Properties

Label 3.3.1129.1-19.1-b1
Base field 3.3.1129.1
Conductor norm \( 19 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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

\({y}^2+\left(a^{2}-5\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-3a^{2}-10a-1\right){x}+11a^{2}+27a+4\)
sage: E = EllipticCurve([K([0,0,0]),K([-4,-1,1]),K([-5,0,1]),K([-1,-10,-3]),K([4,27,11])])
 
gp: E = ellinit([Polrev([0,0,0]),Polrev([-4,-1,1]),Polrev([-5,0,1]),Polrev([-1,-10,-3]),Polrev([4,27,11])], K);
 
magma: E := EllipticCurve([K![0,0,0],K![-4,-1,1],K![-5,0,1],K![-1,-10,-3],K![4,27,11]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^2-a+4)\) = \((-a^2-a+4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 19 \) = \(19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^2-a+4)\) = \((-a^2-a+4)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -19 \) = \(-19\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{11005952}{19} a^{2} - \frac{4861952}{19} a - \frac{74928128}{19} \)
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 $\left(a + 1 : -a^{2} - a + 3 : 1\right)$
Height \(0.035446905571721595264340441305292503156\)
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: \( 0.035446905571721595264340441305292503156 \)
Period: \( 391.85775974254693100758182714862171115 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.2401695485654773274012981536940702295 \)
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\((j)_{-}\)
\((-a^2-a+4)\) \(19\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 19.1-b consists of curves linked by isogenies of degree 3.

Base change

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

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