Invariants
| Base field: | $\F_{97}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 18 x + 97 x^{2} )^{2}$ |
| $1 + 36 x + 518 x^{2} + 3492 x^{3} + 9409 x^{4}$ | |
| Frobenius angles: | $\pm0.866875061252$, $\pm0.866875061252$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $10$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 29$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $13456$ | $86118400$ | $834058439824$ | $7837773373440000$ | $73740830205745549456$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $134$ | $9150$ | $913862$ | $88533118$ | $8587155974$ | $832974949950$ | $80798249343302$ | $7837433941136638$ | $760231055820967814$ | $73742412706861890750$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=13 x^6+72 x^4+72 x^2+13$
- $y^2=30 x^6+52 x^4+52 x^2+30$
- $y^2=43 x^6+84 x^4+84 x^2+43$
- $y^2=43 x^6+9 x^5+4 x^4+19 x^3+47 x^2+24 x+54$
- $y^2=2 x^6+86 x^5+23 x^4+88 x^3+92 x^2+18 x+31$
- $y^2=9 x^6+65 x^5+63 x^4+71 x^3+73 x^2+48 x+81$
- $y^2=66 x^6+71 x^5+55 x^4+42 x^3+7 x^2+37 x+94$
- $y^2=84 x^6+58 x^5+9 x^4+x^3+9 x^2+58 x+84$
- $y^2=86 x^6+86 x^5+32 x^4+48 x^3+32 x^2+86 x+86$
- $y^2=72 x^6+63 x^5+4 x^4+77 x^3+4 x^2+63 x+72$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{97}$.
Endomorphism algebra over $\F_{97}$| The isogeny class factors as 1.97.s 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.