Base field 3.3.316.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -4, -1, 1]))
gp: K = nfinit(Polrev([2, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1]),K([-3,0,1]),K([-3,0,1]),K([-426761105098,-53161745688,100434688208]),K([100184582628814686,12480020414213596,-23577610985047683])])
gp: E = ellinit([Polrev([-3,0,1]),Polrev([-3,0,1]),Polrev([-3,0,1]),Polrev([-426761105098,-53161745688,100434688208]),Polrev([100184582628814686,12480020414213596,-23577610985047683])], K);
magma: E := EllipticCurve([K![-3,0,1],K![-3,0,1],K![-3,0,1],K![-426761105098,-53161745688,100434688208],K![100184582628814686,12480020414213596,-23577610985047683]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^2-a-7)\) | = | \((-a+1)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(2^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-35a^2+8a+147)\) | = | \((-a+1)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 65536 \) | = | \(2^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) | ||
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: | \(0\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/4\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-\frac{274783}{4} a^{2} + \frac{72723}{2} a + \frac{1167589}{4} : -\frac{763473}{8} a^{2} + 50515 a + \frac{3244105}{8} : 1\right)$ | $\left(-3029 a^{2} + 1603 a + 12870 : 72168569 a^{2} - 38200020 a - 306654392 : 1\right)$ |
| 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: | \( 200.59041492749962028237750615230533858 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(8\) | ||
| Leading coefficient: | \( 0.70525577773956474808140129581548975701 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+1)\) | \(2\) | \(4\) | \(I_{8}^{*}\) | Additive | \(-1\) | \(4\) | \(16\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
16.5-b
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.