Properties

Label 4.4.5125.1-41.1-c2
Base field 4.4.5125.1
Conductor \((a^3-a^2-4a-3)\)
Conductor norm \( 41 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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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

\({y}^2+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-10a^{2}+10a\right){x}-31a^{2}+31a+11\)
sage: E = EllipticCurve([K([0,0,0,0]),K([-4,-1,1,0]),K([-4,-1,1,0]),K([0,10,-10,0]),K([11,31,-31,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0,0,0])),Pol(Vecrev([-4,-1,1,0])),Pol(Vecrev([-4,-1,1,0])),Pol(Vecrev([0,10,-10,0])),Pol(Vecrev([11,31,-31,0]))], K);
 
magma: E := EllipticCurve([K![0,0,0,0],K![-4,-1,1,0],K![-4,-1,1,0],K![0,10,-10,0],K![11,31,-31,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-4a-3)\) = \((a^3-a^2-4a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 41 \) = \(41\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((607417a^2-607417a-2935866)\) = \((a^3-a^2-4a-3)^{14}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 37929227194915558802161 \) = \(41^{14}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{7215644871110656}{194754273881} a^{2} + \frac{7215644871110656}{194754273881} a + \frac{16969651353047040}{194754273881} \)
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: \( 0.891323980307425 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 1.22015423553756 \)
Analytic order of Ш: \( 49 \) (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\((j)_{-}\)
\((a^3-a^2-4a-3)\) \(41\) \(2\) \(I_{14}\) Non-split multiplicative \(1\) \(1\) \(14\) \(14\)

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.1.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 41.1-c 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.