Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 1)
gp: K = nfinit(a^2 - a + 1);
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve(K, [1, 1, 1, -2160, -39540])
gp: E = ellinit([1, 1, 1, -2160, -39540],K)
magma: E := ChangeRing(EllipticCurve([1, 1, 1, -2160, -39540]),K);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((-10 a + 5)\) | = | \( \left(-2 a + 1\right) \cdot \left(5\right) \) |
| sage: E.conductor()
magma: Conductor(E);
| ||||
| \(N(\mathfrak{N}) \) | = | \( 75 \) | = | \( 3 \cdot 25 \) |
| sage: E.conductor().norm()
magma: Norm(Conductor(E));
| ||||
| \(\mathfrak{D}\) | = | \((405)\) | = | \( \left(-2 a + 1\right)^{8} \cdot \left(5\right) \) |
| sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| ||||
| \(N(\mathfrak{D})\) | = | \( 164025 \) | = | \( 3^{8} \cdot 25 \) |
| sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| ||||
| \(j\) | = | \( \frac{1114544804970241}{405} \) | ||
| sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| ||||
| \( \text{End} (E) \) | = | \(\Z\) | (no Complex Multiplication ) | |
| sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| ||||
| \( \text{ST} (E) \) | = | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
Rank: \( 0 \)sage: E.rank()
magma: Rank(E);
Regulator: 1
sage: gens = E.gens(); gens
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
sage: E.regulator_of_points(gens)
magma: Regulator(gens);
Torsion subgroup
| Structure: | \(\Z/2\Z\) |
|---|---|
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Generator: | $\left(-\frac{109}{4} : \frac{105}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-2 a + 1\right) \) | \(3\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
| \( \left(5\right) \) | \(25\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 8 and 16.
Its isogeny class
75.1-a
consists of curves linked by isogenies of
degrees dividing 16.