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Properties

Label 5.5.160801.1-9.1-b1
Base field 5.5.160801.1
Conductor norm \( 9 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 5.5.160801.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 4, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 3, 4, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 4, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-a^{3}-5a^{2}+4a+3\right){x}{y}+\left(a^{4}-a^{3}-5a^{2}+4a+2\right){y}={x}^{3}+\left(-2a^{4}+a^{3}+9a^{2}-3a-2\right){x}^{2}+\left(-6a^{4}+a^{3}+26a^{2}-6a-11\right){x}-3a^{4}-4a^{3}+7a^{2}+6a+1\)
sage: E = EllipticCurve([K([3,4,-5,-1,1]),K([-2,-3,9,1,-2]),K([2,4,-5,-1,1]),K([-11,-6,26,1,-6]),K([1,6,7,-4,-3])])
 
gp: E = ellinit([Polrev([3,4,-5,-1,1]),Polrev([-2,-3,9,1,-2]),Polrev([2,4,-5,-1,1]),Polrev([-11,-6,26,1,-6]),Polrev([1,6,7,-4,-3])], K);
 
magma: E := EllipticCurve([K![3,4,-5,-1,1],K![-2,-3,9,1,-2],K![2,4,-5,-1,1],K![-11,-6,26,1,-6],K![1,6,7,-4,-3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-5a^2+3)\) = \((a^4-5a^2+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^4+5a^2-3)\) = \((a^4-5a^2+3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -9 \) = \(-9\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{747793342}{3} a^{4} - 753884124 a^{3} + \frac{839874611}{3} a^{2} + \frac{1290423893}{3} a - \frac{369335632}{3} \)
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 $\left(3 a^{4} + 2 a^{3} - 9 a^{2} - a + 2 : -16 a^{4} - 17 a^{3} + 41 a^{2} + 14 a - 11 : 1\right)$
Height \(0.018468321276315460776173183407387705198\)
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: \( 0.018468321276315460776173183407387705198 \)
Period: \( 10888.105467214731903174295117546705911 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.50729464 \)
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\((j)_{-}\)
\((a^4-5a^2+3)\) \(9\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 9.1-b consists of this curve only.

Base change

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

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