Base field 5.5.81509.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 3 x^{2} + 5 x - 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 5, 3, -5, -1, 1]))
gp: K = nfinit(Polrev([-2, 5, 3, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 5, 3, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,0,1,0,0]),K([-2,6,-2,-2,1]),K([2,-1,-4,0,1]),K([-2,-7,9,2,-3]),K([4,-4,-2,1,0])])
gp: E = ellinit([Polrev([-2,0,1,0,0]),Polrev([-2,6,-2,-2,1]),Polrev([2,-1,-4,0,1]),Polrev([-2,-7,9,2,-3]),Polrev([4,-4,-2,1,0])], K);
magma: E := EllipticCurve([K![-2,0,1,0,0],K![-2,6,-2,-2,1],K![2,-1,-4,0,1],K![-2,-7,9,2,-3],K![4,-4,-2,1,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-a^3-4a^2+a+2)\) | = | \((a^2-2)^{4}\) |
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: | \((a^3-a^2-3a-2)\) | = | \((a^2-2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 256 \) | = | \(2^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -2403655604077 a^{4} + 5146535696608 a^{3} + 6145419251688 a^{2} - 14223680620576 a + 4212769815616 \) | ||
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 | $\left(a^{3} - a^{2} - 3 a + 3 : -a^{4} + 3 a^{2} - 1 : 1\right)$ |
| Height | \(0.045520629997451091339487948291129765063\) |
| 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: | $\left(-\frac{1}{4} a^{4} + a^{3} - 3 a + 2 : -\frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{3}{2} a^{2} - \frac{3}{8} a + \frac{1}{4} : 1\right)$ |
| 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: | \( 0.045520629997451091339487948291129765063 \) | ||
| Period: | \( 6017.8618773791564117083814842195959153 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.39876501 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-2)\) | \(2\) | \(2\) | \(I_0^{*}\) | Additive | \(1\) | \(4\) | \(8\) | \(0\) |
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.
Its isogeny class
16.2-c
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.