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([-5,-6,5,2,-1]),K([2,-2,-5,0,1]),K([0,-3,0,1,0]),K([13,13,-11,-4,2]),K([6,5,-9,-3,2])])
gp: E = ellinit([Polrev([-5,-6,5,2,-1]),Polrev([2,-2,-5,0,1]),Polrev([0,-3,0,1,0]),Polrev([13,13,-11,-4,2]),Polrev([6,5,-9,-3,2])], K);
magma: E := EllipticCurve([K![-5,-6,5,2,-1],K![2,-2,-5,0,1],K![0,-3,0,1,0],K![13,13,-11,-4,2],K![6,5,-9,-3,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3-4a)\) | = | \((-a^4+a^3+4a^2-2a-2)^{2}\cdot(-a^2+a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 45 \) | = | \(3^{2}\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((2a^3-9a^2+a+18)\) | = | \((-a^4+a^3+4a^2-2a-2)^{9}\cdot(-a^2+a+2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -98415 \) | = | \(-3^{9}\cdot5\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{34992}{5} a^{4} - \frac{81598}{5} a^{3} - \frac{54171}{5} a^{2} + \frac{89407}{5} a + \frac{46644}{5} \) | ||
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: | \(\Z/2\Z\) |
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: | $\left(-a^{4} + a^{3} + 4 a^{2} - a - 2 : -a^{4} + 5 a^{2} + a - 3 : 1\right)$ |
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: | \( 887.01171304778400522586123700944601367 \) | ||
Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.73084764 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^4+a^3+4a^2-2a-2)\) | \(3\) | \(2\) | \(III^{*}\) | Additive | \(1\) | \(2\) | \(9\) | \(0\) |
\((-a^2+a+2)\) | \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
45.1-c
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.