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([-2,8,6,-5,-2,1]),K([3,2,-11,0,4,-1]),K([0,-3,-1,1,0,0]),K([0,12,10,-20,-8,5]),K([-6,33,-2,-56,-10,11])])
gp: E = ellinit([Polrev([-2,8,6,-5,-2,1]),Polrev([3,2,-11,0,4,-1]),Polrev([0,-3,-1,1,0,0]),Polrev([0,12,10,-20,-8,5]),Polrev([-6,33,-2,-56,-10,11])], K);
magma: E := EllipticCurve([K![-2,8,6,-5,-2,1],K![3,2,-11,0,4,-1],K![0,-3,-1,1,0,0],K![0,12,10,-20,-8,5],K![-6,33,-2,-56,-10,11]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2+a+1)\) | = | \((-a^2+a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 19 \) | = | \(19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-13a^5+39a^4+42a^3-122a^2-63a+35)\) | = | \((-a^2+a+1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 47045881 \) | = | \(19^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{12210045787883583}{47045881} a^{5} + \frac{3896591398207569}{47045881} a^{4} + \frac{47074450079680857}{47045881} a^{3} + \frac{4102787318035545}{47045881} a^{2} - \frac{25626747801780036}{47045881} a + \frac{4554281634804891}{47045881} \) | ||
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 | $\left(-2 a^{5} + 2 a^{4} + 9 a^{3} + 2 a^{2} - 5 a + 1 : 8 a^{5} - 3 a^{4} - 49 a^{3} - 11 a^{2} + 27 a - 5 : 1\right)$ |
| Height | \(1.3916377339026215591275892209464103828\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-a^{5} + a^{4} + 7 a^{3} - 3 a^{2} - 10 a + 2 : a^{5} - 4 a^{4} - 3 a^{3} + 14 a^{2} + 9 a - 2 : 1\right)$ |
| 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: | \( 1.3916377339026215591275892209464103828 \) | ||
| Period: | \( 297.01761900594489122122193688557328121 \) | ||
| Tamagawa product: | \( 6 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.14685 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^2+a+1)\) | \(19\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
19.2-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.