Base field 4.4.19821.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} + 6 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 6, -8, -1, 1]))
gp: K = nfinit(Polrev([3, 6, -8, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,2,2/3,-1/3]),K([2,-3,-2/3,1/3]),K([0,0,0,0]),K([10,-8,-5/3,4/3]),K([2,-13,41/3,-10/3])])
gp: E = ellinit([Polrev([-3,2,2/3,-1/3]),Polrev([2,-3,-2/3,1/3]),Polrev([0,0,0,0]),Polrev([10,-8,-5/3,4/3]),Polrev([2,-13,41/3,-10/3])], K);
magma: E := EllipticCurve([K![-3,2,2/3,-1/3],K![2,-3,-2/3,1/3],K![0,0,0,0],K![10,-8,-5/3,4/3],K![2,-13,41/3,-10/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/3a^3+1/3a^2-3a+1)\) | = | \((-1/3a^3-1/3a^2+3a+2)\cdot(a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 51 \) | = | \(3\cdot17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-7/3a^3+2/3a^2+17a-10)\) | = | \((-1/3a^3-1/3a^2+3a+2)^{3}\cdot(a+2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 7803 \) | = | \(3^{3}\cdot17^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{27786899}{2601} a^{3} + \frac{37336988}{2601} a^{2} + \frac{209237104}{2601} a - \frac{79584637}{867} \) | ||
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: | \(2\) | |
| Generators | $\left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + \frac{5}{3} a - \frac{4}{3} : -\frac{5}{9} a^{3} + \frac{11}{9} a^{2} + \frac{19}{9} a - \frac{8}{3} : 1\right)$ | $\left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 2 : -\frac{2}{3} a^{3} + \frac{4}{3} a^{2} + 3 a - 4 : 1\right)$ |
| Heights | \(0.18507266902982584312911358219812853672\) | \(0.13281323255354028117506593519225422050\) |
| 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: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 0.019923622853539082706999925784913381461 \) | ||
| Period: | \( 1400.7872184415214370872265434835497494 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 6.34348153477783 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/3a^3-1/3a^2+3a+2)\) | \(3\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((a+2)\) | \(17\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 51.1-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.