Base field 6.6.1397493.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 3 x^{4} + 10 x^{3} + 3 x^{2} - 6 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, 3, 10, -3, -3, 1]))
gp: K = nfinit(Polrev([1, -6, 3, 10, -3, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 3, 10, -3, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,4,9,-3,-3,1]),K([3,4,-4,-2,1,0]),K([0,2,7,-2,-3,1]),K([9,-6,-15,7,5,-2]),K([4,-2,-12,1,4,-1])])
gp: E = ellinit([Polrev([-3,4,9,-3,-3,1]),Polrev([3,4,-4,-2,1,0]),Polrev([0,2,7,-2,-3,1]),Polrev([9,-6,-15,7,5,-2]),Polrev([4,-2,-12,1,4,-1])], K);
magma: E := EllipticCurve([K![-3,4,9,-3,-3,1],K![3,4,-4,-2,1,0],K![0,2,7,-2,-3,1],K![9,-6,-15,7,5,-2],K![4,-2,-12,1,4,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+3a^4+3a^3-8a^2-6a)\) | = | \((a^3-a^2-3a)^{3}\) |
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: | \((-a^5+3a^4+2a^3-8a^2+1)\) | = | \((a^3-a^2-3a)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 27 \) | = | \(3^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -2106 a^{5} + 14985 a^{4} - 32940 a^{3} + 10125 a^{2} + 43254 a - 35397 \) | ||
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 | $\left(a^{5} - 3 a^{4} - 3 a^{3} + 9 a^{2} + 5 a - 3 : a^{5} - 2 a^{4} - 6 a^{3} + 8 a^{2} + 10 a - 4 : 1\right)$ |
| Height | \(0.065830368690868168517650299913733462601\) |
| Torsion structure: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{1}{3} a^{4} + \frac{1}{3} a^{3} + 2 a^{2} - \frac{2}{3} a - \frac{4}{3} : \frac{1}{3} a^{4} - a^{3} + \frac{5}{3} a - 2 : 1\right)$ |
| 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: | \( 0.065830368690868168517650299913733462601 \) | ||
| Period: | \( 96464.232124682849976451093936633584591 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 3.58118 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-3a)\) | \(3\) | \(1\) | \(II\) | Additive | \(-1\) | \(3\) | \(3\) | \(0\) |
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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
27.1-b
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.