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([-3,3,2/3,-1/3]),K([-1,-2,1/3,1/3]),K([172,502,-22/3,-190/3]),K([1095,3179,-134/3,-1208/3])])
gp: E = ellinit([Polrev([-3,2,2/3,-1/3]),Polrev([-3,3,2/3,-1/3]),Polrev([-1,-2,1/3,1/3]),Polrev([172,502,-22/3,-190/3]),Polrev([1095,3179,-134/3,-1208/3])], K);
magma: E := EllipticCurve([K![-3,2,2/3,-1/3],K![-3,3,2/3,-1/3],K![-1,-2,1/3,1/3],K![172,502,-22/3,-190/3],K![1095,3179,-134/3,-1208/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-1/3a^3+2/3a^2+3a-1)\) | = | \((-1/3a^3-1/3a^2+3a+2)\cdot(1/3a^3-2/3a^2-2a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 57 \) | = | \(3\cdot19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((88a^3+6a^2-591a+18)\) | = | \((-1/3a^3-1/3a^2+3a+2)^{17}\cdot(1/3a^3-2/3a^2-2a+5)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 2453663097 \) | = | \(3^{17}\cdot19\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{217217028070}{373977} a^{3} + \frac{1324842416596}{373977} a^{2} - \frac{2515509051217}{373977} a + \frac{161738260330}{41553} \) | ||
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 | $\left(\frac{529517}{729147} a^{3} - \frac{479131}{729147} a^{2} - \frac{1290635}{243049} a + \frac{397469}{243049} : -\frac{1022060203}{359469471} a^{3} - \frac{584877703}{359469471} a^{2} + \frac{2615786956}{119823157} a + \frac{2311774959}{119823157} : 1\right)$ |
| Height | \(3.8997761944364590610742749180106011389\) |
| 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: | $\left(\frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 5 a - 1 : -a + 1 : 1\right)$ |
| 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: | \( 3.8997761944364590610742749180106011389 \) | ||
| Period: | \( 180.83007709824332028664128830587539517 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 5.00896008081805 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/3a^3-1/3a^2+3a+2)\) | \(3\) | \(1\) | \(I_{17}\) | Non-split multiplicative | \(1\) | \(1\) | \(17\) | \(17\) |
| \((1/3a^3-2/3a^2-2a+5)\) | \(19\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
57.1-d
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.