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: | $\pm0.177807684489$, $\pm0.177807684489$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $9$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $5476$ | $61528336$ | $497230881316$ | $3937813504000000$ | $31183288679517607396$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $58$ | $7766$ | $705322$ | $62761758$ | $5584340378$ | $496984048886$ | $44231354019722$ | $3936588866233918$ | $350356402973941498$ | $31181719912842150806$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=15 x^6+40 x^5+77 x^4+13 x^3+54 x^2+54 x+48$
- $y^2=65 x^6+71 x^5+61 x^4+37 x^3+63 x^2+9 x+62$
- $y^2=5 x^6+51 x^5+41 x^4+71 x^3+22 x^2+25 x+24$
- $y^2=14 x^6+73 x^5+61 x^4+29 x^3+12 x^2+57 x+77$
- $y^2=35 x^6+27 x^4+27 x^2+35$
- $y^2=19 x^6+51 x^5+10 x^4+19 x^3+21 x^2+30 x+13$
- $y^2=26 x^6+18 x^5+85 x^4+87 x^3+72 x^2+47 x+24$
- $y^2=70 x^6+78 x^5+34 x^4+28 x^3+44 x^2+42 x+86$
- $y^2=25 x^6+83 x^5+15 x^4+62 x^3+84 x^2+26 x+56$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{89}$.
Endomorphism algebra over $\F_{89}$| The isogeny class factors as 1.89.aq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.