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([-2,1,1,-4,0,1]),K([-4,5,5,-5,-1,1]),K([-2,1,1,0,0,0]),K([580,2039,-690,-1548,118,258]),K([-29121,-105796,-21228,107566,9446,-20998])])
gp: E = ellinit([Polrev([-2,1,1,-4,0,1]),Polrev([-4,5,5,-5,-1,1]),Polrev([-2,1,1,0,0,0]),Polrev([580,2039,-690,-1548,118,258]),Polrev([-29121,-105796,-21228,107566,9446,-20998])], K);
magma: E := EllipticCurve([K![-2,1,1,-4,0,1],K![-4,5,5,-5,-1,1],K![-2,1,1,0,0,0],K![580,2039,-690,-1548,118,258],K![-29121,-105796,-21228,107566,9446,-20998]]);
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: | \((-119a^5-289a^4+940a^3+967a^2-1541a+168)\) | = | \((a^4-4a^2+a+2)\cdot(-a^3+3a)^{14}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -27561634699895023 \) | = | \(-7\cdot13^{14}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{210494899395329104316882674818601470}{27561634699895023} a^{5} + \frac{220326152901046187397217035121037707}{27561634699895023} a^{4} - \frac{601531540208629141835154188949660818}{27561634699895023} a^{3} - \frac{389178106994375647928627220188772756}{27561634699895023} a^{2} + \frac{255940859747009559786272783613073517}{27561634699895023} a + \frac{102845559693375533397370969339933439}{27561634699895023} \) | ||
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 | $\left(466 a^{5} - 133 a^{4} - 2457 a^{3} + 118 a^{2} + 2582 a + 893 : -22649 a^{5} + 8332 a^{4} + 118044 a^{3} - 15657 a^{2} - 120764 a - 32128 : 1\right)$ |
| Height | \(2.7558158298354997392694944934465794881\) |
| 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: | $\left(7 a^{5} - 10 a^{4} - 32 a^{3} + 31 a^{2} + 21 a + 16 : -a^{5} + a^{4} + a^{3} - 6 a^{2} + 19 a + 22 : 1\right)$ |
| 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: | \( 2.7558158298354997392694944934465794881 \) | ||
| Period: | \( 0.13046906259494527507272951778204633242 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-4a^2+a+2)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((-a^3+3a)\) | \(13\) | \(2\) | \(I_{14}\) | Non-split multiplicative | \(1\) | \(1\) | \(14\) | \(14\) |
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.