Base field 6.6.1241125.1
Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
gp: K = nfinit(Polrev([1, 7, 11, -2, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-10,-30,-5,20,2,-3]),K([-21,-65,-1,41,2,-6]),K([-12,-29,0,20,1,-3]),K([-16,-36,16,18,-3,-2]),K([-20,-40,16,17,-2,-2])])
gp: E = ellinit([Polrev([-10,-30,-5,20,2,-3]),Polrev([-21,-65,-1,41,2,-6]),Polrev([-12,-29,0,20,1,-3]),Polrev([-16,-36,16,18,-3,-2]),Polrev([-20,-40,16,17,-2,-2])], K);
magma: E := EllipticCurve([K![-10,-30,-5,20,2,-3],K![-21,-65,-1,41,2,-6],K![-12,-29,0,20,1,-3],K![-16,-36,16,18,-3,-2],K![-20,-40,16,17,-2,-2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+a^4+6a^3-4a^2-8a)\) | = | \((-2a^5+a^4+13a^3-2a^2-19a-5)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-8a^5-a^4+57a^3+18a^2-99a-38)\) | = | \((-2a^5+a^4+13a^3-2a^2-19a-5)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1953125 \) | = | \(5^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{3288568389824}{25} a^{5} - \frac{1515811489972}{25} a^{4} - \frac{22321293042632}{25} a^{3} + \frac{3711500587103}{25} a^{2} + \frac{6892701907756}{5} a + \frac{7134572445643}{25} \) | ||
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: | \(\Z/5\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(2 a^{5} - 15 a^{3} - 3 a^{2} + 26 a + 11 : 9 a^{5} - a^{4} - 63 a^{3} - 10 a^{2} + 103 a + 44 : 1\right)$ |
| 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: | \( 25263.390455299292956200275628171386533 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(5\) | ||
| Leading coefficient: | \( 1.81415 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^5+a^4+13a^3-2a^2-19a-5)\) | \(5\) | \(2\) | \(I_{3}^{*}\) | Additive | \(1\) | \(2\) | \(9\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
| \(5\) | 5B.1.1[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
25.2-b
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.