Invariants
| Base field: | $\F_{67}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 14 x + 67 x^{2} )^{2}$ |
| $1 + 28 x + 330 x^{2} + 1876 x^{3} + 4489 x^{4}$ | |
| Frobenius angles: | $\pm0.826557230848$, $\pm0.826557230848$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $11$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6724$ | $19607184$ | $90416881636$ | $406274655937536$ | $1822656392754958084$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $96$ | $4366$ | $300624$ | $20161390$ | $1349990736$ | $90459575422$ | $6060703902720$ | $406067705445214$ | $27206534521929408$ | $1822837800924344686$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):
- $y^2=17 x^6+33 x^5+21 x^4+42 x^3+49 x^2+x+10$
- $y^2=14 x^6+17 x^5+61 x^4+6 x^3+19 x^2+15 x+24$
- $y^2=2 x^6+57 x^4+57 x^2+2$
- $y^2=15 x^6+33 x^5+20 x^4+8 x^3+29 x^2+27 x+28$
- $y^2=21 x^6+51 x^5+47 x^4+9 x^3+47 x^2+51 x+21$
- $y^2=2 x^6+20 x^3+57$
- $y^2=29 x^6+60 x^5+2 x^4+48 x^3+2 x^2+60 x+29$
- $y^2=21 x^6+45 x^4+45 x^2+21$
- $y^2=2 x^6+41 x^3+28$
- $y^2=23 x^6+64 x^5+56 x^4+56 x^3+46 x^2+28 x+35$
- $y^2=12 x^6+48 x^5+49 x^3+48 x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{67}$.
Endomorphism algebra over $\F_{67}$| The isogeny class factors as 1.67.o 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.