Properties

Label 3.3.1129.1-9.4-b2
Base field 3.3.1129.1
Conductor norm \( 9 \)
CM no
Base change no
Q-curve no
Torsion order \( 6 \)
Rank \( 0 \)

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Base field 3.3.1129.1

Generator \(a\), with minimal polynomial \( x^{3} - 7 x - 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, -7, 0, 1]))
 
gp: K = nfinit(Polrev([-3, -7, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -7, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(-3a^{2}+16\right){x}-2a^{2}+12\)
sage: E = EllipticCurve([K([-4,0,1]),K([5,0,-1]),K([-4,0,1]),K([16,0,-3]),K([12,0,-2])])
 
gp: E = ellinit([Polrev([-4,0,1]),Polrev([5,0,-1]),Polrev([-4,0,1]),Polrev([16,0,-3]),Polrev([12,0,-2])], K);
 
magma: E := EllipticCurve([K![-4,0,1],K![5,0,-1],K![-4,0,1],K![16,0,-3],K![12,0,-2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a+3)\) = \((a)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((4a^2-12a+9)\) = \((a)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 6561 \) = \(3^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1364027}{9} a^{2} + \frac{1290553}{3} a + \frac{1441468}{9} \)
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: \(0\)
Torsion structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a^{2} + 8 : -8 a^{2} + 3 a + 54 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 108.99747129905421374411438722967960141 \)
Tamagawa product: \( 4 \)
Torsion order: \(6\)
Leading coefficient: \( 0.36043498814297315896331029698181954661 \)
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\((j)_{-}\)
\((a)\) \(3\) \(4\) \(I_{2}^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 9.4-b consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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