Base field 6.6.1081856.1
Generator \(a\), with minimal polynomial \( x^{6} - 6 x^{4} - 2 x^{3} + 7 x^{2} + 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 2, 7, -2, -6, 0, 1]))
gp: K = nfinit(Polrev([-1, 2, 7, -2, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 7, -2, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,2,2,-4,-1,1]),K([-1,4,-2,-2,1,0]),K([2,2,-4,-1,1,0]),K([4,-15,-18,24,8,-6]),K([6,-22,-37,44,15,-11])])
gp: E = ellinit([Polrev([-1,2,2,-4,-1,1]),Polrev([-1,4,-2,-2,1,0]),Polrev([2,2,-4,-1,1,0]),Polrev([4,-15,-18,24,8,-6]),Polrev([6,-22,-37,44,15,-11])], K);
magma: E := EllipticCurve([K![-1,2,2,-4,-1,1],K![-1,4,-2,-2,1,0],K![2,2,-4,-1,1,0],K![4,-15,-18,24,8,-6],K![6,-22,-37,44,15,-11]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+5a^3+3a^2-4a-1)\) | = | \((-a^5+5a^3+3a^2-4a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-7a^5+7a^4+33a^3-19a^2-26a+16)\) | = | \((-a^5+5a^3+3a^2-4a-1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 390625 \) | = | \(25^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{209401250304}{625} a^{5} + \frac{57428760576}{625} a^{4} - \frac{1240676953472}{625} a^{3} - \frac{759032698048}{625} a^{2} + \frac{1257708985088}{625} a + \frac{763688663168}{625} \) | ||
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 | $\left(19 a^{5} - 28 a^{4} - 76 a^{3} + 76 a^{2} + 34 a - 14 : 166 a^{5} - 232 a^{4} - 679 a^{3} + 621 a^{2} + 325 a - 122 : 1\right)$ |
| Height | \(0.086102996611054630800552116210381866158\) |
| 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: | $\left(a^{5} - 2 a^{4} - 3 a^{3} + \frac{11}{2} a^{2} - 1 : -a^{5} + \frac{1}{2} a^{4} + \frac{21}{4} a^{3} - a^{2} - \frac{15}{4} a - \frac{1}{4} : 1\right)$ |
| 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: | \( 0.086102996611054630800552116210381866158 \) | ||
| Period: | \( 5453.7797117806048225805228823441435660 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.70883 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^5+5a^3+3a^2-4a-1)\) | \(25\) | \(4\) | \(I_{4}\) | 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
25.1-a
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.