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([0,-2,0,1,0]),K([2,1,-1,0,0]),K([0,3,-3,-1,1]),K([-7,-110,-69,34,8]),K([411,-503,-806,107,137])])
gp: E = ellinit([Polrev([0,-2,0,1,0]),Polrev([2,1,-1,0,0]),Polrev([0,3,-3,-1,1]),Polrev([-7,-110,-69,34,8]),Polrev([411,-503,-806,107,137])], K);
magma: E := EllipticCurve([K![0,-2,0,1,0],K![2,1,-1,0,0],K![0,3,-3,-1,1],K![-7,-110,-69,34,8],K![411,-503,-806,107,137]]);
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: | \((5a^4-11a^3-21a^2+12a+36)\) | = | \((a^2-2)^{24}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 16777216 \) | = | \(2^{24}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2632818763157728469}{4096} a^{4} - \frac{7144835376719340663}{4096} a^{3} - \frac{919502457946488339}{4096} a^{2} + \frac{9474232411249232493}{4096} a - \frac{3072546256788093161}{4096} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{3}{2} a^{4} + \frac{3}{2} a^{3} - 10 a^{2} - \frac{25}{4} a + \frac{15}{2} : \frac{1}{8} a^{4} + \frac{5}{4} a^{3} - \frac{5}{2} a^{2} - \frac{13}{4} a + \frac{5}{2} : 1\right)$ | $\left(-a^{4} + a^{3} + 4 a^{2} - 3 a - 9 : 4 a^{3} + a^{2} - 8 a - 1 : 1\right)$ |
| 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: | \( 358.50418826009461156411031926921761510 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.25571607 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-2)\) | \(2\) | \(4\) | \(I_{16}^{*}\) | Additive | \(-1\) | \(4\) | \(24\) | \(12\) |
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 |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
16.2-a
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.