Base field 5.5.173513.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 5 x^{3} + 3 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 3, -5, -2, 1]))
gp: K = nfinit(Polrev([-1, 3, 3, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([4,-1,-7,-1,1]),K([4,-1,-11,-3,2]),K([3,4,-5,-2,1]),K([-4,-29,7,39,-12]),K([-69,78,80,-120,28])])
gp: E = ellinit([Polrev([4,-1,-7,-1,1]),Polrev([4,-1,-11,-3,2]),Polrev([3,4,-5,-2,1]),Polrev([-4,-29,7,39,-12]),Polrev([-69,78,80,-120,28])], K);
magma: E := EllipticCurve([K![4,-1,-7,-1,1],K![4,-1,-11,-3,2],K![3,4,-5,-2,1],K![-4,-29,7,39,-12],K![-69,78,80,-120,28]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-2a^3-5a^2+a+2)\) | = | \((a^4-2a^3-5a^2+a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 17 \) | = | \(17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((14a^4-38a^3-35a^2+47a-30)\) | = | \((a^4-2a^3-5a^2+a+2)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 24137569 \) | = | \(17^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{3015149053665871}{24137569} a^{4} - \frac{764713166787431}{24137569} a^{3} + \frac{28972470067066415}{24137569} a^{2} + \frac{15096145370756503}{24137569} a - \frac{5961152725283797}{24137569} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(0\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(a^{4} - 2 a^{3} - 4 a^{2} + 3 : -a^{4} + a^{3} + 8 a^{2} - 2 a - 6 : 1\right)$ | $\left(-\frac{1}{2} a^{4} + \frac{11}{2} a^{2} - \frac{1}{4} a - \frac{17}{4} : \frac{7}{4} a^{4} - \frac{27}{8} a^{3} - \frac{63}{8} a^{2} + \frac{3}{4} a + \frac{37}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 650.76109411303198030872709142609685233 \) | ||
| Tamagawa product: | \( 6 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 2.34340218 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-2a^3-5a^2+a+2)\) | \(17\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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 |
| \(3\) | 3Ns |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
17.1-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.