Invariants
| Base field: | $\F_{83}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 9 x - 2 x^{2} - 747 x^{3} + 6889 x^{4}$ |
| Frobenius angles: | $\pm0.00222320941996$, $\pm0.668889876087$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-251})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $7$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6132$ | $46873008$ | $325214716176$ | $2251981233238464$ | $15515779166646991932$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $75$ | $6805$ | $568764$ | $47451769$ | $3938974125$ | $326938088230$ | $27136046227575$ | $2252292180164209$ | $186940253541492612$ | $15516041183752549525$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=72 x^6+16 x^5+67 x^4+27 x^3+71 x^2+x+72$
- $y^2=52 x^6+25 x^5+44 x^4+68 x^3+35 x^2+38 x+52$
- $y^2=79 x^6+62 x^5+22 x^4+26 x^3+67 x^2+80 x+79$
- $y^2=3 x^6+23 x^5+27 x^4+52 x^3+40 x^2+78 x+3$
- $y^2=28 x^5+15 x^4+56 x^3+41 x^2+55 x$
- $y^2=79 x^6+8 x^5+53 x^4+26 x^3+36 x^2+51 x+79$
- $y^2=79 x^6+53 x^5+41 x^4+26 x^3+48 x^2+6 x+79$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{83^{3}}$.
Endomorphism algebra over $\F_{83}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-251})\). |
| The base change of $A$ to $\F_{83^{3}}$ is 1.571787.acge 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-251}) \)$)$ |
Base change
This is a primitive isogeny class.