Properties

Label 4.4.5125.1-29.1-b2
Base field 4.4.5125.1
Conductor \((a^3-4a-4)\)
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 7 \)
Rank \( 1 \)

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Base field 4.4.5125.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 7, -6, -2, 1]))
 
gp: K = nfinit(Pol(Vecrev([11, 7, -6, -2, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 7, -6, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+3\right){x}^{2}+\left(-3a^{3}+7a^{2}+8a-18\right){x}+4a^{3}-14a^{2}-9a+43\)
sage: E = EllipticCurve([K([-3,-1,1,0]),K([3,-3,-2,1]),K([0,-3,-1,1]),K([-18,8,7,-3]),K([43,-9,-14,4])])
 
gp: E = ellinit([Pol(Vecrev([-3,-1,1,0])),Pol(Vecrev([3,-3,-2,1])),Pol(Vecrev([0,-3,-1,1])),Pol(Vecrev([-18,8,7,-3])),Pol(Vecrev([43,-9,-14,4]))], K);
 
magma: E := EllipticCurve([K![-3,-1,1,0],K![3,-3,-2,1],K![0,-3,-1,1],K![-18,8,7,-3],K![43,-9,-14,4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-4a-4)\) = \((a^3-4a-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 29 \) = \(29\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^2+2a+5)\) = \((a^3-4a-4)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -29 \) = \(-29\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1067314607}{29} a^{3} + \frac{130599537}{29} a^{2} - \frac{6126735575}{29} a - \frac{5531882051}{29} \)
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 $\left(-\frac{2}{5} a^{3} + \frac{4}{5} a^{2} + a - \frac{2}{5} : \frac{1}{5} a^{3} - \frac{9}{5} a^{2} - \frac{1}{5} a + \frac{37}{5} : 1\right)$
Height \(1.05649217369876\)
Torsion structure: \(\Z/7\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(1 : -2 a^{2} + a + 8 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.05649217369876 \)
Period: \( 1661.60091271852 \)
Tamagawa product: \( 1 \)
Torsion order: \(7\)
Leading coefficient: \( 2.00175067095063 \)
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\((j)_{-}\)
\((a^3-4a-4)\) \(29\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 29.1-b consists of curves linked by isogenies of degree 7.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.