Base field 3.3.564.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 5 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -5, -1, 1]))
gp: K = nfinit(Polrev([3, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,1,1]),K([1,-1,0]),K([1,0,0]),K([-737039,-59482,140847]),K([-242274553,-19512288,46317141])])
gp: E = ellinit([Polrev([-4,1,1]),Polrev([1,-1,0]),Polrev([1,0,0]),Polrev([-737039,-59482,140847]),Polrev([-242274553,-19512288,46317141])], K);
magma: E := EllipticCurve([K![-4,1,1],K![1,-1,0],K![1,0,0],K![-737039,-59482,140847],K![-242274553,-19512288,46317141]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2+a+5)\) | = | \((-a+2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-17a^2+5a+79)\) | = | \((-a+2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -6561 \) | = | \(-3^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{777670594336}{3} a^{2} + 110949412416 a + 1359579986768 \) | ||
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: | \(0\) |
| 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: | $\left(\frac{335}{4} a^{2} - 58 a - \frac{1947}{4} : \frac{703}{4} a^{2} - \frac{83}{4} a - \frac{3239}{4} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 34.553464418685919032447935848497562513 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.4549633219849977639021151262733998667 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+2)\) | \(3\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 6, 9 and 18.
Its isogeny class
9.2-b
consists of curves linked by isogenies of
degrees dividing 18.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.