Base field \(\Q(\sqrt{209}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 52 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-52, -1, 1]))
gp: K = nfinit(Polrev([-52, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-52, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([1,1]),K([1,0]),K([-3754149800,-557954461]),K([131220496498681,19502434692506])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([1,0]),Polrev([-3754149800,-557954461]),Polrev([131220496498681,19502434692506])], K);
magma: E := EllipticCurve([K![1,1],K![1,1],K![1,0],K![-3754149800,-557954461],K![131220496498681,19502434692506]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a-2)\) | = | \((11a+74)\cdot(-4a-27)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 50 \) | = | \(2\cdot5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((3657a-28264)\) | = | \((11a+74)^{2}\cdot(-4a-27)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 62500 \) | = | \(2^{2}\cdot5^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2361203}{4} a - \frac{18549919}{4} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(\frac{5935}{2} a + \frac{79863}{4} : \frac{1016445}{4} a + \frac{13678107}{8} : 1\right)$ |
| Height | \(1.7860224180822910825924916781699557713\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 1.7860224180822910825924916781699557713 \) | ||
| Period: | \( 5.8294206499100773885886556036022435267 \) | ||
| Tamagawa product: | \( 8 \) = \(2\cdot2^{2}\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 11.522829624484711247198792760069142759 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((11a+74)\) | \(2\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((-4a-27)\) | \(5\) | \(4\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
50.6-g
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.