Base field 5.5.89417.1
Generator \(a\), with minimal polynomial \( x^{5} - 6 x^{3} - x^{2} + 8 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 8, -1, -6, 0, 1]))
gp: K = nfinit(Polrev([3, 8, -1, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 8, -1, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([6,4,-5,-1,1]),K([4,2,-4,-1,1]),K([5,1,-5,0,1]),K([2327,4612,-4424,-1249,978]),K([-69785,-130011,129230,35329,-28873])])
gp: E = ellinit([Polrev([6,4,-5,-1,1]),Polrev([4,2,-4,-1,1]),Polrev([5,1,-5,0,1]),Polrev([2327,4612,-4424,-1249,978]),Polrev([-69785,-130011,129230,35329,-28873])], K);
magma: E := EllipticCurve([K![6,4,-5,-1,1],K![4,2,-4,-1,1],K![5,1,-5,0,1],K![2327,4612,-4424,-1249,978],K![-69785,-130011,129230,35329,-28873]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^2+3)\) | = | \((-a^2+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 3 \) | = | \(3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((a^2+12a+36)\) | = | \((-a^2+3)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 43046721 \) | = | \(3^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{7189868035226731598462231824}{43046721} a^{4} + \frac{4952838052023862893752165536}{14348907} a^{3} - \frac{4144247278847983173902029664}{14348907} a^{2} - \frac{32883257167577010709120092559}{43046721} a - \frac{10437288928816976802328054390}{43046721} \) | ||
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: | \(\Z/2\Z\) |
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: | $\left(-\frac{29}{4} a^{4} + 7 a^{3} + \frac{127}{4} a^{2} - \frac{121}{4} a - \frac{47}{4} : -\frac{45}{4} a^{4} + \frac{117}{8} a^{3} + \frac{429}{8} a^{2} - 51 a - \frac{311}{8} : 1\right)$ |
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: | \( 111.64915042654719024601979513014851747 \) | ||
Tamagawa product: | \( 16 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.49350048 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^2+3)\) | \(3\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
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
3.1-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.