Properties

Label 4.4.2624.1-41.1-b1
Base field 4.4.2624.1
Conductor norm \( 41 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.2624.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 2, -3, -2, 1]))
 
gp: K = nfinit(Polrev([1, 2, -3, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -3, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-2a^{2}-a+1\right){y}={x}^{3}+\left(-a^{3}+3a^{2}-a-2\right){x}^{2}+\left(-6a^{3}+16a^{2}+6a-17\right){x}-a^{3}+2a^{2}+a+1\)
sage: E = EllipticCurve([K([0,-3,-1,1]),K([-2,-1,3,-1]),K([1,-1,-2,1]),K([-17,6,16,-6]),K([1,1,2,-1])])
 
gp: E = ellinit([Polrev([0,-3,-1,1]),Polrev([-2,-1,3,-1]),Polrev([1,-1,-2,1]),Polrev([-17,6,16,-6]),Polrev([1,1,2,-1])], K);
 
magma: E := EllipticCurve([K![0,-3,-1,1],K![-2,-1,3,-1],K![1,-1,-2,1],K![-17,6,16,-6],K![1,1,2,-1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+2a^2+4a-2)\) = \((-a^3+2a^2+4a-2)\)
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: \((-29a^3+58a^2+58a-28)\) = \((-a^3+2a^2+4a-2)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2825761 \) = \(41^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{8803072}{1681} a^{3} - \frac{119358784}{1681} a^{2} + \frac{227100416}{1681} a + \frac{100750080}{1681} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{1}{2} a^{2} + a + \frac{1}{2} : -\frac{3}{4} a^{3} + \frac{3}{2} a^{2} + \frac{3}{2} a - \frac{1}{4} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 174.39113818561653904961436137891573658 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.70220750565522 \)
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\((j)_{-}\)
\((-a^3+2a^2+4a-2)\) \(41\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 41.1-b consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.