Base field 5.5.157457.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 4 x^{3} + 5 x^{2} + 4 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 4, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([-1, 4, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-3,-1,1,0]),K([1,1,-4,-1,1]),K([0,-2,-1,1,0]),K([1,0,-1,1,0]),K([23,-131,33,58,-20])])
gp: E = ellinit([Polrev([1,-3,-1,1,0]),Polrev([1,1,-4,-1,1]),Polrev([0,-2,-1,1,0]),Polrev([1,0,-1,1,0]),Polrev([23,-131,33,58,-20])], K);
magma: E := EllipticCurve([K![1,-3,-1,1,0],K![1,1,-4,-1,1],K![0,-2,-1,1,0],K![1,0,-1,1,0],K![23,-131,33,58,-20]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^4+2a^3+3a^2-3a-1)\) | = | \((a-1)\cdot(a^2-a-2)\) |
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: | \((-a^4+3a^3-2a-9)\) | = | \((a-1)^{6}\cdot(a^2-a-2)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -91125 \) | = | \(-3^{6}\cdot5^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{815895463534687}{91125} a^{4} + \frac{571481037512924}{18225} a^{3} - \frac{1028729291749087}{91125} a^{2} - \frac{844716480359038}{30375} a + \frac{543144154205414}{91125} \) | ||
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: | \( 121.98084393238314947554561176305488811 \) | ||
Tamagawa product: | \( 6 \) = \(2\cdot3\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.84442881 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a-1)\) | \(3\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
\((a^2-a-2)\) | \(5\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3Ns |
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 15.1-b 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.