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 \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1)
gp (2.8): K = nfinit(a^5 - a^4 - 4*a^3 + 3*a^2 + 3*a - 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^4 + a^3 - 3*a^2 - 2*a, -a^4 - a^3 + 5*a^2 + 3*a - 5, a^4 + a^3 - 4*a^2 - 2*a + 3, -2681*a^4 - 1331*a^3 + 7125*a^2 + 587*a - 4049, -146736*a^4 - 108973*a^3 + 354020*a^2 + 126719*a - 132130]),K);
sage: E = EllipticCurve(K, [a^4 + a^3 - 3*a^2 - 2*a, -a^4 - a^3 + 5*a^2 + 3*a - 5, a^4 + a^3 - 4*a^2 - 2*a + 3, -2681*a^4 - 1331*a^3 + 7125*a^2 + 587*a - 4049, -146736*a^4 - 108973*a^3 + 354020*a^2 + 126719*a - 132130])
gp (2.8): E = ellinit([a^4 + a^3 - 3*a^2 - 2*a, -a^4 - a^3 + 5*a^2 + 3*a - 5, a^4 + a^3 - 4*a^2 - 2*a + 3, -2681*a^4 - 1331*a^3 + 7125*a^2 + 587*a - 4049, -146736*a^4 - 108973*a^3 + 354020*a^2 + 126719*a - 132130],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((23,-a^{4} + a^{3} + 4 a^{2} - 3 a - 1)\) | = | \( \left(-a^{4} + 2 a^{2} + 1\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 23 \) | = | \( 23 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((529,a + 517,a^{2} + 385,a^{3} - 3 a + 424,a^{4} - 4 a^{2} + 471)\) | = | \( \left(-a^{4} + 2 a^{2} + 1\right)^{2} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 529 \) | = | \( 23^{2} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{1665599353208772437646840723861648544}{529} a^{4} - \frac{515871876593405186030634717346598316}{529} a^{3} + \frac{5986748941379712064585808940738486553}{529} a^{2} + \frac{2844175551769835948036386085428144797}{529} a - \frac{1271720281023855543211042033983127776}{529} \) | ||
| magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
| ||||
| \( \text{End} (E) \) | = | \(\Z\) | (no Complex Multiplication ) | |
| magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
| ||||
| \( \text{ST} (E) \) | = | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
Rank not available.magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()
Regulator: not available
magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())
Torsion subgroup
| Structure: | \(\Z/2\Z\) |
|---|---|
| magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[2]
| |
| magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp (2.8): elltors(E)[1]
| |
| Generator: | $\left(-\frac{27}{2} a^{4} + \frac{9}{2} a^{3} + \frac{79}{2} a^{2} - \frac{115}{4} a - \frac{109}{4} : \frac{239}{8} a^{4} + 29 a^{3} - \frac{281}{4} a^{2} - \frac{449}{8} a + \frac{57}{8} : 1\right)$ |
| magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[3]
| |
Local data at primes of bad reduction
magma: LocalInformation(E);
sage: E.local_data()
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a^{4} + 2 a^{2} + 1\right) \) | \(23\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
23.4-a
consists of curves linked by isogenies of
degrees dividing 10.