Base field 3.3.321.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 1]))
gp: K = nfinit(Polrev([1, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0]),K([-4,0,1]),K([-3,-1,1]),K([-15,46,-15]),K([42,-217,82])])
gp: E = ellinit([Polrev([1,1,0]),Polrev([-4,0,1]),Polrev([-3,-1,1]),Polrev([-15,46,-15]),Polrev([42,-217,82])], K);
magma: E := EllipticCurve([K![1,1,0],K![-4,0,1],K![-3,-1,1],K![-15,46,-15],K![42,-217,82]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2a+1)\) | = | \((-a^2+a+3)\cdot(-a^2+2)\) |
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: | \((-2a^2+a+22)\) | = | \((-a^2+a+3)^{4}\cdot(-a^2+2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 3969 \) | = | \(3^{4}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{404487996}{49} a^{2} + \frac{5401742792}{441} a - \frac{1320930403}{441} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(a^{2} - 3 a : 2 : 1\right)$ | $\left(\frac{3}{4} a^{2} - \frac{5}{2} a - \frac{1}{4} : \frac{3}{8} a + 2 : 1\right)$ |
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: | \( 51.927772453420463252271804837689503497 \) | ||
Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 1.4491628134178025612611910074517195378 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^2+a+3)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
\((-a^2+2)\) | \(7\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
21.2-b
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.