Properties

Label 5.5.70601.1-9.1-b3
Base field 5.5.70601.1
Conductor norm \( 9 \)
CM no
Base change no
Q-curve no
Torsion order \( 10 \)
Rank \( 0 \)

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Base field 5.5.70601.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 2, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 3, 2, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{4}-2a^{3}-9a^{2}+4a+3\right){x}{y}+\left(a^{4}-6a^{2}-a+3\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+a+1\right){x}^{2}+\left(-2a^{4}+5a^{3}+a^{2}-5a+2\right){x}-2a^{4}+6a^{3}-a^{2}-5a+2\)
sage: E = EllipticCurve([K([3,4,-9,-2,2]),K([1,1,-5,-1,1]),K([3,-1,-6,0,1]),K([2,-5,1,5,-2]),K([2,-5,-1,6,-2])])
 
gp: E = ellinit([Polrev([3,4,-9,-2,2]),Polrev([1,1,-5,-1,1]),Polrev([3,-1,-6,0,1]),Polrev([2,-5,1,5,-2]),Polrev([2,-5,-1,6,-2])], K);
 
magma: E := EllipticCurve([K![3,4,-9,-2,2],K![1,1,-5,-1,1],K![3,-1,-6,0,1],K![2,-5,1,5,-2],K![2,-5,-1,6,-2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+a^2+4a)\) = \((-a^3+a^2+4a)\)
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: \((6a^4-2a^3-37a^2-4a+30)\) = \((-a^3+a^2+4a)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 59049 \) = \(9^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1226891}{243} a^{4} - \frac{2337049}{243} a^{3} - \frac{4118459}{243} a^{2} + \frac{2062492}{81} a - \frac{1078507}{243} \)
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: \(\Z/10\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(4 a^{4} - 12 a^{3} + 2 a^{2} + 9 a - 4 : 45 a^{4} - 125 a^{3} - 2 a^{2} + 90 a - 24 : 1\right)$
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: \( 6799.5960135103644891984531684452265081 \)
Tamagawa product: \( 5 \)
Torsion order: \(10\)
Leading coefficient: \( 1.27952180 \)
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\((j)_{-}\)
\((-a^3+a^2+4a)\) \(9\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)

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
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 9.1-b consists of curves linked by isogenies of degrees dividing 10.

Base change

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

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