Properties

Label 2.0.7.1-28672.7-n1
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 28672 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 2 \)

Related objects

Downloads

Learn more

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

Weierstrass equation

\({y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(16a-12\right){x}-30a+10\)
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([-12,16]),K([10,-30])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([0,0]),Polrev([-12,16]),Polrev([10,-30])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![-12,16],K![10,-30]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-128a+64)\) = \((a)^{6}\cdot(-a+1)^{6}\cdot(-2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 28672 \) = \(2^{6}\cdot2^{6}\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((320a+3648)\) = \((a)^{6}\cdot(-a+1)^{15}\cdot(-2a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 14680064 \) = \(2^{6}\cdot2^{15}\cdot7\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1111000}{7} a - \frac{1378000}{7} \)
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: \(2\)
Generators $\left(-\frac{13}{16} a + \frac{47}{16} : -\frac{5}{64} a + \frac{7}{64} : 1\right)$ $\left(-\frac{1}{4} a + \frac{11}{4} : -\frac{13}{8} a - \frac{1}{8} : 1\right)$
Heights \(1.7268971738497312273265166539376039151\) \(1.2240108915806313033127343119381942016\)
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: $\left(-a + 3 : 0 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 1.7391902837548789866204613996892659974 \)
Period: \( 2.4952796273851326316735427653386684542 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 6.5611096024795975879012166099658174382 \)
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\((j)_{-}\)
\((a)\) \(2\) \(1\) \(II\) Additive \(-1\) \(6\) \(6\) \(0\)
\((-a+1)\) \(2\) \(2\) \(I_{5}^{*}\) Additive \(-1\) \(6\) \(15\) \(0\)
\((-2a+1)\) \(7\) \(1\) \(I_{1}\) 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 and 4.
Its isogeny class 28672.7-n consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a \(\Q\)-curve.

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