Base field 5.5.65657.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([1, 5, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-1,1,0,0]),K([7,4,-9,-2,2]),K([1,1,0,0,0]),K([16,36,-24,-15,6]),K([21,70,-60,-27,18])])
gp: E = ellinit([Polrev([-1,-1,1,0,0]),Polrev([7,4,-9,-2,2]),Polrev([1,1,0,0,0]),Polrev([16,36,-24,-15,6]),Polrev([21,70,-60,-27,18])], K);
magma: E := EllipticCurve([K![-1,-1,1,0,0],K![7,4,-9,-2,2],K![1,1,0,0,0],K![16,36,-24,-15,6],K![21,70,-60,-27,18]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^4-3a^3-9a^2+9a+7)\) | = | \((-a^4+a^3+4a^2-2a-2)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 27 \) | = | \(3^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^2-5a-9)\) | = | \((-a^4+a^3+4a^2-2a-2)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 19683 \) | = | \(3^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -24302019083166184075 a^{4} + 29987786156907253441 a^{3} + 114494070674765766206 a^{2} - 75391385133403896914 a - 103871320062709432359 \) | ||
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 | $\left(-\frac{44}{9} a^{4} + \frac{23}{3} a^{3} + \frac{163}{9} a^{2} - 21 a - \frac{58}{9} : \frac{499}{27} a^{4} - \frac{280}{9} a^{3} - \frac{1760}{27} a^{2} + \frac{251}{3} a + \frac{596}{27} : 1\right)$ |
| Height | \(0.21057210269045551345077967528216568677\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.21057210269045551345077967528216568677 \) | ||
| Period: | \( 635.62176719593238319185778691048885378 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.61173484 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+a^3+4a^2-2a-2)\) | \(3\) | \(1\) | \(IV^{*}\) | Additive | \(1\) | \(3\) | \(9\) | \(0\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
27.1-c
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.