Base field 4.4.15317.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([2, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-4,-1,1]),K([0,-5,-1,1]),K([-3,0,1,0]),K([-63,-69,1,12]),K([-24,-339,-68,73])])
gp: E = ellinit([Polrev([1,-4,-1,1]),Polrev([0,-5,-1,1]),Polrev([-3,0,1,0]),Polrev([-63,-69,1,12]),Polrev([-24,-339,-68,73])], K);
magma: E := EllipticCurve([K![1,-4,-1,1],K![0,-5,-1,1],K![-3,0,1,0],K![-63,-69,1,12],K![-24,-339,-68,73]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-5a)\) | = | \((-a)\cdot(-a^2+a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 8 \) | = | \(2\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((1635a^3-605a^2-7083a-1154)\) | = | \((-a)^{40}\cdot(-a^2+a+1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -70368744177664 \) | = | \(-2^{40}\cdot4^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{133425326437522565}{1099511627776} a^{3} - \frac{82385904144071959}{274877906944} a^{2} - \frac{101331943756558911}{274877906944} a + \frac{887810214505548273}{1099511627776} \) | ||
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: | $\left(\frac{3}{2} a^{3} - 2 a^{2} - \frac{23}{4} a + \frac{3}{2} : \frac{9}{8} a^{3} - \frac{1}{2} a^{2} - \frac{13}{2} a - 4 : 1\right)$ |
| 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: | \( 225.23375383717647691659884721483792274 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.909948101392787 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(2\) | \(2\) | \(I_{40}\) | Non-split multiplicative | \(1\) | \(1\) | \(40\) | \(40\) |
| \((-a^2+a+1)\) | \(4\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
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
8.1-a
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.