Properties

Label 4.4.8768.1-28.1-b2
Base field 4.4.8768.1
Conductor norm \( 28 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 4.4.8768.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 5 x^{2} + 6 x + 7 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, 6, -5, -2, 1]))
 
gp: K = nfinit(Polrev([7, 6, -5, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, 6, -5, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(4a^{3}-18a^{2}+10a+19\right){x}+714a^{3}-2617a^{2}+775a+3011\)
sage: E = EllipticCurve([K([-2,-1,1,0]),K([3,1,-1,0]),K([1,-3,-1,1]),K([19,10,-18,4]),K([3011,775,-2617,714])])
 
gp: E = ellinit([Polrev([-2,-1,1,0]),Polrev([3,1,-1,0]),Polrev([1,-3,-1,1]),Polrev([19,10,-18,4]),Polrev([3011,775,-2617,714])], K);
 
magma: E := EllipticCurve([K![-2,-1,1,0],K![3,1,-1,0],K![1,-3,-1,1],K![19,10,-18,4],K![3011,775,-2617,714]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+a^2+3a)\) = \((a^2-a-3)\cdot(-a)\)
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: \((-2a^3-2a^2+14a)\) = \((a^2-a-3)^{3}\cdot(-a)^{3}\)
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: \( \frac{661426621493919350957}{343} a^{3} - \frac{7553259107304969906021}{1372} a^{2} - \frac{6771160831123005516347}{1372} a + \frac{21662985339131396964743}{1372} \)
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: \( 13.294085852533414794657947584849817059 \)
Tamagawa product: \( 9 \)  =  \(3\cdot3\)
Torsion order: \(1\)
Leading coefficient: \( 1.27776420567630 \)
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\((j)_{-}\)
\((a^2-a-3)\) \(4\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((-a)\) \(7\) \(3\) \(I_{3}\) 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.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 28.1-b 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.