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,-4,0,1,0]),K([-10,-11,13,4,-3]),K([-2,0,1,0,0]),K([-16,-251,-506,247,17]),K([1457,5378,-5396,-6557,3325])])
gp: E = ellinit([Polrev([-1,-4,0,1,0]),Polrev([-10,-11,13,4,-3]),Polrev([-2,0,1,0,0]),Polrev([-16,-251,-506,247,17]),Polrev([1457,5378,-5396,-6557,3325])], K);
magma: E := EllipticCurve([K![-1,-4,0,1,0],K![-10,-11,13,4,-3],K![-2,0,1,0,0],K![-16,-251,-506,247,17],K![1457,5378,-5396,-6557,3325]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^4-2a^3-4a^2+5a+3)\) | = | \((-a^4+a^3+4a^2-2a-2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 9 \) | = | \(3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-3a^4+2a^3+15a^2-6a-8)\) | = | \((-a^4+a^3+4a^2-2a-2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -6561 \) | = | \(-3^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{2521498274075249905348265612}{9} a^{4} - \frac{1488610650200171884252903124}{3} a^{3} - \frac{9163877093686100970416615593}{9} a^{2} + 1345476142192101801059271874 a + \frac{3269990960361707860438009855}{9} \) | ||
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: | \( 5.3659745798086978234956901075095161344 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.04707574 \) | ||
Analytic order of Ш: | \( 25 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^4+a^3+4a^2-2a-2)\) | \(3\) | \(2\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(5\) | 5B.4.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
9.1-b
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.