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([1,4,-3,-2,1,0]),K([-3,1,2,-1,0,0]),K([-1,4,-2,-2,1,0]),K([-101,90,197,-65,-65,22]),K([-396,403,692,-292,-194,72])])
gp: E = ellinit([Polrev([1,4,-3,-2,1,0]),Polrev([-3,1,2,-1,0,0]),Polrev([-1,4,-2,-2,1,0]),Polrev([-101,90,197,-65,-65,22]),Polrev([-396,403,692,-292,-194,72])], K);
magma: E := EllipticCurve([K![1,4,-3,-2,1,0],K![-3,1,2,-1,0,0],K![-1,4,-2,-2,1,0],K![-101,90,197,-65,-65,22],K![-396,403,692,-292,-194,72]]);
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: | \((-3a^5+9a^4+9a^3-24a^2-18a)\) | = | \((a^3-a^2-3a)^{9}\) |
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: | \( -20571839670924 a^{5} + 84686501192643 a^{4} - 32847346580697 a^{3} - 169040305433673 a^{2} + 127038715216011 a - 18423269152983 \) | ||
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: | \(0\) |
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: | $\left(-\frac{13}{3} a^{5} + \frac{37}{3} a^{4} + 15 a^{3} - \frac{122}{3} a^{2} - \frac{61}{3} a + 22 : 16 a^{5} - \frac{133}{3} a^{4} - 60 a^{3} + 152 a^{2} + \frac{244}{3} a - 84 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
|||
Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 20586.052569035867445014846452322846201 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 1.93489 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3-a^2-3a)\) | \(3\) | \(1\) | \(IV^{*}\) | 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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
27.1-g
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.