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 - 3*a + 1, -a^4 + 4*a^2 + a - 3, a^4 + a^3 - 3*a^2 - 3*a + 1, -30*a^4 + 30*a^3 + 25*a^2 + 160*a - 215, -226*a^4 + 563*a^3 - 458*a^2 + 913*a - 1012]),K);
sage: E = EllipticCurve(K, [a^4 + a^3 - 3*a^2 - 3*a + 1, -a^4 + 4*a^2 + a - 3, a^4 + a^3 - 3*a^2 - 3*a + 1, -30*a^4 + 30*a^3 + 25*a^2 + 160*a - 215, -226*a^4 + 563*a^3 - 458*a^2 + 913*a - 1012])
gp (2.8): E = ellinit([a^4 + a^3 - 3*a^2 - 3*a + 1, -a^4 + 4*a^2 + a - 3, a^4 + a^3 - 3*a^2 - 3*a + 1, -30*a^4 + 30*a^3 + 25*a^2 + 160*a - 215, -226*a^4 + 563*a^3 - 458*a^2 + 913*a - 1012],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((43,a^{4} - 2 a^{2} - a - 1)\) | = | \( \left(-a^{4} + 2 a^{2} + a + 1\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 43 \) | = | \( 43 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((271818611107,a + 150821116011,a^{2} + 242741157992,a^{3} - 3 a + 73434615221,a^{4} - 4 a^{2} + 250768899967)\) | = | \( \left(-a^{4} + 2 a^{2} + a + 1\right)^{7} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 271818611107 \) | = | \( 43^{7} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{70548898331353450135096255010}{271818611107} a^{4} + \frac{189247840974084378816217200685}{271818611107} a^{3} - \frac{842216448583260989645063883}{6321363049} a^{2} - \frac{150714011772341855278459782553}{271818611107} a + \frac{41930794528416841925025385323}{271818611107} \) | ||
| 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: | 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a^{4} + 2 a^{2} + a + 1\right) \) | \(43\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
43.2-a
consists of curves linked by isogenies of
degree 7.