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([1,0]),K([4,0]),K([-6,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([4,0]),Polrev([-6,0])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![4,0],K![-6,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-21a-98)\) | = | \((-3a-14)\cdot(2a+9)\cdot(-2a+9)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 98 \) | = | \(2\cdot7\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-21952)\) | = | \((-3a-14)^{12}\cdot(2a+9)^{3}\cdot(-2a+9)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 481890304 \) | = | \(2^{12}\cdot7^{3}\cdot7^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{9938375}{21952} \) | ||
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: | \(2\) | |
| Generators | $\left(-\frac{32}{9} a + \frac{161}{9} : \frac{640}{27} a - \frac{2995}{27} : 1\right)$ | $\left(\frac{16}{81} a + \frac{185}{81} : \frac{304}{729} a + \frac{1895}{729} : 1\right)$ |
| Heights | \(1.7481422386464015851358481404006088041\) | \(1.8561356318911055905495812903251367229\) |
| Torsion structure: | \(\Z/6\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generator: | $\left(9 : 23 : 1\right)$ | |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 2.7824060904205084947737266287851853000 \) | ||
| Period: | \( 3.9257159468709430520307637303495930307 \) | ||
| Tamagawa product: | \( 18 \) = \(2\cdot3\cdot3\) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 2.3287777713431993551346132703946248932 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-3a-14)\) | \(2\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
| \((2a+9)\) | \(7\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
| \((-2a+9)\) | \(7\) | \(3\) | \(I_{3}\) | 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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
98.1-b
consists of curves linked by isogenies of
degrees dividing 18.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 14.a6 |
| \(\Q\) | 54208.m6 |