Base field 4.4.14272.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 5 x^{2} + 2 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 2, -5, -2, 1]))
gp: K = nfinit(Polrev([3, 2, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 2, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([5,-6,-5,2]),K([0,0,0,0]),K([5,-6,-5,2]),K([-215,144,130,-48]),K([-1971,1414,1261,-466])])
gp: E = ellinit([Polrev([5,-6,-5,2]),Polrev([0,0,0,0]),Polrev([5,-6,-5,2]),Polrev([-215,144,130,-48]),Polrev([-1971,1414,1261,-466])], K);
magma: E := EllipticCurve([K![5,-6,-5,2],K![0,0,0,0],K![5,-6,-5,2],K![-215,144,130,-48],K![-1971,1414,1261,-466]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-3a^2-2a+2)\) | = | \((a^3-3a^2-2a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 17 \) | = | \(17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^3-3a^2-2a+2)\) | = | \((a^3-3a^2-2a+2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 17 \) | = | \(17\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{64062278444570}{17} a^{3} + \frac{159030677714743}{17} a^{2} + \frac{222959298472246}{17} a - \frac{198097673386670}{17} \) | ||
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 | $\left(\frac{192593}{27889} a^{3} - \frac{1489767}{223112} a^{2} - \frac{557813}{55778} a + \frac{3935795}{223112} : -\frac{2824317709}{37259704} a^{3} + \frac{2531447425}{149038816} a^{2} + \frac{21748131897}{149038816} a - \frac{7749410225}{149038816} : 1\right)$ |
| Height | \(6.9814890990538660613675457511891729565\) |
| 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: | $\left(4 a^{3} - 10 a^{2} - 12 a + 13 : -10 a^{3} + 27 a^{2} + 30 a - 44 : 1\right)$ |
| 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: | \( 6.9814890990538660613675457511891729565 \) | ||
| Period: | \( 4.2073408616396067836267906567506340506 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.21286975699569 \) | ||
| Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-3a^2-2a+2)\) | \(17\) | \(1\) | \(I_{1}\) | 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 |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
17.1-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.