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
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);
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Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((6a^4-2a^3-37a^2-4a+30)\) | = | \((-a^3+a^2+4a)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 59049 \) | = | \(9^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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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);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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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);
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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)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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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.