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Properties

Label 4.4.15125.1-19.3-a1
Base field 4.4.15125.1
Conductor norm \( 19 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 4.4.15125.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 14 x^{2} + 14 x + 31 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([31, 14, -14, -1, 1]))
 
gp: K = nfinit(Polrev([31, 14, -14, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 14, -14, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+2a^{2}-8a-13\right){x}{y}+\left(-2a^{3}-3a^{2}+18a+17\right){y}={x}^{3}+\left(4a^{3}+7a^{2}-37a-43\right){x}^{2}+\left(-6a^{3}+a^{2}+39a+27\right){x}-a^{3}+41a^{2}-62a-132\)
sage: E = EllipticCurve([K([-13,-8,2,1]),K([-43,-37,7,4]),K([17,18,-3,-2]),K([27,39,1,-6]),K([-132,-62,41,-1])])
 
gp: E = ellinit([Polrev([-13,-8,2,1]),Polrev([-43,-37,7,4]),Polrev([17,18,-3,-2]),Polrev([27,39,1,-6]),Polrev([-132,-62,41,-1])], K);
 
magma: E := EllipticCurve([K![-13,-8,2,1],K![-43,-37,7,4],K![17,18,-3,-2],K![27,39,1,-6],K![-132,-62,41,-1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((6a^3+11a^2-53a-65)\) = \((6a^3+11a^2-53a-65)\)
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: \((-23a^3-41a^2+210a+256)\) = \((6a^3+11a^2-53a-65)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -6859 \) = \(-19^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{39656807629}{6859} a^{3} + \frac{69710284663}{6859} a^{2} - \frac{362938149300}{6859} a - \frac{445724222643}{6859} \)
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 $\left(-2 a^{3} - 3 a^{2} + 18 a + 18 : 3 a^{3} + 4 a^{2} - 23 a - 33 : 1\right)$
Height \(0.019605404074072226327155177560972062445\)
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: \( 0.019605404074072226327155177560972062445 \)
Period: \( 1650.7340530112620363747476189554049418 \)
Tamagawa product: \( 3 \)
Torsion order: \(1\)
Leading coefficient: \( 3.15781339423595 \)
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\((j)_{-}\)
\((6a^3+11a^2-53a-65)\) \(19\) \(3\) \(I_{3}\) 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 19.3-a 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.