↑

Properties

Label 4.4.6125.1-71.3-d1
Base field 4.4.6125.1
Conductor norm \( 71 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field 4.4.6125.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 9 x^{2} + 9 x + 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 9, -9, -1, 1]))
 
gp: K = nfinit(Polrev([11, 9, -9, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 9, -9, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{3}+3a^{2}-12a-12\right){x}{y}+\left(a^{3}+2a^{2}-6a-7\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(10a^{3}+12a^{2}-62a-46\right){x}+35a^{3}+41a^{2}-221a-165\)
sage: E = EllipticCurve([K([-12,-12,3,2]),K([-4,1,1,0]),K([-7,-6,2,1]),K([-46,-62,12,10]),K([-165,-221,41,35])])
 
gp: E = ellinit([Polrev([-12,-12,3,2]),Polrev([-4,1,1,0]),Polrev([-7,-6,2,1]),Polrev([-46,-62,12,10]),Polrev([-165,-221,41,35])], K);
 
magma: E := EllipticCurve([K![-12,-12,3,2],K![-4,1,1,0],K![-7,-6,2,1],K![-46,-62,12,10],K![-165,-221,41,35]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((5a^3+7a^2-28a-24)\) = \((5a^3+7a^2-28a-24)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 71 \) = \(71\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3a^3-4a^2+19a+19)\) = \((5a^3+7a^2-28a-24)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 71 \) = \(71\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{5520752072949}{71} a^{3} + \frac{7531137764446}{71} a^{2} - \frac{31882024190872}{71} a - \frac{25687141194430}{71} \)
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 $\left(a^{3} + 2 a^{2} - 8 a - 7 : 3 a^{3} - 14 a - 6 : 1\right)$
Height \(0.077631787363044595628465340549283820168\)
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: \( 0.077631787363044595628465340549283820168 \)
Period: \( 821.98230113981957842434399457010672579 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 3.26143702046240 \)
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\((j)_{-}\)
\((5a^3+7a^2-28a-24)\) \(71\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 71.3-d consists of this curve only.

Base change

This elliptic curve is not a \(\Q\)-curve.

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