Properties

Label 2.0.7.1-6300.2-d6
Base field \(\Q(\sqrt{-7}) \)
Conductor \(\left(-60a + 30\right)\)
Conductor norm \( 6300 \)
CM no
Base change yes: 1470.m5,210.d5
Q-curve yes
Torsion order \( 12 \)
Rank \( 0 \)

Related objects

Downloads

Learn more about

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

\(y^2+xy=x^{3}-361x+2585\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-361,0]),K([2585,0])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-361,0])),Pol(Vecrev([2585,0]))], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-361,0],K![2585,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \(\left(-60a + 30\right)\) = \(\left(a\right)\cdot\left(-a + 1\right)\cdot\left(-2a + 1\right)\cdot\left(3\right)\cdot\left(5\right)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 6300 \) = \(2\cdot2\cdot7\cdot9\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \(( 57153600 )\) = \(\left(a\right)^{6}\cdot\left(-a + 1\right)^{6}\cdot\left(-2a + 1\right)^{4}\cdot\left(3\right)^{6}\cdot\left(5\right)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3266533992960000 \) = \(2^{6}\cdot2^{6}\cdot7^{4}\cdot9^{6}\cdot25^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{5203798902289}{57153600} \)
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: \(\Z/2\Z\times\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(8 : 11 : 1\right)$ $\left(10 : -5 : 1\right)$
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: \( 0.545369050370003 \)
Tamagawa product: \( 1728 \)  =  \(( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2\)
Torsion order: \(12\)
Leading coefficient: \( 4.94712301724739 \)
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\((j)_{-}\)
\(\left(a\right)\) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\(\left(-a + 1\right)\) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\(\left(-2a + 1\right)\) \(7\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\(\left(3\right)\) \(9\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\(\left(5\right)\) \(25\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 6300.2-d consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base change of elliptic curves 1470.m5, 210.d5, defined over \(\Q\), so it is also a \(\Q\)-curve.