Properties

Label 2.0.7.1-26244.2-l3
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 26244 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 3 \)
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

\({y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-95{x}-697\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([-95,0]),K([-697,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-95,0]),Polrev([-697,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![-95,0],K![-697,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((162)\) = \((a)\cdot(-a+1)\cdot(3)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 26244 \) = \(2\cdot2\cdot9^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-169869312)\) = \((a)^{21}\cdot(-a+1)^{21}\cdot(3)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 28855583159353344 \) = \(2^{21}\cdot2^{21}\cdot9^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1159088625}{2097152} \)
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 $\left(-4 a - 1 : -12 a - 6 : 1\right)$
Height \(0.23990086462566904313136453079413008001\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(19 : -74 : 1\right)$
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: \( 0.23990086462566904313136453079413008001 \)
Period: \( 0.58348521984058875136882165159873123061 \)
Tamagawa product: \( 441 \)  =  \(( 3 \cdot 7 )\cdot( 3 \cdot 7 )\cdot1\)
Torsion order: \(3\)
Leading coefficient: \( 10.369760452363677327351334332217861112 \)
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\((j)_{-}\)
\((a)\) \(2\) \(21\) \(I_{21}\) Split multiplicative \(-1\) \(1\) \(21\) \(21\)
\((-a+1)\) \(2\) \(21\) \(I_{21}\) Split multiplicative \(-1\) \(1\) \(21\) \(21\)
\((3)\) \(9\) \(1\) \(II\) Additive \(1\) \(4\) \(4\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.1
\(7\) 7B.2.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 7 and 21.
Its isogeny class 26244.2-l consists of curves linked by isogenies of degrees dividing 21.

Base change

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

Base field Curve
\(\Q\) 162.c2
\(\Q\) 7938.x2