Base field 4.4.10025.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 11 x^{2} + 10 x + 20 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([20, 10, -11, -1, 1]))
gp: K = nfinit(Polrev([20, 10, -11, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, 10, -11, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-11,-7,2,1]),K([-9,-7/2,3/2,1/2]),K([0,0,0,0]),K([-12,-21/2,5/2,3/2]),K([-1,-7/2,1/2,1/2])])
gp: E = ellinit([Polrev([-11,-7,2,1]),Polrev([-9,-7/2,3/2,1/2]),Polrev([0,0,0,0]),Polrev([-12,-21/2,5/2,3/2]),Polrev([-1,-7/2,1/2,1/2])], K);
magma: E := EllipticCurve([K![-11,-7,2,1],K![-9,-7/2,3/2,1/2],K![0,0,0,0],K![-12,-21/2,5/2,3/2],K![-1,-7/2,1/2,1/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^3-2a^2+7a+10)\) | = | \((-2a^3-3a^2+13a+14)\cdot(-a^3-a^2+8a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 20 \) | = | \(4\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((25a^3+47a^2-180a-240)\) | = | \((-2a^3-3a^2+13a+14)^{2}\cdot(-a^3-a^2+8a+5)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -6250000 \) | = | \(-4^{2}\cdot5^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{3863203}{5000} a^{3} - \frac{1529269}{5000} a^{2} + \frac{5761753}{1000} a + \frac{897131}{2500} \) | ||
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 | $\left(\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - 2 : \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - 1 : 1\right)$ |
Height | \(0.015784360789844940291021672563736334233\) |
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: | $\left(-\frac{1}{2} a^{3} - \frac{3}{4} a^{2} + \frac{7}{2} a + 3 : \frac{5}{8} a^{3} + \frac{5}{4} a^{2} - 4 a - 6 : 1\right)$ |
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: | \( 0.015784360789844940291021672563736334233 \) | ||
Period: | \( 767.75667596073972283377396705602038334 \) | ||
Tamagawa product: | \( 16 \) = \(2\cdot2^{3}\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.93654856488247 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-2a^3-3a^2+13a+14)\) | \(4\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
\((-a^3-a^2+8a+5)\) | \(5\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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.
Its isogeny class
20.1-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.