Base field \(\Q(\zeta_{11})^+\)
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 3, -4, -1, 1]))
gp: K = nfinit(Polrev([-1, 3, 3, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-2,-4,1,1]),K([-1,4,3,-1,-1]),K([3,-2,-4,1,1]),K([-38813,-22250,43749,8933,-10566]),K([3073152,1606180,-3613623,-616448,848205])])
gp: E = ellinit([Polrev([3,-2,-4,1,1]),Polrev([-1,4,3,-1,-1]),Polrev([3,-2,-4,1,1]),Polrev([-38813,-22250,43749,8933,-10566]),Polrev([3073152,1606180,-3613623,-616448,848205])], K);
magma: E := EllipticCurve([K![3,-2,-4,1,1],K![-1,4,3,-1,-1],K![3,-2,-4,1,1],K![-38813,-22250,43749,8933,-10566],K![3073152,1606180,-3613623,-616448,848205]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^3+3a^2+5a-4)\) | = | \((-a^4+3a^2+a-2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 529 \) | = | \(23^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((43a^4-136a^3-55a^2+299a+67)\) | = | \((-a^4+3a^2+a-2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -78310985281 \) | = | \(-23^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1191520348048782872032188672054705939}{529} a^{4} + \frac{2181471229802177623677475441208246860}{529} a^{3} + \frac{2953649593612105912153638037483875591}{529} a^{2} - \frac{6028541812813127685001791606278142264}{529} a + \frac{1434132506958014181934752744337840202}{529} \) | ||
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: | $\left(\frac{113}{4} a^{4} - \frac{63}{2} a^{3} - 101 a^{2} + \frac{275}{4} a + \frac{211}{2} : -\frac{329}{8} a^{4} + \frac{21}{8} a^{3} + 177 a^{2} - \frac{203}{8} a - \frac{529}{4} : 1\right)$ |
| 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: | \( 6.9977529328633457676973266563056879650 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.44581672 \) | ||
| Analytic order of Ш: | \( 25 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+3a^2+a-2)\) | \(23\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
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 |
| \(5\) | 5B.4.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
529.14-a
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.