Base field 4.4.17609.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 10 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 10, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, 10, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 10, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([-2,12,-1,-2]),K([2,-11,1,2]),K([-29,47,0,-7]),K([17,-13,-3,2])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([-2,12,-1,-2]),Polrev([2,-11,1,2]),Polrev([-29,47,0,-7]),Polrev([17,-13,-3,2])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![-2,12,-1,-2],K![2,-11,1,2],K![-29,47,0,-7],K![17,-13,-3,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-6a)\) | = | \((a^3+a^2-5a+1)\cdot(-a^3+6a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 14 \) | = | \(2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((73a^3-516a+213)\) | = | \((a^3+a^2-5a+1)^{17}\cdot(-a^3+6a-2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 314703872 \) | = | \(2^{17}\cdot7^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{705345550725079}{314703872} a^{3} + \frac{1343251625907103}{157351936} a^{2} - \frac{2604379864571517}{314703872} a + \frac{252085497925447}{314703872} \) | ||
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: | \(1\) |
| Generator | $\left(-a^{3} + 6 a - 2 : 2 a^{3} + 2 a^{2} - 13 a - 5 : 1\right)$ |
| Height | \(0.81061125319007566211241264253938667950\) |
| 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);
| |
| Torsion generator: | $\left(\frac{3}{4} a^{3} - \frac{1}{2} a^{2} - \frac{19}{4} a + \frac{21}{4} : -\frac{11}{8} a^{3} - \frac{1}{4} a^{2} + \frac{63}{8} a - \frac{29}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.81061125319007566211241264253938667950 \) | ||
| Period: | \( 76.919557953144244620820028509712581672 \) | ||
| Tamagawa product: | \( 4 \) = \(1\cdot2^{2}\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.87949880020666 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-5a+1)\) | \(2\) | \(1\) | \(I_{17}\) | Non-split multiplicative | \(1\) | \(1\) | \(17\) | \(17\) |
| \((-a^3+6a-2)\) | \(7\) | \(4\) | \(I_{4}\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
14.1-d
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.