Invariants
| Base field: | $\F_{89}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 89 x^{2} + 7921 x^{4}$ |
| Frobenius angles: | $\pm0.166666666667$, $\pm0.833333333333$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-89})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $11$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7833$ | $61355889$ | $496982700900$ | $3937582956526569$ | $31181719924382124153$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $90$ | $7744$ | $704970$ | $62758084$ | $5584059450$ | $496984110838$ | $44231334895530$ | $3936588931186564$ | $350356403707485210$ | $31181719918798064704$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):
- $y^2=33 x^6+59 x^5+84 x^4+25 x^3+17 x^2+50 x+33$
- $y^2=10 x^6+88 x^5+74 x^4+75 x^3+51 x^2+61 x+10$
- $y^2=82 x^6+69 x^5+86 x^4+24 x^3+81 x^2+67 x+82$
- $y^2=68 x^6+29 x^5+80 x^4+72 x^3+65 x^2+23 x+68$
- $y^2=44 x^6+61 x^5+13 x^4+72 x^3+12 x^2+25 x+44$
- $y^2=43 x^6+5 x^5+39 x^4+38 x^3+36 x^2+75 x+43$
- $y^2=x^6+42 x^5+67 x^4+57 x^3+76 x^2+28 x+59$
- $y^2=55 x^6+37 x^5+55 x^4+30 x^3+72 x^2+26 x+55$
- $y^2=76 x^6+22 x^5+76 x^4+x^3+38 x^2+78 x+76$
- $y^2=3 x^6+54 x^5+32 x^4+2 x^3+74 x^2+53 x+3$
- $y^2=9 x^6+73 x^5+7 x^4+6 x^3+44 x^2+70 x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{89^{6}}$.
Endomorphism algebra over $\F_{89}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-89})\). |
| The base change of $A$ to $\F_{89^{6}}$ is 1.496981290961.dcfsk 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $89$ and $\infty$. |
- Endomorphism algebra over $\F_{89^{2}}$
The base change of $A$ to $\F_{89^{2}}$ is 1.7921.adl 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{89^{3}}$
The base change of $A$ to $\F_{89^{3}}$ is 1.704969.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-89}) \)$)$
Base change
This is a primitive isogeny class.