Properties

Label 4.4.13888.1-28.1-c2
Base field 4.4.13888.1
Conductor norm \( 28 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 4.4.13888.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 6 x + 9 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 6, -7, -2, 1]))
 
gp: K = nfinit(Polrev([9, 6, -7, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 6, -7, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{3}a^{3}-\frac{2}{3}a^{2}-\frac{1}{3}a+1\right){x}{y}+{y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{10}{3}a-2\right){x}^{2}+\left(-\frac{2}{3}a^{3}+\frac{10}{3}a^{2}+\frac{5}{3}a\right){x}+\frac{1}{3}a^{3}+\frac{4}{3}a^{2}-\frac{13}{3}a-2\)
sage: E = EllipticCurve([K([1,-1/3,-2/3,1/3]),K([-2,-10/3,1/3,1/3]),K([1,0,0,0]),K([0,5/3,10/3,-2/3]),K([-2,-13/3,4/3,1/3])])
 
gp: E = ellinit([Polrev([1,-1/3,-2/3,1/3]),Polrev([-2,-10/3,1/3,1/3]),Polrev([1,0,0,0]),Polrev([0,5/3,10/3,-2/3]),Polrev([-2,-13/3,4/3,1/3])], K);
 
magma: E := EllipticCurve([K![1,-1/3,-2/3,1/3],K![-2,-10/3,1/3,1/3],K![1,0,0,0],K![0,5/3,10/3,-2/3],K![-2,-13/3,4/3,1/3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^2+a+5)\) = \((1/3a^3-2/3a^2-4/3a+1)\cdot(1/3a^3-2/3a^2-7/3a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 28 \) = \(4\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^3+10a^2+2a-40)\) = \((1/3a^3-2/3a^2-4/3a+1)^{3}\cdot(1/3a^3-2/3a^2-7/3a+1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -21952 \) = \(-4^{3}\cdot7^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{8642508041}{4116} a^{3} - \frac{3993611659}{4116} a^{2} - \frac{66629144975}{4116} a - \frac{4215560890}{343} \)
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 $\left(-a^{2} + a + 4 : \frac{2}{3} a^{3} - \frac{4}{3} a^{2} - \frac{5}{3} a + 4 : 1\right)$
Height \(0.20458886450976619600303360201698569564\)
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: \( 0.20458886450976619600303360201698569564 \)
Period: \( 147.71138861156726004162252886888230309 \)
Tamagawa product: \( 3 \)  =  \(1\cdot3\)
Torsion order: \(1\)
Leading coefficient: \( 3.07721162897075 \)
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\((j)_{-}\)
\((1/3a^3-2/3a^2-4/3a+1)\) \(4\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((1/3a^3-2/3a^2-7/3a+1)\) \(7\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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 28.1-c 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.