Base field \(\Q(\sqrt{-11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 3)
gp (2.8): K = nfinit(a^2 - a + 3);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a + 1, a - 1, a + 1, 3*a - 14, 2*a - 11]),K);
sage: E = EllipticCurve(K, [a + 1, a - 1, a + 1, 3*a - 14, 2*a - 11])
gp (2.8): E = ellinit([a + 1, a - 1, a + 1, 3*a - 14, 2*a - 11],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((4 a + 6)\) | = | \( \left(2\right) \cdot \left(-a\right)^{3} \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 108 \) | = | \( 3^{3} \cdot 4 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((37888 a - 53760)\) | = | \( \left(2\right)^{9} \cdot \left(-a\right)^{9} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 5159780352 \) | = | \( 3^{9} \cdot 4^{9} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{488881}{256} a + \frac{1381533}{512} \) | ||
| 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: \( 1 \)magma: Rank(E);
sage: E.rank()
Generator: $\left(2 a - 5 : 2 a + 17 : 1\right)$
Height: 0.13074274971306377
magma: Generators(E); // includes torsion
sage: E.gens()
Regulator: 0.130742749713
magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())
Torsion subgroup
| Structure: | \(\Z/3\Z\) |
|---|---|
| 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: | $\left(-2 a + 7 : 2 a - 19 : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a\right) \) | \(3\) | \(3\) | \(IV^*\) | Additive | \(-1\) | \(3\) | \(9\) | \(0\) |
| \( \left(2\right) \) | \(4\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
108.1-a
consists of curves linked by isogenies of
degrees dividing 27.