Base field 4.4.2777.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 1, -4, -1, 1]))
gp: K = nfinit(Polrev([2, 1, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 1, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0]),K([2,5,0,-1]),K([-1,-1,1,0]),K([-15,32,12,-10]),K([56,-45,-29,16])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([2,5,0,-1]),Polrev([-1,-1,1,0]),Polrev([-15,32,12,-10]),Polrev([56,-45,-29,16])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![2,5,0,-1],K![-1,-1,1,0],K![-15,32,12,-10],K![56,-45,-29,16]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^3+3a)\) | = | \((-a)\cdot(-a^3+2a^2+2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 22 \) | = | \(2\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-9a^3+a^2+64a-12)\) | = | \((-a)^{8}\cdot(-a^3+2a^2+2a-1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 3748096 \) | = | \(2^{8}\cdot11^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{1878243643917}{3748096} a^{3} + \frac{1364709508625}{3748096} a^{2} - \frac{1139759245535}{1874048} a - \frac{14164000839}{3748096} \) | ||
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: | \(\Z/2\Z\oplus\Z/6\Z\) | |
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: | $\left(\frac{3}{4} a^{3} - \frac{5}{4} a^{2} - \frac{7}{2} a + \frac{7}{4} : -\frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{7}{4} a + \frac{3}{8} : 1\right)$ | $\left(2 a^{3} - 2 a^{2} - 8 a + 5 : -8 a^{3} + 13 a^{2} + 19 a - 18 : 1\right)$ |
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: | \( 1596.6509244728143287827094472859731704 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(12\) | ||
Leading coefficient: | \( 0.841626760920221 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(2\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
\((-a^3+2a^2+2a-1)\) | \(11\) | \(2\) | \(I_{4}\) | 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 |
---|---|
\(2\) | 2Cs |
\(3\) | 3B.1.1 |
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
22.1-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.