Base field \(\Q(\sqrt{89}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 22 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-22, -1, 1]))
gp: K = nfinit(Polrev([-22, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,1]),K([1,0]),K([-235181,45081]),K([59424228,-11390518])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,1]),Polrev([1,0]),Polrev([-235181,45081]),Polrev([59424228,-11390518])], K);
magma: E := EllipticCurve([K![1,0],K![-1,1],K![1,0],K![-235181,45081],K![59424228,-11390518]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-3a-13)\) | = | \((a-5)\cdot(4a-21)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 10 \) | = | \(2\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((41a-219)\) | = | \((a-5)^{4}\cdot(4a-21)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -2000 \) | = | \(-2^{4}\cdot5^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{4602102825621}{2000} a + \frac{12004564630091}{1000} \) | ||
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 | $\left(-\frac{1008}{25} a + \frac{5222}{25} : \frac{2847}{125} a - \frac{14823}{125} : 1\right)$ |
| Height | \(0.81024245637692225382094345115082338007\) |
| Torsion structure: | \(\Z/4\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-39 a + 202 : -22 a + 115 : 1\right)$ |
| 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: | \( 0.81024245637692225382094345115082338007 \) | ||
| Period: | \( 16.766373360148804988638334227228522153 \) | ||
| Tamagawa product: | \( 6 \) = \(2\cdot3\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.0799916291197367208890640082028865891 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a-5)\) | \(2\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((4a-21)\) | \(5\) | \(3\) | \(I_{3}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
10.1-b
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.