Base field 4.4.18097.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 6 x + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 6, -7, -1, 1]))
gp: K = nfinit(Polrev([4, 6, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 6, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-5/2,1/2,1/2]),K([0,5,0,-1]),K([1,1,0,0]),K([543,-360,-90,56]),K([4081,-5773/2,-1333/2,911/2])])
gp: E = ellinit([Polrev([-1,-5/2,1/2,1/2]),Polrev([0,5,0,-1]),Polrev([1,1,0,0]),Polrev([543,-360,-90,56]),Polrev([4081,-5773/2,-1333/2,911/2])], K);
magma: E := EllipticCurve([K![-1,-5/2,1/2,1/2],K![0,5,0,-1],K![1,1,0,0],K![543,-360,-90,56],K![4081,-5773/2,-1333/2,911/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-1/2a^3+1/2a^2+5/2a)\) | = | \((-1/2a^3+1/2a^2+5/2a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 17 \) | = | \(17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((a^3+3a^2-8a-13)\) | = | \((-1/2a^3+1/2a^2+5/2a)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 4913 \) | = | \(17^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{4417791485045697}{9826} a^{3} + \frac{15496172499429131}{9826} a^{2} - \frac{7797203413682701}{9826} a - \frac{3540429105928813}{4913} \) | ||
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 | $\left(-\frac{298}{441} a^{3} + \frac{110}{147} a^{2} + \frac{2092}{441} a - \frac{1619}{441} : -\frac{19669}{9261} a^{3} + \frac{13388}{3087} a^{2} + \frac{109636}{9261} a - \frac{271715}{9261} : 1\right)$ |
Height | \(2.6236193484478016946397502438536009716\) |
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: | $\left(-\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + \frac{5}{2} a - 6 : a^{3} + a^{2} - 5 a - 2 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 2.6236193484478016946397502438536009716 \) | ||
Period: | \( 110.58566307374659565160603479894165231 \) | ||
Tamagawa product: | \( 3 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 6.47019862275563 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-1/2a^3+1/2a^2+5/2a)\) | \(17\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
17.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.