Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 116 x^{2} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.0293443469575$, $\pm0.970655653043$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{-13})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $0$ |
| Isomorphism classes: | 14 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3366$ | $11329956$ | $42180184134$ | $146673123713424$ | $511116752436550326$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $60$ | $3250$ | $205380$ | $12104374$ | $714924300$ | $42179834626$ | $2488651484820$ | $146830401729694$ | $8662995818654940$ | $511116751572459250$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59^{2}}$.
Endomorphism algebra over $\F_{59}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-13})\). |
| The base change of $A$ to $\F_{59^{2}}$ is 1.3481.aem 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$ |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.59.a_em | $4$ | (not in LMFDB) |
| 2.59.ac_c | $8$ | (not in LMFDB) |
| 2.59.c_c | $8$ | (not in LMFDB) |