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([0, -a^4 + 4*a^2 + a - 2, a^4 - 4*a^2 + 3, -4*a^4 + 9*a^3 - a^2 - 3*a, -13*a^4 + 34*a^3 - 6*a^2 - 25*a + 5]),K);
sage: E = EllipticCurve(K, [0, -a^4 + 4*a^2 + a - 2, a^4 - 4*a^2 + 3, -4*a^4 + 9*a^3 - a^2 - 3*a, -13*a^4 + 34*a^3 - 6*a^2 - 25*a + 5])
gp (2.8): E = ellinit([0, -a^4 + 4*a^2 + a - 2, a^4 - 4*a^2 + 3, -4*a^4 + 9*a^3 - a^2 - 3*a, -13*a^4 + 34*a^3 - 6*a^2 - 25*a + 5],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((11,a^{4} + a^{3} - 3 a^{2} - 3 a - 2)\) | = | \( \left(a^{2} + a - 2\right)^{2} \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 121 \) | = | \( 11^{2} \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((121,121 a,121 a^{2} - 242,121 a^{3} - 363 a,11 a^{4} + 88 a^{3} + 22 a^{2} - 253 a - 66)\) | = | \( \left(a^{2} + a - 2\right)^{9} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 2357947691 \) | = | \( 11^{9} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -32768 \) | ||
| magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
| ||||
| \( \text{End} (E) \) | = | \(\Z[(1+\sqrt{-11})/2]\) | ( Complex Multiplication ) | |
| magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
| ||||
| \( \text{ST} (E) \) | = | $N(\mathrm{U}(1))$ | ||
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: | Trivial |
|---|---|
| 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]
| |
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{2} + a - 2\right) \) | \(11\) | \(2\) | \(III^*\) | Additive | \(1\) | \(2\) | \(9\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(11\) | 11B.1.10[5] |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
11.
Its isogeny class
121.1-b
consists of curves linked by isogenies of
degree 11.