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([3,-7,-6,5,2,-1]),K([3,-4,-10,7,4,-2]),K([2,-6,-6,5,2,-1]),K([16,-42,-79,50,30,-14]),K([31,-82,-142,88,52,-24])])
gp: E = ellinit([Polrev([3,-7,-6,5,2,-1]),Polrev([3,-4,-10,7,4,-2]),Polrev([2,-6,-6,5,2,-1]),Polrev([16,-42,-79,50,30,-14]),Polrev([31,-82,-142,88,52,-24])], K);
magma: E := EllipticCurve([K![3,-7,-6,5,2,-1],K![3,-4,-10,7,4,-2],K![2,-6,-6,5,2,-1],K![16,-42,-79,50,30,-14],K![31,-82,-142,88,52,-24]]);
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: | \((-3a^5+7a^4+12a^3-22a^2-10a+2)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)^{9}\) |
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: | \( -132954280 a^{5} + 362404713 a^{4} + 365155892 a^{3} - 1096319488 a^{2} - 167101878 a + 485890469 \) | ||
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 | $\left(20 a^{5} - 45 a^{4} - 73 a^{3} + 125 a^{2} + 71 a - 27 : -115 a^{5} + 260 a^{4} + 419 a^{3} - 725 a^{2} - 412 a + 157 : 1\right)$ |
Height | \(0.070638582562865943734858713191628668396\) |
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: | \( 0.070638582562865943734858713191628668396 \) | ||
Period: | \( 3401.7406834986274071696672721333210339 \) | ||
Tamagawa product: | \( 3 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.67259 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^5-3a^4-2a^3+8a^2+a-2)\) | \(3\) | \(3\) | \(IV^{*}\) | 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-m
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.