Properties

Label 2.0.7.1-6300.2-c10
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 6300 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}-1920800{x}-1024800150\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-1920800,0]),K([-1024800150,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-1920800,0]),Polrev([-1024800150,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-1920800,0],K![-1024800150,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-60a+30)\) = \((a)\cdot(-a+1)\cdot(-2a+1)\cdot(3)\cdot(5)\)
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: \((360150)\) = \((a)\cdot(-a+1)\cdot(-2a+1)^{8}\cdot(3)\cdot(5)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 129708022500 \) = \(2\cdot2\cdot7^{8}\cdot9\cdot25^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{783736670177727068275201}{360150} \)
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 $\left(-\frac{11853787}{14812} : \frac{1443607}{1192366} a + \frac{1905572493}{4769464} : 1\right)$
Height \(9.9367211786574160460566975739379693160\)
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: $\left(-\frac{3201}{4} : \frac{3201}{8} : 1\right)$
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: \( 9.9367211786574160460566975739379693160 \)
Period: \( 0.027629842682292462198446106697647857105 \)
Tamagawa product: \( 4 \)  =  \(1\cdot1\cdot2\cdot1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 6.6412903869459543008370323249614582323 \)
Analytic order of Ш: \( 16 \) (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\((j)_{-}\)
\((a)\) \(2\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((-a+1)\) \(2\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((-2a+1)\) \(7\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((3)\) \(9\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((5)\) \(25\) \(2\) \(I_{2}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8, 16 and 32.
Its isogeny class 6300.2-c consists of curves linked by isogenies of degrees dividing 32.

Base change

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

Base field Curve
\(\Q\) 210.e1
\(\Q\) 1470.j1