Base field 4.4.2624.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 2, -3, -2, 1]))
gp: K = nfinit(Polrev([1, 2, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-2,1,0]),K([0,0,0,0]),K([1,0,0,0]),K([5,10,-5,0]),K([-2,10,-10,2])])
gp: E = ellinit([Polrev([-1,-2,1,0]),Polrev([0,0,0,0]),Polrev([1,0,0,0]),Polrev([5,10,-5,0]),Polrev([-2,10,-10,2])], K);
magma: E := EllipticCurve([K![-1,-2,1,0],K![0,0,0,0],K![1,0,0,0],K![5,10,-5,0],K![-2,10,-10,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-2a^2-4a+1)\) | = | \((a^3-2a^2-2a+1)\cdot(-a^2+a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 28 \) | = | \(4\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1424a^3+2288a^2+3824a+192)\) | = | \((a^3-2a^2-2a+1)^{9}\cdot(-a^2+a+2)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 10578455953408 \) | = | \(4^{9}\cdot7^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{85827882383798417}{1291315424} a^{3} - \frac{19791550232988209}{92236816} a^{2} + \frac{20716144036860169}{322828856} a + \frac{69874727355884387}{1291315424} \) | ||
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: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(a^{2} - 2 a + 2 : 2 a^{3} - 8 a^{2} + 4 a - 2 : 1\right)$ |
| 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: | \( 6.9822003183713645023550057808571385299 \) | ||
| Tamagawa product: | \( 81 \) = \(3^{2}\cdot3^{2}\) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 1.22674105122736 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-2a^2-2a+1)\) | \(4\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
| \((-a^2+a+2)\) | \(7\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
28.2-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.