Properties

Label 4.4.6125.1-71.1-d1
Base field 4.4.6125.1
Conductor norm \( 71 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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

\({y}^2+\left(a^{3}+2a^{2}-6a-8\right){x}{y}+\left(2a^{3}+3a^{2}-12a-11\right){y}={x}^{3}+\left(-2a^{3}-3a^{2}+11a+13\right){x}^{2}+\left(a^{3}-2a^{2}-4a+8\right){x}+4a^{3}-3a^{2}-13a\)
sage: E = EllipticCurve([K([-8,-6,2,1]),K([13,11,-3,-2]),K([-11,-12,3,2]),K([8,-4,-2,1]),K([0,-13,-3,4])])
 
gp: E = ellinit([Polrev([-8,-6,2,1]),Polrev([13,11,-3,-2]),Polrev([-11,-12,3,2]),Polrev([8,-4,-2,1]),Polrev([0,-13,-3,4])], K);
 
magma: E := EllipticCurve([K![-8,-6,2,1],K![13,11,-3,-2],K![-11,-12,3,2],K![8,-4,-2,1],K![0,-13,-3,4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^3-2a^2+14a+5)\) = \((-2a^3-2a^2+14a+5)\)
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: \((-a^3-2a^2+7a+4)\) = \((-2a^3-2a^2+14a+5)\)
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: \( \frac{732083245280}{71} a^{3} - \frac{1278302446217}{71} a^{2} - \frac{5634987718502}{71} a + \frac{10793107895044}{71} \)
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 $\left(-4 a^{3} - 5 a^{2} + 26 a + 19 : 11 a^{3} + 14 a^{2} - 70 a - 57 : 1\right)$
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\((j)_{-}\)
\((-2a^3-2a^2+14a+5)\) \(71\) \(1\) \(I_{1}\) 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 71.1-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.