Base field 4.4.1957.1
Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -4, 0, 1]))
gp: K = nfinit(Polrev([1, -1, -4, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -4, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-1,1,0]),K([0,8,1,-2]),K([-1,4,1,-1]),K([21,-26,-58,32]),K([-53,233,305,-197])])
gp: E = ellinit([Polrev([-1,-1,1,0]),Polrev([0,8,1,-2]),Polrev([-1,4,1,-1]),Polrev([21,-26,-58,32]),Polrev([-53,233,305,-197])], K);
magma: E := EllipticCurve([K![-1,-1,1,0],K![0,8,1,-2],K![-1,4,1,-1],K![21,-26,-58,32],K![-53,233,305,-197]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2a^3-6a-1)\) | = | \((2a^3-6a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 47 \) | = | \(47\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((143a^3-160a^2-507a-48)\) | = | \((2a^3-6a-1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -10779215329 \) | = | \(-47^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{11148830379801159533743}{10779215329} a^{3} + \frac{19666703347863657066005}{10779215329} a^{2} + \frac{9902963271520642369814}{10779215329} a - \frac{6320145061844672607397}{10779215329} \) | ||
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: | $\left(\frac{1}{2} a^{3} + \frac{9}{4} a^{2} - \frac{19}{4} a - 4 : \frac{13}{4} a^{3} - \frac{7}{2} a^{2} - 7 a - \frac{5}{8} : 1\right)$ |
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: | \( 151.57094150224645908578392365247192544 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.71313086533475 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2a^3-6a-1)\) | \(47\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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 |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
47.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.