Base field \(\Q(\sqrt{209}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 52 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-52, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 52)
gp: K = nfinit(a^2 - a - 52);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([1, a, a + 1, 12491185*a - 96537059, 61050477245*a - 471823494147]),K);
sage: E = EllipticCurve(K, [1, a, a + 1, 12491185*a - 96537059, 61050477245*a - 471823494147])
gp: E = ellinit([1, a, a + 1, 12491185*a - 96537059, 61050477245*a - 471823494147],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((7 a + 47)\) | = | \( \left(-11 a + 85\right) \cdot \left(4 a - 31\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 10 \) | = | \( 2 \cdot 5 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((-a - 1)\) | = | \( \left(-11 a + 85\right) \cdot \left(4 a - 31\right)^{2} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp: E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 50 \) | = | \( 2 \cdot 5^{2} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp: norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{56211}{50} a + \frac{217211}{25} \) | ||
| magma: jInvariant(E);
sage: E.j_invariant()
gp: 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()
Regulator: not available
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
sage: gens = E.gens(); gens
magma: Regulator(gens);
sage: E.regulator_of_points(gens)
Torsion subgroup
| Structure: | \(\Z/2\Z\) |
|---|---|
| magma: T,piT := TorsionSubgroup(E); Invariants(T);
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
| |
| Generator: | $\left(\frac{2543}{4} a - 4916 : -\frac{2547}{8} a + \frac{4915}{2} : 1\right)$ |
| magma: [piT(P) : P in Generators(T)];
sage: T.gens()
gp: T[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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-11 a + 85\right) \) | \(2\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \( \left(4 a - 31\right) \) | \(5\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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.
Its isogeny class
10.1-b
consists of curves linked by isogenies of
degree 2.