Base field \(\Q(\sqrt{22}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 22 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-22, 0, 1]))
gp: K = nfinit(Polrev([-22, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,1]),K([15105,-3221]),K([264264,-56343])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,1]),Polrev([15105,-3221]),Polrev([264264,-56343])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,1],K![15105,-3221],K![264264,-56343]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a+6)\) | = | \((-3a-14)\cdot(2a+9)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 14 \) | = | \(2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((6016a+28416)\) | = | \((-3a-14)^{15}\cdot(2a+9)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 11239424 \) | = | \(2^{15}\cdot7^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{19062299657}{87808} a - \frac{44634002097}{43904} \) | ||
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 | $\left(0 : 77 a - 364 : 1\right)$ |
| Height | \(0.90994530392902025057533254172483023523\) |
| 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: | \( 0.90994530392902025057533254172483023523 \) | ||
| Period: | \( 2.6263188535998233839890951063285048668 \) | ||
| Tamagawa product: | \( 3 \) = \(1\cdot3\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.5285253780210977277870001422781042938 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-3a-14)\) | \(2\) | \(1\) | \(I_{15}\) | Non-split multiplicative | \(1\) | \(1\) | \(15\) | \(15\) |
| \((2a+9)\) | \(7\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
14.2-b
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.