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
sage: E = EllipticCurve([K([-2,0,1,0,0]),K([2,2,-4,-1,1]),K([-2,0,1,0,0]),K([5,16,-40,-10,10]),K([-9,-13,100,18,-25])])
gp: E = ellinit([Polrev([-2,0,1,0,0]),Polrev([2,2,-4,-1,1]),Polrev([-2,0,1,0,0]),Polrev([5,16,-40,-10,10]),Polrev([-9,-13,100,18,-25])], K);
magma: E := EllipticCurve([K![-2,0,1,0,0],K![2,2,-4,-1,1],K![-2,0,1,0,0],K![5,16,-40,-10,10],K![-9,-13,100,18,-25]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2a^4-3a^3-8a^2+7a+5)\) | = | \((-a^4+a^3+4a^2-2a-2)\cdot(-a^2+a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 15 \) | = | \(3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((94a^4-101a^3-392a^2+276a+150)\) | = | \((-a^4+a^3+4a^2-2a-2)^{2}\cdot(-a^2+a+2)^{14}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 54931640625 \) | = | \(3^{2}\cdot5^{14}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{810387038438023516}{54931640625} a^{4} + \frac{363395431073753132}{18310546875} a^{3} - \frac{1491237718490098883}{54931640625} a^{2} - \frac{207634903981355686}{6103515625} a - \frac{338342690307372463}{54931640625} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(2 a^{4} - 2 a^{3} - 8 a^{2} + 4 a + 1 : a^{4} - 2 a^{3} - 4 a^{2} + 5 a + 2 : 1\right)$ | $\left(-\frac{5}{4} a^{4} + a^{3} + 5 a^{2} - 2 a - 1 : -\frac{1}{2} a^{4} + \frac{11}{8} a^{3} + \frac{13}{8} a^{2} - \frac{13}{4} a - \frac{1}{8} : 1\right)$ |
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: | \( 1587.3957610643180385564014192988980181 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 1.54876208 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^4+a^3+4a^2-2a-2)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((-a^2+a+2)\) | \(5\) | \(2\) | \(I_{14}\) | Non-split multiplicative | \(1\) | \(1\) | \(14\) | \(14\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
15.1-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.