Base field \(\Q(\zeta_{21})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -8, 8, 6, -6, -1, 1]))
gp: K = nfinit(Polrev([1, -8, 8, 6, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 8, 6, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0,0,0,0,0]),K([2,-5,-1,5,0,-1]),K([-2,5,1,-5,0,1]),K([522,-3911,-391,2607,0,-391]),K([8968,-61479,-5979,40423,0,-5979])])
gp: E = ellinit([Polrev([0,0,0,0,0,0]),Polrev([2,-5,-1,5,0,-1]),Polrev([-2,5,1,-5,0,1]),Polrev([522,-3911,-391,2607,0,-391]),Polrev([8968,-61479,-5979,40423,0,-5979])], K);
magma: E := EllipticCurve([K![0,0,0,0,0,0],K![2,-5,-1,5,0,-1],K![-2,5,1,-5,0,1],K![522,-3911,-391,2607,0,-391],K![8968,-61479,-5979,40423,0,-5979]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-2a-1)\) | = | \((a^3+a^2-2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 27 \) | = | \(27\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1594323)\) | = | \((a^3+a^2-2a-1)^{26}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 16423203268260658146231467800709255289 \) | = | \(27^{26}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1713910976512}{1594323} \) | ||
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 | $\left(\frac{159}{7} a^{5} - \frac{1030}{7} a^{3} + \frac{159}{7} a^{2} + \frac{1500}{7} a - \frac{83}{7} : -\frac{1826}{7} a^{5} + \frac{1458}{7} a^{4} + \frac{10588}{7} a^{3} - \frac{5471}{7} a^{2} - \frac{14962}{7} a + \frac{2923}{7} : 1\right)$ |
| Height | \(0.99151044712942316397777192312828373436\) |
| 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: | \( 0.99151044712942316397777192312828373436 \) | ||
| Period: | \( 0.046982890125726965539285719434734178110 \) | ||
| Tamagawa product: | \( 26 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.82314 \) | ||
| Analytic order of Ш: | \( 169 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-2a-1)\) | \(27\) | \(26\) | \(I_{26}\) | Split multiplicative | \(-1\) | \(1\) | \(26\) | \(26\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(13\) | 13B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
13.
Its isogeny class
27.1-b
consists of curves linked by isogenies of
degree 13.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 5 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 147.c1 |
| \(\Q\) | 441.a1 |
| \(\Q(\sqrt{21}) \) | 2.2.21.1-147.1-e1 |
| \(\Q(\zeta_{7})^+\) | 3.3.49.1-27.1-a1 |
| \(\Q(\zeta_{7})^+\) | a curve with conductor norm 35721 (not in the database) |