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Properties

Label 4.4.6125.1-59.1-b1
Base field 4.4.6125.1
Conductor norm \( 59 \)
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

\({y}^2+\left(2a^{3}+3a^{2}-11a-11\right){x}{y}+\left(a^{3}+a^{2}-6a-4\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(21a^{3}+29a^{2}-121a-99\right){x}+26a^{3}+36a^{2}-150a-124\)
sage: E = EllipticCurve([K([-11,-11,3,2]),K([-3,-5,1,1]),K([-4,-6,1,1]),K([-99,-121,29,21]),K([-124,-150,36,26])])
 
gp: E = ellinit([Polrev([-11,-11,3,2]),Polrev([-3,-5,1,1]),Polrev([-4,-6,1,1]),Polrev([-99,-121,29,21]),Polrev([-124,-150,36,26])], K);
 
magma: E := EllipticCurve([K![-11,-11,3,2],K![-3,-5,1,1],K![-4,-6,1,1],K![-99,-121,29,21],K![-124,-150,36,26]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3+a^2-5a-2)\) = \((a^3+a^2-5a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 59 \) = \(59\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((28a^3+29a^2-181a-120)\) = \((a^3+a^2-5a-2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -205379 \) = \(-59^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{573485304627}{205379} a^{3} - \frac{806332617709}{205379} a^{2} + \frac{3346741850839}{205379} a + \frac{2708257735800}{205379} \)
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 $\left(-a^{3} - a^{2} + 6 a + 3 : a^{3} + 2 a^{2} - 6 a - 8 : 1\right)$
Height \(0.062194364252432044227068110986727272931\)
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: \( 0.062194364252432044227068110986727272931 \)
Period: \( 271.92996515815070178103777634240379646 \)
Tamagawa product: \( 3 \)
Torsion order: \(1\)
Leading coefficient: \( 2.59320171061156 \)
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\((j)_{-}\)
\((a^3+a^2-5a-2)\) \(59\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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 59.1-b 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.