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Properties

Label 4.4.13888.1-28.2-c1
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

\({y}^2+a{x}{y}+\left(\frac{1}{3}a^{3}-\frac{2}{3}a^{2}-\frac{4}{3}a+2\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{5}{3}a^{2}-\frac{2}{3}a-6\right){x}^{2}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{2}{3}a+4\right){x}-\frac{2}{3}a^{3}+\frac{7}{3}a^{2}+\frac{2}{3}a-5\)
sage: E = EllipticCurve([K([0,1,0,0]),K([-6,-2/3,5/3,-1/3]),K([2,-4/3,-2/3,1/3]),K([4,-2/3,2/3,-1/3]),K([-5,2/3,7/3,-2/3])])
 
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([-6,-2/3,5/3,-1/3]),Polrev([2,-4/3,-2/3,1/3]),Polrev([4,-2/3,2/3,-1/3]),Polrev([-5,2/3,7/3,-2/3])], K);
 
magma: E := EllipticCurve([K![0,1,0,0],K![-6,-2/3,5/3,-1/3],K![2,-4/3,-2/3,1/3],K![4,-2/3,2/3,-1/3],K![-5,2/3,7/3,-2/3]]);
 

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: \((-19/3a^3+38/3a^2+76/3a-25)\) = \((1/3a^3-2/3a^2-4/3a+1)\cdot(1/3a^3+1/3a^2-1/3a-1)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 470596 \) = \(4\cdot7^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{557101}{2058} a^{3} + \frac{132200}{1029} a^{2} - \frac{2506865}{1029} a - \frac{1440367}{686} \)
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\((j)_{-}\)
\((1/3a^3-2/3a^2-4/3a+1)\) \(4\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((1/3a^3+1/3a^2-1/3a-1)\) \(7\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

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-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.