Base field 6.6.1387029.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
gp: K = nfinit(Polrev([1, -4, -1, 9, -2, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,4,-2,-2,1,0]),K([-3,3,10,-3,-3,1]),K([3,-2,-9,3,3,-1]),K([-1,1,4,10,-1,-3]),K([14,-56,-31,129,17,-41])])
gp: E = ellinit([Polrev([0,4,-2,-2,1,0]),Polrev([-3,3,10,-3,-3,1]),Polrev([3,-2,-9,3,3,-1]),Polrev([-1,1,4,10,-1,-3]),Polrev([14,-56,-31,129,17,-41])], K);
magma: E := EllipticCurve([K![0,4,-2,-2,1,0],K![-3,3,10,-3,-3,1],K![3,-2,-9,3,3,-1],K![-1,1,4,10,-1,-3],K![14,-56,-31,129,17,-41]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-3a^4-3a^3+8a^2+2a-2)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)^{3}\) |
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: | \((-5a^5+17a^4+9a^3-54a^2+3a+23)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)^{11}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -177147 \) | = | \(-3^{11}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 3426222102 a^{5} - 12812570435 a^{4} + 2623723832 a^{3} + 28894116691 a^{2} - 24794656162 a + 4632631517 \) | ||
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 | $\left(5 a^{4} - a^{3} - 16 a^{2} + 7 : 12 a^{5} + 2 a^{4} - 39 a^{3} - 13 a^{2} + 17 a + 6 : 1\right)$ |
| Height | \(1.0892209521803822950343361535160671132\) |
| 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: | $\left(a^{4} - 3 a^{2} - a + 1 : 2 a^{5} - 4 a^{4} - 6 a^{3} + 11 a^{2} + 4 a - 3 : 1\right)$ |
| 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: | \( 1.0892209521803822950343361535160671132 \) | ||
| Period: | \( 4883.5774974182382192752844078147739879 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 3.01106 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-3a^4-2a^3+8a^2+a-2)\) | \(3\) | \(1\) | \(II^{*}\) | Additive | \(-1\) | \(3\) | \(11\) | \(0\) |
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-a
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.