Base field 3.1.23.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 + 1)
gp (2.8): K = nfinit(a^3 - a^2 + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^2 + a + 1, a^2 + a, a^2 + a + 1, -1407*a^2 + 2472*a - 1882, -36680*a^2 + 64355*a - 48601]),K);
sage: E = EllipticCurve(K, [a^2 + a + 1, a^2 + a, a^2 + a + 1, -1407*a^2 + 2472*a - 1882, -36680*a^2 + 64355*a - 48601])
gp (2.8): E = ellinit([a^2 + a + 1, a^2 + a, a^2 + a + 1, -1407*a^2 + 2472*a - 1882, -36680*a^2 + 64355*a - 48601],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((185,-6 a^{2} + 4 a + 1)\) | = | \( \left(a^{2} + 1\right) \cdot \left(3 a^{2} - a + 1\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 185 \) | = | \( 5 \cdot 37 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((64143160225,a + 63556347608,a^{2} - a + 45053684278)\) | = | \( \left(a^{2} + 1\right)^{2} \cdot \left(3 a^{2} - a + 1\right)^{6} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 64143160225 \) | = | \( 5^{2} \cdot 37^{6} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{263188403436724755665988}{64143160225} a^{2} - \frac{461863452181436668806417}{64143160225} a + \frac{348650404446288146701261}{64143160225} \) | ||
| 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: \( 0 \)magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()
Regulator: 1
magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())
Torsion subgroup
| Structure: | \(\Z/2\Z\times\Z/2\Z\) |
|---|---|
| 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]
| |
| Generators: | $\left(24 a^{2} - 44 a + 31 : -8 a^{2} + 18 a - 14 : 1\right)$,$\left(-12 a^{2} + 20 a - 17 : 6 a^{2} - 8 a + 6 : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{2} + 1\right) \) | \(5\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \( \left(3 a^{2} - a + 1\right) \) | \(37\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
185.1-A
consists of curves linked by isogenies of
degrees dividing 12.