Base field 4.4.16225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 13 x^{2} + 6 x + 36 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([36, 6, -13, -1, 1]))
gp: K = nfinit(Polrev([36, 6, -13, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, 6, -13, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([8,19/6,-5/6,-1/6]),K([0,-7/6,-1/6,1/6]),K([-65,95/2,21/2,-11/2]),K([299,-289/3,-118/3,37/3])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([8,19/6,-5/6,-1/6]),Polrev([0,-7/6,-1/6,1/6]),Polrev([-65,95/2,21/2,-11/2]),Polrev([299,-289/3,-118/3,37/3])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![8,19/6,-5/6,-1/6],K![0,-7/6,-1/6,1/6],K![-65,95/2,21/2,-11/2],K![299,-289/3,-118/3,37/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((1/3a^3-1/3a^2-10/3a+2)\cdot(1/3a^3+2/3a^2-4/3a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(4\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4/3a^3+8/3a^2-40/3a-32)\) | = | \((1/3a^3-1/3a^2-10/3a+2)^{4}\cdot(1/3a^3+2/3a^2-4/3a-3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 4096 \) | = | \(4^{4}\cdot4^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{31557625}{96} a^{3} + \frac{87395431}{96} a^{2} + \frac{255480511}{96} a - \frac{53452375}{8} \) | ||
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 | $\left(\frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{13}{6} a + 2 : -\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{10}{3} a - 2 : 1\right)$ |
| Height | \(0.024957901544779698778428016661819533668\) |
| 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: | \( 0.024957901544779698778428016661819533668 \) | ||
| Period: | \( 530.44367836262550327383635852588566088 \) | ||
| Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.32586771044100 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/3a^3-1/3a^2-10/3a+2)\) | \(4\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((1/3a^3+2/3a^2-4/3a-3)\) | \(4\) | \(2\) | \(I_{2}\) | 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 16.1-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.