Invariants
| Base field: | $\F_{89}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 174 x^{2} + 7921 x^{4}$ |
| Frobenius angles: | $\pm0.0338042870639$, $\pm0.966195712936$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{22})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $9$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7748$ | $60031504$ | $496980157700$ | $3934777896484864$ | $31181719924525829828$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $90$ | $7574$ | $704970$ | $62713374$ | $5584059450$ | $496979024438$ | $44231334895530$ | $3936588639990334$ | $350356403707485210$ | $31181719919085476054$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=78 x^6+42 x^5+37 x^4+74 x^2+10 x+1$
- $y^2=60 x^6+77 x^5+76 x^4+x^3+9 x^2+19 x+11$
- $y^2=44 x^6+28 x^5+88 x^4+34 x^2+28 x+72$
- $y^2=35 x^6+14 x^5+82 x^4+87 x^3+44 x^2+70 x+57$
- $y^2=23 x^6+5 x^5+71 x^4+x^3+27 x^2+78 x+67$
- $y^2=82 x^6+48 x^5+30 x^4+23 x^3+42 x^2+30 x+5$
- $y^2=35 x^6+8 x^5+78 x^4+11 x^2+81 x+54$
- $y^2=12 x^6+13 x^5+43 x^4+62 x^2+60 x+30$
- $y^2=43 x^6+77 x^5+82 x^4+29 x^2+77 x+51$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{89^{2}}$.
Endomorphism algebra over $\F_{89}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{22})\). |
| The base change of $A$ to $\F_{89^{2}}$ is 1.7921.ags 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-22}) \)$)$ |
Base change
This is a primitive isogeny class.