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([-1,3,1,-1]),K([-1,2,1,-1]),K([-1,-3,0,1]),K([2,-9,-17,1]),K([-9,11,41,8])])
gp: E = ellinit([Polrev([-1,3,1,-1]),Polrev([-1,2,1,-1]),Polrev([-1,-3,0,1]),Polrev([2,-9,-17,1]),Polrev([-9,11,41,8])], K);
magma: E := EllipticCurve([K![-1,3,1,-1],K![-1,2,1,-1],K![-1,-3,0,1],K![2,-9,-17,1],K![-9,11,41,8]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a^3+a^2+7a)\) | = | \((a^3-4a)\cdot(a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 21 \) | = | \(3\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-69a^3-190a^2-85a+425)\) | = | \((a^3-4a)^{12}\cdot(a^2-2)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 62523502209 \) | = | \(3^{12}\cdot7^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{84413562014875846901}{62523502209} a^{3} + \frac{148907375867724857512}{62523502209} a^{2} + \frac{24992949793645640572}{20841167403} a - \frac{47851964758998446149}{62523502209} \) | ||
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: | \(\Z/2\Z\oplus\Z/4\Z\) | |
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 generators: | $\left(-3 a^{2} + a + 1 : -2 a^{3} + 3 a^{2} + 4 a - 1 : 1\right)$ | $\left(a^{3} + 2 a^{2} - a : 2 a^{3} + 4 a^{2} + 3 a + 1 : 1\right)$ |
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: | \( 525.49285223579858306090972175503041554 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(8\) | ||
Leading coefficient: | \( 0.742422999880055 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3-4a)\) | \(3\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
\((a^2-2)\) | \(7\) | \(2\) | \(I_{6}\) | 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 |
---|---|
\(2\) | 2Cs |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
21.1-a
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.