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([5,2,-15,-2,3]),K([-2,4,1,-1,0]),K([1,1,-4,-1,1]),K([-22,46,92,0,-17]),K([8,-11,-30,-2,6])])
gp: E = ellinit([Polrev([5,2,-15,-2,3]),Polrev([-2,4,1,-1,0]),Polrev([1,1,-4,-1,1]),Polrev([-22,46,92,0,-17]),Polrev([8,-11,-30,-2,6])], K);
magma: E := EllipticCurve([K![5,2,-15,-2,3],K![-2,4,1,-1,0],K![1,1,-4,-1,1],K![-22,46,92,0,-17],K![8,-11,-30,-2,6]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^4-a^3-5a^2+3a+2)\) | = | \((-a^4+6a^2+2a-4)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 49 \) | = | \(7^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((16a^4+13a^3-98a^2-35a-47)\) | = | \((-a^4+6a^2+2a-4)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -13841287201 \) | = | \(-7^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{3986622664}{117649} a^{4} + \frac{2668349896}{117649} a^{3} + \frac{20834045607}{117649} a^{2} - \frac{162996513}{16807} a - \frac{12309440397}{117649} \) | ||
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: | 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: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 273.48172823831008147497246538935382366 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.05850954 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^4+6a^2+2a-4)\) | \(7\) | \(2\) | \(I_{6}^{*}\) | Additive | \(-1\) | \(2\) | \(12\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
49.1-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.