Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([0,1,0,0]),K([-2,0,1,0]),K([-123,227,60,-50]),K([949,-1736,-462,382])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([0,1,0,0]),Polrev([-2,0,1,0]),Polrev([-123,227,60,-50]),Polrev([949,-1736,-462,382])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![0,1,0,0],K![-2,0,1,0],K![-123,227,60,-50],K![949,-1736,-462,382]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a-3)\) | = | \((-a^3+a^2+4a)\cdot(a^3-a^2-4a+1)\) |
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: | \((-9a^3+9a^2+81a-18)\) | = | \((-a^3+a^2+4a)^{8}\cdot(a^3-a^2-4a+1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 15752961 \) | = | \(3^{8}\cdot7^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{25780022}{21609} a^{3} - \frac{18946955}{21609} a^{2} + \frac{5547721}{2401} a - \frac{16556678}{21609} \) | ||
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 | $\left(a^{3} - 4 a + 2 : 4 a^{3} - 9 a^{2} - 24 a + 15 : 1\right)$ |
Height | \(1.0640843313305757325177113777707671830\) |
Torsion structure: | \(\Z/4\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(2 a^{3} - 2 a^{2} - 9 a + 5 : 3 a^{3} - 5 a^{2} - 15 a + 9 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(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: | \( 1.0640843313305757325177113777707671830 \) | ||
Period: | \( 87.105030326823751242452985106402193751 \) | ||
Tamagawa product: | \( 16 \) = \(2^{3}\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 3.72446898122785 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^3+a^2+4a)\) | \(3\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
\((a^3-a^2-4a+1)\) | \(7\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
21.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.