Properties

Label 5.5.65657.1-43.1-a1
Base field 5.5.65657.1
Conductor norm \( 43 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 5.5.65657.1

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

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

Weierstrass equation

\({y}^2+{x}{y}+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){y}={x}^{3}+\left(-2a^{4}+3a^{3}+9a^{2}-10a-8\right){x}^{2}+\left(-a^{4}-2a^{3}+4a^{2}+11a+10\right){x}+2a^{4}+3a^{3}-9a^{2}-20a-9\)
sage: E = EllipticCurve([K([1,0,0,0,0]),K([-8,-10,9,3,-2]),K([2,3,-4,-1,1]),K([10,11,4,-2,-1]),K([-9,-20,-9,3,2])])
 
gp: E = ellinit([Polrev([1,0,0,0,0]),Polrev([-8,-10,9,3,-2]),Polrev([2,3,-4,-1,1]),Polrev([10,11,4,-2,-1]),Polrev([-9,-20,-9,3,2])], K);
 
magma: E := EllipticCurve([K![1,0,0,0,0],K![-8,-10,9,3,-2],K![2,3,-4,-1,1],K![10,11,4,-2,-1],K![-9,-20,-9,3,2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-2a^3-4a^2+4a+2)\) = \((a^4-2a^3-4a^2+4a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 43 \) = \(43\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2a^4-3a^3-8a^2+8a+6)\) = \((a^4-2a^3-4a^2+4a+2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -43 \) = \(-43\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{12952}{43} a^{4} + \frac{5048}{43} a^{3} + \frac{43307}{43} a^{2} + \frac{10999}{43} a + \frac{393}{43} \)
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 $\left(-a^{3} + 5 a + 3 : -2 a^{4} + 9 a^{2} + 4 a - 1 : 1\right)$
Height \(0.078230725386790549855472709437068289805\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.078230725386790549855472709437068289805 \)
Period: \( 1874.3328348886477397131694007505805665 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.86123519 \)
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\((j)_{-}\)
\((a^4-2a^3-4a^2+4a+2)\) \(43\) \(1\) \(I_{1}\) Non-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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 43.1-a 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.