Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 113 x^{2} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.0464969758871$, $\pm0.953503024113$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-5}, \sqrt{231})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 32 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3369$ | $11350161$ | $42180270804$ | $146689764518025$ | $511116753143602929$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $60$ | $3256$ | $205380$ | $12105748$ | $714924300$ | $42180007966$ | $2488651484820$ | $146830418631268$ | $8662995818654940$ | $511116752986564456$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=37 x^6+57 x^5+45 x^4+49 x^3+12 x^2+49 x+50$
- $y^2=32 x^6+18 x^5+2 x^4+51 x^3+44 x^2+6 x+46$
- $y^2=54 x^6+44 x^5+44 x^4+16 x^2+53 x+37$
- $y^2=38 x^6+20 x^5+8 x^4+28 x^3+38 x^2+14 x+48$
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 \(\Q(\sqrt{-5}, \sqrt{231})\). |
| The base change of $A$ to $\F_{59^{2}}$ is 1.3481.aej 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1155}) \)$)$ |
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_ej | $4$ | (not in LMFDB) |