Base field 5.5.36497.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 1, 5, -3, -2, 1]))
gp: K = nfinit(Polrev([-1, 1, 5, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 5, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,0,1,0,0]),K([0,5,-3,-2,1]),K([1,6,-3,-2,1]),K([28,-40,-214,48,21]),K([20,-141,155,696,-319])])
gp: E = ellinit([Polrev([-2,0,1,0,0]),Polrev([0,5,-3,-2,1]),Polrev([1,6,-3,-2,1]),Polrev([28,-40,-214,48,21]),Polrev([20,-141,155,696,-319])], K);
magma: E := EllipticCurve([K![-2,0,1,0,0],K![0,5,-3,-2,1],K![1,6,-3,-2,1],K![28,-40,-214,48,21],K![20,-141,155,696,-319]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^4+3a^3+7a^2-6a-2)\) | = | \((a^2-1)\cdot(a^3-3a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 39 \) | = | \(3\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-7a^4+10a^3+21a^2-19a-5)\) | = | \((a^2-1)^{7}\cdot(a^3-3a)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 28431 \) | = | \(3^{7}\cdot13\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{250577525153856331}{28431} a^{4} + \frac{23688817181716957}{9477} a^{3} - \frac{188375931275690315}{9477} a^{2} - \frac{51297605922912091}{28431} a + \frac{111644917373722610}{28431} \) | ||
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: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{13}{4} a^{4} + 4 a^{3} + 8 a^{2} - a + 1 : -\frac{3}{8} a^{4} + \frac{9}{8} a^{3} + \frac{9}{8} a^{2} - \frac{29}{8} a + \frac{7}{4} : 1\right)$ |
| 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: | \( 182.19707208243321532320834017688435341 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.953702256 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-1)\) | \(3\) | \(1\) | \(I_{7}\) | Non-split multiplicative | \(1\) | \(1\) | \(7\) | \(7\) |
| \((a^3-3a)\) | \(13\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 |
| \(7\) | 7B.6.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 7, 14 and 28.
Its isogeny class
39.1-a
consists of curves linked by isogenies of
degrees dividing 28.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.