Properties

Label 4.4.13888.1-28.2-a1
Base field 4.4.13888.1
Conductor norm \( 28 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/3a^3+4/3a^2-4/3a-3)\) = \((1/3a^3-2/3a^2-4/3a+1)\cdot(1/3a^3+1/3a^2-1/3a-1)\)
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: \((-2/3a^3+4/3a^2+8/3a-10)\) = \((1/3a^3-2/3a^2-4/3a+1)^{3}\cdot(1/3a^3+1/3a^2-1/3a-1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3136 \) = \(4^{3}\cdot7^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{9073387}{84} a^{3} - \frac{3135904}{21} a^{2} + \frac{10933271}{42} a + \frac{7712361}{28} \)
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: 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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 100.33966736223458822502390614163465279 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 1.70287511400767 \)
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 \(1\) \(1\) \(3\) \(3\)
\((1/3a^3+1/3a^2-1/3a-1)\) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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 28.2-a 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.