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([-1,3,-2,-2,1,0]),K([2,-3,2,2,-1,0]),K([1,-2,-1,1,0,0]),K([2,-12,9,8,-4,0]),K([0,-4,6,3,-2,0])])
gp: E = ellinit([Polrev([-1,3,-2,-2,1,0]),Polrev([2,-3,2,2,-1,0]),Polrev([1,-2,-1,1,0,0]),Polrev([2,-12,9,8,-4,0]),Polrev([0,-4,6,3,-2,0])], K);
magma: E := EllipticCurve([K![-1,3,-2,-2,1,0],K![2,-3,2,2,-1,0],K![1,-2,-1,1,0,0],K![2,-12,9,8,-4,0],K![0,-4,6,3,-2,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4+2a^3+3a^2-3a-2)\) | = | \((-a^4+2a^3+3a^2-3a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 37 \) | = | \(37\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^4-2a^3-3a^2+3a+2)\) | = | \((-a^4+2a^3+3a^2-3a-2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 37 \) | = | \(37\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{262024065}{37} a^{5} - \frac{632903382}{37} a^{4} - \frac{1156055733}{37} a^{3} + \frac{1944483489}{37} a^{2} + \frac{1922787234}{37} a - \frac{448205859}{37} \) | ||
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 | $\left(-1 : a^{4} - 3 a^{3} - a^{2} + 5 a - 1 : 1\right)$ |
| Height | \(0.072226456404329310128921794799595925436\) |
| 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: | $\left(-\frac{1}{3} a^{5} + \frac{4}{3} a^{4} - \frac{11}{3} a^{2} + \frac{2}{3} a + 1 : \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{8}{3} a^{3} + \frac{5}{3} a^{2} + \frac{13}{3} a - \frac{7}{3} : 1\right)$ |
| 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: | \( 0.072226456404329310128921794799595925436 \) | ||
| Period: | \( 56104.431173629457530300597071681500294 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 2.28521 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+2a^3+3a^2-3a-2)\) | \(37\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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
37.1-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.