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: | $\pm0.173442769152$, $\pm0.173442769152$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $11$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2916$ | $19607184$ | $90501095556$ | $406274655937536$ | $1823019230776982436$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $40$ | $4366$ | $300904$ | $20161390$ | $1350259480$ | $90459575422$ | $6060719307928$ | $406067705445214$ | $27206534270660488$ | $1822837800924344686$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):
- $y^2=34 x^6+66 x^5+42 x^4+17 x^3+31 x^2+2 x+20$
- $y^2=28 x^6+34 x^5+55 x^4+12 x^3+38 x^2+30 x+48$
- $y^2=x^6+62 x^4+62 x^2+1$
- $y^2=30 x^6+66 x^5+40 x^4+16 x^3+58 x^2+54 x+56$
- $y^2=44 x^6+59 x^5+57 x^4+38 x^3+57 x^2+59 x+44$
- $y^2=x^6+10 x^3+62$
- $y^2=58 x^6+53 x^5+4 x^4+29 x^3+4 x^2+53 x+58$
- $y^2=44 x^6+56 x^4+56 x^2+44$
- $y^2=x^6+54 x^3+14$
- $y^2=46 x^6+61 x^5+45 x^4+45 x^3+25 x^2+56 x+3$
- $y^2=24 x^6+29 x^5+31 x^3+29 x+24$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{67}$.
Endomorphism algebra over $\F_{67}$| The isogeny class factors as 1.67.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.