Base field 3.3.1016.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 6 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -6, -1, 1]))
gp: K = nfinit(Polrev([2, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1]),K([5,0,-1]),K([1,1,0]),K([-1169,116,102]),K([26719,110,-3315])])
gp: E = ellinit([Polrev([-3,0,1]),Polrev([5,0,-1]),Polrev([1,1,0]),Polrev([-1169,116,102]),Polrev([26719,110,-3315])], K);
magma: E := EllipticCurve([K![-3,0,1],K![5,0,-1],K![1,1,0],K![-1169,116,102],K![26719,110,-3315]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2+a-2)\) | = | \((-a)\cdot(-a+3)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 8 \) | = | \(2\cdot2^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2304a^2+3072a-6656)\) | = | \((-a)^{30}\cdot(-a+3)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -274877906944 \) | = | \(-2^{30}\cdot2^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{2133363366805}{16384} a^{2} + \frac{13557156967317}{32768} a - \frac{3919110612215}{32768} \) | ||
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 | $\left(8 a^{2} - 11 a - 35 : 38 a^{2} - 36 a - 206 : 1\right)$ |
| Height | \(0.025771459138905171505857507715159978128\) |
| 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: | \( 0.025771459138905171505857507715159978128 \) | ||
| Period: | \( 37.381408042719105220749657013643713373 \) | ||
| Tamagawa product: | \( 30 \) = \(( 2 \cdot 3 \cdot 5 )\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.7201341300579085325520673750481054976 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(2\) | \(30\) | \(I_{30}\) | Split multiplicative | \(-1\) | \(1\) | \(30\) | \(30\) |
| \((-a+3)\) | \(2\) | \(1\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
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
8.3-b
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.