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([-1,-3,0,1,0]),K([-10,-10,13,4,-3]),K([-7,-8,9,3,-2]),K([19,12,-12,-3,2]),K([-25,-21,37,9,-8])])
gp: E = ellinit([Polrev([-1,-3,0,1,0]),Polrev([-10,-10,13,4,-3]),Polrev([-7,-8,9,3,-2]),Polrev([19,12,-12,-3,2]),Polrev([-25,-21,37,9,-8])], K);
magma: E := EllipticCurve([K![-1,-3,0,1,0],K![-10,-10,13,4,-3],K![-7,-8,9,3,-2],K![19,12,-12,-3,2],K![-25,-21,37,9,-8]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2a^4-3a^3-9a^2+9a+7)\) | = | \((-a^4+a^3+4a^2-2a-2)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 27 \) | = | \(3^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-a^4+2a^3+5a^2-5a-7)\) | = | \((-a^4+a^3+4a^2-2a-2)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 243 \) | = | \(3^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -169969542 a^{4} + 461869972 a^{3} + 56647563 a^{2} - 437225155 a - 98969253 \) | ||
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: | \(1\) |
Generator | $\left(a^{4} - 2 a^{3} - 4 a^{2} + 5 a + 4 : a^{4} - 3 a^{3} - 3 a^{2} + 10 a + 5 : 1\right)$ |
Height | \(0.0065708466567544774170425584794010717493\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.0065708466567544774170425584794010717493 \) | ||
Period: | \( 5293.2848538602579329391040923669598226 \) | ||
Tamagawa product: | \( 3 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.03609173 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^4+a^3+4a^2-2a-2)\) | \(3\) | \(3\) | \(IV\) | Additive | \(1\) | \(3\) | \(5\) | \(0\) |
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 27.1-f 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.