Base field 6.6.592661.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 5, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 5, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 5, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,2,0,-4,0,1]),K([-1,0,-3,4,1,-1]),K([-2,1,4,-4,-1,1]),K([-1839,-1685,1760,370,-1196,144]),K([-32680,-43051,54926,26550,-49271,2258])])
gp: E = ellinit([Polrev([-1,2,0,-4,0,1]),Polrev([-1,0,-3,4,1,-1]),Polrev([-2,1,4,-4,-1,1]),Polrev([-1839,-1685,1760,370,-1196,144]),Polrev([-32680,-43051,54926,26550,-49271,2258])], K);
magma: E := EllipticCurve([K![-1,2,0,-4,0,1],K![-1,0,-3,4,1,-1],K![-2,1,4,-4,-1,1],K![-1839,-1685,1760,370,-1196,144],K![-32680,-43051,54926,26550,-49271,2258]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^5-a^4-10a^3+3a^2+9a-1)\) | = | \((a^4-4a^2+a+2)\cdot(-a^3+3a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 91 \) | = | \(7\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-15a^5-2a^4+77a^3+34a^2-94a-79)\) | = | \((a^4-4a^2+a+2)^{2}\cdot(-a^3+3a)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -3074677333 \) | = | \(-7^{2}\cdot13^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{21597161649790866133606261509753255}{3074677333} a^{5} + \frac{28681417201034137650056150261573678}{3074677333} a^{4} + \frac{98577790071360663809442695639115372}{3074677333} a^{3} - \frac{118723926019387924147249784736135276}{3074677333} a^{2} - \frac{69042235603570084841387147093553420}{3074677333} a + \frac{65841412268351842671192507620649913}{3074677333} \) | ||
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 | $\left(119 a^{5} + 27 a^{4} - 747 a^{3} - 32 a^{2} + 769 a + 344 : -3821 a^{5} - 208 a^{4} + 24250 a^{3} - 2951 a^{2} - 23899 a - 5429 : 1\right)$ |
| Height | \(5.5116316596709994785389889868931589763\) |
| 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: | $\left(\frac{23}{2} a^{5} - \frac{5}{4} a^{4} - \frac{69}{2} a^{3} + 7 a^{2} + \frac{21}{4} a - \frac{77}{4} : \frac{13}{4} a^{5} - \frac{61}{4} a^{4} - \frac{199}{8} a^{3} + \frac{155}{8} a^{2} + \frac{85}{8} a - \frac{103}{8} : 1\right)$ |
| 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: | \( 5.5116316596709994785389889868931589763 \) | ||
| Period: | \( 0.065234531297472637536364758891023166213 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.36409 \) | ||
| Analytic order of Ш: | \( 2401 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-4a^2+a+2)\) | \(7\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((-a^3+3a)\) | \(13\) | \(1\) | \(I_{7}\) | Non-split multiplicative | \(1\) | \(1\) | \(7\) | \(7\) |
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 |
| \(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 7 and 14.
Its isogeny class
91.1-b
consists of curves linked by isogenies of
degrees dividing 14.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.