Properties

Label 4.4.18097.1-12.1-d1
Base field 4.4.18097.1
Conductor norm \( 12 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 4.4.18097.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 6 x + 4 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 6, -7, -1, 1]))
 
gp: K = nfinit(Polrev([4, 6, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 6, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{3}{2}a-1\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{5}{2}a-3\right){x}^{2}+\left(4a^{3}+6a^{2}-14a-2\right){x}+13a^{3}+14a^{2}-31a-18\)
sage: E = EllipticCurve([K([-1,-3/2,1/2,1/2]),K([-3,5/2,1/2,-1/2]),K([0,1,0,0]),K([-2,-14,6,4]),K([-18,-31,14,13])])
 
gp: E = ellinit([Polrev([-1,-3/2,1/2,1/2]),Polrev([-3,5/2,1/2,-1/2]),Polrev([0,1,0,0]),Polrev([-2,-14,6,4]),Polrev([-18,-31,14,13])], K);
 
magma: E := EllipticCurve([K![-1,-3/2,1/2,1/2],K![-3,5/2,1/2,-1/2],K![0,1,0,0],K![-2,-14,6,4],K![-18,-31,14,13]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a-2)\) = \((1/2a^3-1/2a^2-5/2a+1)\cdot(a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 12 \) = \(3\cdot4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-12a^3-7a^2+85a)\) = \((1/2a^3-1/2a^2-5/2a+1)^{13}\cdot(a)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 6377292 \) = \(3^{13}\cdot4\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1049416907}{6377292} a^{3} + \frac{395116529}{2125764} a^{2} - \frac{4876120775}{6377292} a - \frac{743685596}{1594323} \)
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 $\left(-\frac{1}{2} a^{3} + \frac{7}{2} a^{2} - \frac{7}{2} a - 1 : \frac{5}{2} a^{3} - \frac{43}{2} a^{2} + \frac{31}{2} a + 12 : 1\right)$
Height \(0.018723521992003206299655056793235152724\)
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: \( 0.018723521992003206299655056793235152724 \)
Period: \( 369.26099002554251500994446555103185733 \)
Tamagawa product: \( 13 \)  =  \(13\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 2.67252037755543 \)
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\((j)_{-}\)
\((1/2a^3-1/2a^2-5/2a+1)\) \(3\) \(13\) \(I_{13}\) Split multiplicative \(-1\) \(1\) \(13\) \(13\)
\((a)\) \(4\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 12.1-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.