Base field \(\Q(\sqrt{229}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 57 \); class number \(3\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-57, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 57)
gp (2.8): K = nfinit(a^2 - a - 57);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a, 0, a, 22*a - 285, 307*a - 3606]),K);
sage: E = EllipticCurve(K, [a, 0, a, 22*a - 285, 307*a - 3606])
gp (2.8): E = ellinit([a, 0, a, 22*a - 285, 307*a - 3606],K)
This is not a global minimal model: it is minimal at all primes except \((3,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((1)\) | = | \((1)\) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 1 \) | = | 1 |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \((\Delta)\) | = | \((146 a - 903)\) | = | \( \left(3, a\right)^{12} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\Delta)\) | = | \( 531441 \) | = | \( 3^{12} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(\mathfrak{D}\) | = | \((1)\) | = | \((1)\) |
| \(N(\mathfrak{D})\) | = | \( 1 \) | = | \((1)\) |
| \(j\) | = | \( -299510191348095 a + 2415960913292737 \) | ||
| 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()
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(3, a\right) \) | \(3\) | \(1\) | \(I_{0}\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cn |
| \(3\) | 3B.1.2 |
| \(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degrees dividing 15.