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([0,1]),K([1,0]),K([0,0]),K([304014,-64813]),K([-667307485,142270436])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,0]),Polrev([304014,-64813]),Polrev([-667307485,142270436])], K);
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![304014,-64813],K![-667307485,142270436]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-18a-84)\) | = | \((-3a-14)^{3}\cdot(-a+5)\cdot(a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 72 \) | = | \(2^{3}\cdot3\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-209952)\) | = | \((-3a-14)^{10}\cdot(-a+5)^{8}\cdot(a+5)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 44079842304 \) | = | \(2^{10}\cdot3^{8}\cdot3^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{207646}{6561} \) | ||
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 | $\left(-\frac{929}{7} a + \frac{30396}{49} : -\frac{207168}{49} a + \frac{6803665}{343} : 1\right)$ |
| Height | \(2.8323195521147206903631952969770154702\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-98 a + \frac{915}{2} : -\frac{915}{4} a + 1078 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 2.8323195521147206903631952969770154702 \) | ||
| Period: | \( 2.3252798689554027597660292252204142920 \) | ||
| Tamagawa product: | \( 8 \) = \(2\cdot2\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.8082523913526462959824794750276748193 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-3a-14)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(-1\) | \(3\) | \(10\) | \(0\) |
| \((-a+5)\) | \(3\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
| \((a+5)\) | \(3\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
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, 4 and 8.
Its isogeny class
72.1-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 24.a6 |
| \(\Q\) | 23232.bo6 |