Base field 6.6.703493.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,10,6,-11,-1,2]),K([2,-15,-10,17,2,-3]),K([-3,10,10,-11,-2,2]),K([50,-431,-240,413,52,-70]),K([192,-1603,-780,1478,166,-248])])
gp: E = ellinit([Polrev([-2,10,6,-11,-1,2]),Polrev([2,-15,-10,17,2,-3]),Polrev([-3,10,10,-11,-2,2]),Polrev([50,-431,-240,413,52,-70]),Polrev([192,-1603,-780,1478,166,-248])], K);
magma: E := EllipticCurve([K![-2,10,6,-11,-1,2],K![2,-15,-10,17,2,-3],K![-3,10,10,-11,-2,2],K![50,-431,-240,413,52,-70],K![192,-1603,-780,1478,166,-248]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-4a+3)\) | = | \((a^3-a^2-4a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 41 \) | = | \(41\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((36a^5-23a^4-227a^3+120a^2+273a-89)\) | = | \((a^3-a^2-4a+3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -4750104241 \) | = | \(-41^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{3868146036397569720}{4750104241} a^{5} + \frac{7289390284716297722}{4750104241} a^{4} + \frac{20185766230579865977}{4750104241} a^{3} - \frac{980914258788784182}{115856201} a^{2} - \frac{12396489983066410677}{4750104241} a + \frac{33384677007660926967}{4750104241} \) | ||
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 | $\left(15 a^{5} - 11 a^{4} - 88 a^{3} + 52 a^{2} + 91 a - 12 : 52 a^{5} - 35 a^{4} - 307 a^{3} + 165 a^{2} + 326 a - 39 : 1\right)$ |
| Height | \(0.49259532543800074765744061664164434652\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-9 a^{5} + 4 a^{4} + 53 a^{3} - 19 a^{2} - 58 a + 6 : -13 a^{5} + 5 a^{4} + 78 a^{3} - 25 a^{2} - 80 a + 10 : 1\right)$ |
| 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: | \( 0.49259532543800074765744061664164434652 \) | ||
| Period: | \( 4356.8269132172565763499349366187791454 \) | ||
| Tamagawa product: | \( 6 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 2.55877 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-4a+3)\) | \(41\) | \(6\) | \(I_{6}\) | 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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
41.3-e
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.