Invariants
| Base field: | $\F_{89}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 16 x + 89 x^{2} )^{2}$ |
| $1 + 32 x + 434 x^{2} + 2848 x^{3} + 7921 x^{4}$ | |
| Frobenius angles: | $\pm0.822192315511$, $\pm0.822192315511$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $9$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 53$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $11236$ | $61528336$ | $496734582436$ | $3937813504000000$ | $31180151242211268196$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $122$ | $7766$ | $704618$ | $62761758$ | $5583778522$ | $496984048886$ | $44231315771338$ | $3936588866233918$ | $350356404441028922$ | $31181719912842150806$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=45 x^6+31 x^5+53 x^4+39 x^3+73 x^2+73 x+55$
- $y^2=81 x^6+83 x^5+50 x^4+42 x^3+21 x^2+3 x+80$
- $y^2=15 x^6+64 x^5+34 x^4+35 x^3+66 x^2+75 x+72$
- $y^2=64 x^6+54 x^5+50 x^4+69 x^3+4 x^2+19 x+85$
- $y^2=71 x^6+9 x^4+9 x^2+71$
- $y^2=36 x^6+17 x^5+33 x^4+36 x^3+7 x^2+10 x+34$
- $y^2=68 x^6+6 x^5+58 x^4+29 x^3+24 x^2+75 x+8$
- $y^2=32 x^6+56 x^5+13 x^4+84 x^3+43 x^2+37 x+80$
- $y^2=38 x^6+87 x^5+5 x^4+80 x^3+28 x^2+68 x+78$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{89}$.
Endomorphism algebra over $\F_{89}$| The isogeny class factors as 1.89.q 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.