Properties

 Label 2.0.3.1-60025.3-i1 Base field $$\Q(\sqrt{-3})$$ Conductor norm $$60025$$ CM no Base change yes Q-curve yes Torsion order $$1$$ Rank $$0$$

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

$${y}^2+{y}={x}^{3}-343{x}-4202$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([1,0]),K([-343,0]),K([-4202,0])])

gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-343,0]),Polrev([-4202,0])], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![1,0],K![-343,0],K![-4202,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(245)$$ = $$(-3a+1)^{2}\cdot(3a-2)^{2}\cdot(5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$60025$$ = $$7^{2}\cdot7^{2}\cdot25$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-5044200875)$$ = $$(-3a+1)^{9}\cdot(3a-2)^{9}\cdot(5)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$25443962467350765625$$ = $$7^{9}\cdot7^{9}\cdot25^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{110592}{125}$$ 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: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$0.32877274630903451828095573016616402771$$ Tamagawa product: $$12$$  =  $$2\cdot2\cdot3$$ Torsion order: $$1$$ Leading coefficient: $$4.5556088060096068710820392447707705343$$ 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$$(j)_{-}$$
$$(-3a+1)$$ $$7$$ $$2$$ $$III^{*}$$ Additive $$-1$$ $$2$$ $$9$$ $$0$$
$$(3a-2)$$ $$7$$ $$2$$ $$III^{*}$$ Additive $$-1$$ $$2$$ $$9$$ $$0$$
$$(5)$$ $$25$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Nn[2]

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 60025.3-i consists of this curve only.

Base change

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

Base field Curve
$$\Q$$ 245.b1
$$\Q$$ 2205.l1