Base field 5.5.24217.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} - x^{2} + 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([4,3,-9,-1,2]),K([3,-1,-14,0,3]),K([0,1,-4,0,1]),K([9,14,-5,-4,1]),K([0,-6,-10,1,2])])
gp: E = ellinit([Polrev([4,3,-9,-1,2]),Polrev([3,-1,-14,0,3]),Polrev([0,1,-4,0,1]),Polrev([9,14,-5,-4,1]),Polrev([0,-6,-10,1,2])], K);
magma: E := EllipticCurve([K![4,3,-9,-1,2],K![3,-1,-14,0,3],K![0,1,-4,0,1],K![9,14,-5,-4,1],K![0,-6,-10,1,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3+a^2-3a)\) | = | \((a^3+a^2-3a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 61 \) | = | \(61\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((3a^4-a^3-15a^2+2a+7)\) | = | \((a^3+a^2-3a)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 61 \) | = | \(61\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{23855802}{61} a^{4} + \frac{45097325}{61} a^{3} + \frac{29308428}{61} a^{2} - \frac{26625007}{61} a - \frac{11020762}{61} \) | ||
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 | $\left(-4 a^{4} + 2 a^{3} + 18 a^{2} - 6 a - 5 : -11 a^{4} + 8 a^{3} + 49 a^{2} - 25 a - 14 : 1\right)$ |
Height | \(0.0041711455719915583450742334589629378207\) |
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: | \( 0.0041711455719915583450742334589629378207 \) | ||
Period: | \( 13410.496083925149190455205925848115937 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.79725634 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3+a^2-3a)\) | \(61\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
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 61.2-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.