Base field 6.6.300125.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 7, 2, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 7, 2, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([4,13,-9,-21,-1,3]),K([-1,-11,4,14,1,-2]),K([-6,-17,14,23,0,-3]),K([-25,-32,44,55,-2,-7]),K([-25,98,186,-254,-195,88])])
gp: E = ellinit([Polrev([4,13,-9,-21,-1,3]),Polrev([-1,-11,4,14,1,-2]),Polrev([-6,-17,14,23,0,-3]),Polrev([-25,-32,44,55,-2,-7]),Polrev([-25,98,186,-254,-195,88])], K);
magma: E := EllipticCurve([K![4,13,-9,-21,-1,3],K![-1,-11,4,14,1,-2],K![-6,-17,14,23,0,-3],K![-25,-32,44,55,-2,-7],K![-25,98,186,-254,-195,88]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+a^4+7a^3-2a^2-7a)\) | = | \((-a^5+a^4+7a^3-2a^2-7a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 71 \) | = | \(71\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-416a^5-570a^4+3584a^3+5465a^2-1377a-1806)\) | = | \((-a^5+a^4+7a^3-2a^2-7a)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -3255243551009881201 \) | = | \(-71^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{7245137537592799459668364}{3255243551009881201} a^{5} - \frac{17175864433733831866928686}{3255243551009881201} a^{4} - \frac{29002126063391338539879632}{3255243551009881201} a^{3} + \frac{59053137968044220795988828}{3255243551009881201} a^{2} - \frac{333346305063766203121436}{45848500718449031} a + \frac{536537380242287614298139}{3255243551009881201} \) | ||
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: | \(0\) |
| 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: | $\left(\frac{7}{2} a^{5} - a^{4} - \frac{105}{4} a^{3} - \frac{33}{4} a^{2} + \frac{37}{2} a + \frac{9}{4} : -a^{5} + \frac{7}{4} a^{4} + \frac{33}{8} a^{3} - 2 a^{2} - \frac{5}{4} a + \frac{1}{2} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 1348.5051518367487080175587843280379234 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.23075 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^5+a^4+7a^3-2a^2-7a)\) | \(71\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
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
71.3-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.