Properties

Label 2.0.7.1-14641.3-d1
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 14641 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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

\({y}^2+{y}={x}^{3}-{x}^{2}-946260{x}+354609639\)
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([-946260,0]),K([354609639,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-946260,0]),Polrev([354609639,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![-946260,0],K![354609639,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((121)\) = \((-2a+3)^{2}\cdot(2a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 14641 \) = \(11^{2}\cdot11^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-19487171)\) = \((-2a+3)^{7}\cdot(2a+1)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 379749833583241 \) = \(11^{7}\cdot11^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{52893159101157376}{11} \)
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 $\left(-\frac{46345648416205066273008150278875}{8167677748751632240041796} : -\frac{119251666241478182303874377246406471153745679479}{11671264091961017238413879716533116828} a + \frac{119251666229806918211913360007992591437212562651}{23342528183922034476827759433066233656} : 1\right)$
Height \(70.515883950173463768076651064157069986\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 70.515883950173463768076651064157069986 \)
Period: \( 0.033664429517381078503781940215004525249 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 14.355858739519966086011636946829310334 \)
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\((j)_{-}\)
\((-2a+3)\) \(11\) \(2\) \(I_{1}^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)
\((2a+1)\) \(11\) \(2\) \(I_{1}^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5 and 25.
Its isogeny class 14641.3-d consists of curves linked by isogenies of degrees dividing 25.

Base change

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

Base field Curve
\(\Q\) 121.d1
\(\Q\) 5929.h1