Invariants
| Base field: | $\F_{97}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 11 x + 97 x^{2} )^{2}$ |
| $1 - 22 x + 315 x^{2} - 2134 x^{3} + 9409 x^{4}$ | |
| Frobenius angles: | $\pm0.311399950814$, $\pm0.311399950814$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $24$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3, 29$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7569$ | $89927289$ | $836390727936$ | $7839822296358441$ | $73741845754378719009$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $76$ | $9556$ | $916414$ | $88556260$ | $8587274236$ | $832968661822$ | $80798254108060$ | $7837433584587844$ | $760231061492780158$ | $73742412721662734836$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=79 x^6+91 x^5+95 x^4+71 x^3+65 x^2+51 x+95$
- $y^2=96 x^6+75 x^5+37 x^4+52 x^3+64 x^2+47 x+19$
- $y^2=25 x^6+24 x^5+35 x^4+20 x^3+45 x^2+2 x+50$
- $y^2=72 x^6+2 x^5+45 x^4+35 x^3+68 x^2+5 x+6$
- $y^2=17 x^6+22 x^5+89 x^4+31 x^3+87 x^2+54 x+80$
- $y^2=x^6+x^3+1$
- $y^2=72 x^6+15 x^5+78 x^4+32 x^3+76 x^2+46 x+48$
- $y^2=17 x^6+73 x^5+72 x^4+31 x^3+4 x^2+41 x+93$
- $y^2=71 x^6+5 x^5+80 x^4+35 x^3+10 x^2+96 x+53$
- $y^2=x^6+84 x^5+94 x^4+63 x^3+65 x^2+79 x+59$
- $y^2=69 x^6+52 x^5+84 x^4+52 x^3+86 x^2+53 x+10$
- $y^2=94 x^6+87 x^5+54 x^4+24 x^3+86 x^2+83 x+31$
- $y^2=41 x^6+23 x^5+3 x^4+54 x^3+96 x^2+20 x+71$
- $y^2=92 x^6+84 x^5+14 x^4+7 x^3+92 x^2+80 x+5$
- $y^2=92 x^6+37 x^5+21 x^4+77 x^3+33 x^2+61 x+17$
- $y^2=67 x^6+52 x^5+9 x^4+76 x^3+34 x^2+73 x+57$
- $y^2=2 x^6+93 x^5+55 x^4+30 x^3+14 x^2+75 x+61$
- $y^2=x^6+84 x^5+57 x^4+55 x^3+46 x^2+75 x+19$
- $y^2=92 x^6+22 x^5+37 x^4+40 x^3+27 x^2+33 x+19$
- $y^2=20 x^6+24 x^5+94 x^4+95 x^3+72 x^2+50 x+67$
- $y^2=90 x^6+2 x^4+70 x^3+82 x^2+57 x+10$
- $y^2=57 x^6+92 x^5+80 x^4+76 x^3+53 x^2+25 x+22$
- $y^2=28 x^6+20 x^5+35 x^4+49 x^3+37 x^2+25 x+46$
- $y^2=13 x^6+86 x^5+x^4+42 x^3+71 x^2+12 x+20$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{97}$.
Endomorphism algebra over $\F_{97}$| The isogeny class factors as 1.97.al 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-267}) \)$)$ |
Base change
This is a primitive isogeny class.