401079907991677525174992602532079$$ 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: $\left(-a - \frac{57}{4} : \frac{1}{2} a + \frac{53}{8} : 1\right)$ 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: $$2.7690401079907991677525174992602532079$$ Period: $$0.35616692849373169400489067692545008420$$ Tamagawa product: $$8$$  =  $$2\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$4.5552497921158811988318787349431652850$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$2$$ $$III$$ Additive $$1$$ $$2$$ $$3$$ $$0$$
$$(-5a+3)$$ $$19$$ $$2$$ $$III^{*}$$ Additive $$1$$ $$2$$ $$9$$ $$0$$
$$(-5a+2)$$ $$19$$ $$2$$ $$I_{4}$$ Non-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
$$2$$ 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 61731.2-g consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is not a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.