Base field 6.6.1767625.1
Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 1, 11, -1, -7, 0, 1]))
gp: K = nfinit(Polrev([-1, 1, 11, -1, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 11, -1, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1/2,0,3/2,-2,-1/2,1/2]),K([-2,-6,2,6,0,-1]),K([-2,11,0,-7,0,1]),K([2210,-3589,-22286,15663,6122,-3685]),K([-302659/2,242726,3037731/2,-1066837,-833779/2,502583/2])])
gp: E = ellinit([Polrev([1/2,0,3/2,-2,-1/2,1/2]),Polrev([-2,-6,2,6,0,-1]),Polrev([-2,11,0,-7,0,1]),Polrev([2210,-3589,-22286,15663,6122,-3685]),Polrev([-302659/2,242726,3037731/2,-1066837,-833779/2,502583/2])], K);
magma: E := EllipticCurve([K![1/2,0,3/2,-2,-1/2,1/2],K![-2,-6,2,6,0,-1],K![-2,11,0,-7,0,1],K![2210,-3589,-22286,15663,6122,-3685],K![-302659/2,242726,3037731/2,-1066837,-833779/2,502583/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^5+a^4+12a^3-2a^2-15a)\) | = | \((a+1)\cdot(1/2a^5-1/2a^4-2a^3+3/2a^2-3/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 36 \) | = | \(4\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^5+a^4+5a^3-2a^2-3a-2)\) | = | \((a+1)\cdot(1/2a^5-1/2a^4-2a^3+3/2a^2-3/2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -324 \) | = | \(-4\cdot9^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{24440219738032611356663}{36} a^{5} - \frac{40561631579796278917201}{36} a^{4} - \frac{51882193713025061416439}{18} a^{3} + \frac{147769887187775885902283}{36} a^{2} + \frac{5899906308447709939721}{9} a - \frac{1636259899263628386437}{4} \) | ||
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 | $\left(13 a^{5} - 15 a^{4} - 57 a^{3} + 57 a^{2} + 22 a - 1 : -31 a^{5} + 6 a^{4} + 119 a^{3} - 58 a^{2} - 49 a - 2 : 1\right)$ |
| Height | \(0.15059167830814870066176330447486379170\) |
| 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: | $\left(\frac{93}{8} a^{5} - \frac{163}{8} a^{4} - \frac{97}{2} a^{3} + \frac{597}{8} a^{2} + \frac{31}{4} a - \frac{87}{8} : -22 a^{5} + \frac{293}{8} a^{4} + \frac{759}{8} a^{3} - \frac{1069}{8} a^{2} - \frac{207}{8} a + \frac{123}{8} : 1\right)$ |
| 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: | \( 0.15059167830814870066176330447486379170 \) | ||
| Period: | \( 587.78079242422115000937139977926595672 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.19567 \) | ||
| Analytic order of Ш: | \( 16 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+1)\) | \(4\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((1/2a^5-1/2a^4-2a^3+3/2a^2-3/2)\) | \(9\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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
36.1-c
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.