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(Pol(Vecrev([2, -4, -1, 1])));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0]),K([-2,0,1]),K([-2,0,1]),K([-210859010,-26266736,49623936]),K([1590406466852,198117361452,-374288977427])])
gp: E = ellinit([Pol(Vecrev([0,1,0])),Pol(Vecrev([-2,0,1])),Pol(Vecrev([-2,0,1])),Pol(Vecrev([-210859010,-26266736,49623936])),Pol(Vecrev([1590406466852,198117361452,-374288977427]))], K);
magma: E := EllipticCurve([K![0,1,0],K![-2,0,1],K![-2,0,1],K![-210859010,-26266736,49623936],K![1590406466852,198117361452,-374288977427]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a)\) | = | \((a)^{3}\cdot(-a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(2^{3}\cdot2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((32a^2-32)\) | = | \((a)^{10}\cdot(-a+1)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 262144 \) | = | \(2^{10}\cdot2^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{285267441}{256} a^{2} - \frac{401089939}{128} a + \frac{156913533}{128} \) | ||
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: | \(\Z/8\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(489 a^{2} - 242 a - 2048 : -333020 a^{2} + 176399 a + 1415276 : 1\right)$ |
| 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: | \( 117.905611502659 \) | ||
| Tamagawa product: | \( 16 \) = \(2\cdot2^{3}\) | ||
| Torsion order: | \(8\) | ||
| Leading coefficient: | \( 1.65817721191157 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(1\) | \(3\) | \(10\) | \(0\) |
| \((-a+1)\) | \(2\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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, 4 and 8.
Its isogeny class
16.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.