Base field 5.5.181057.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 4 x^{3} + 7 x^{2} + 2 x - 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 2, 7, -4, -2, 1]))
gp: K = nfinit(Polrev([-3, 2, 7, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 2, 7, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([5,-1,-6,0,1]),K([2,0,-1,0,0]),K([-1,0,1,0,0]),K([-1007,-819,1068,299,-221]),K([-13016,-10778,14030,3879,-2885])])
gp: E = ellinit([Polrev([5,-1,-6,0,1]),Polrev([2,0,-1,0,0]),Polrev([-1,0,1,0,0]),Polrev([-1007,-819,1068,299,-221]),Polrev([-13016,-10778,14030,3879,-2885])], K);
magma: E := EllipticCurve([K![5,-1,-6,0,1],K![2,0,-1,0,0],K![-1,0,1,0,0],K![-1007,-819,1068,299,-221],K![-13016,-10778,14030,3879,-2885]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-a^3-4a^2+a+3)\) | = | \((a)^{2}\cdot(a^4-a^3-5a^2+2a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 27 \) | = | \(3^{2}\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((86a^4-85a^3-401a^2+25a+498)\) | = | \((a)^{13}\cdot(a^4-a^3-5a^2+2a+5)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -847288609443 \) | = | \(-3^{13}\cdot3^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{34495656525909971}{531441} a^{4} + \frac{28517667503038951}{177147} a^{3} + \frac{81066375780822875}{531441} a^{2} - \frac{258890063675704426}{531441} a + \frac{39365969687406722}{177147} \) | ||
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: | $\left(-3 a^{4} + \frac{13}{4} a^{3} + \frac{37}{2} a^{2} - 13 a - \frac{43}{2} : \frac{25}{4} a^{4} - \frac{53}{8} a^{3} - \frac{239}{8} a^{2} + \frac{47}{4} a + \frac{53}{2} : 1\right)$ |
| 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: | \( 7.1889095445860672287664172467836528266 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.08127294 \) | ||
| Analytic order of Ш: | \( 64 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(3\) | \(2\) | \(I_{7}^{*}\) | Additive | \(-1\) | \(2\) | \(13\) | \(7\) |
| \((a^4-a^3-5a^2+2a+5)\) | \(3\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
27.2-c
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.