Invariants
| Base field: | $\F_{97}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 19 x + 97 x^{2} )( 1 - 14 x + 97 x^{2} )$ |
| $1 - 33 x + 460 x^{2} - 3201 x^{3} + 9409 x^{4}$ | |
| Frobenius angles: | $\pm0.0849741350078$, $\pm0.248359198326$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $30$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6636$ | $86958144$ | $832972061376$ | $7838296141568256$ | $73743194552411821836$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $65$ | $9241$ | $912674$ | $88539025$ | $8587431305$ | $832972117822$ | $80798277509273$ | $7837433512244449$ | $760231058654565218$ | $73742412703904321161$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=93 x^6+82 x^5+67 x^4+69 x^3+68 x^2+76 x+20$
- $y^2=9 x^6+70 x^5+43 x^4+36 x^3+18 x^2+36 x+52$
- $y^2=94 x^6+59 x^5+30 x^4+91 x^3+28 x^2+17 x+34$
- $y^2=56 x^6+76 x^5+11 x^4+18 x^3+9 x^2+66 x+15$
- $y^2=3 x^6+84 x^5+14 x^4+44 x^3+17 x^2+6 x+48$
- $y^2=83 x^6+49 x^5+3 x^4+17 x^3+27 x^2+38 x+74$
- $y^2=24 x^6+58 x^5+49 x^4+94 x^3+71 x^2+26 x+38$
- $y^2=60 x^6+86 x^5+41 x^4+10 x^3+21 x^2+96 x+22$
- $y^2=90 x^6+30 x^5+56 x^4+18 x^3+75 x^2+21 x+26$
- $y^2=13 x^6+30 x^5+67 x^4+67 x^3+13 x^2+56 x+67$
- $y^2=15 x^6+44 x^5+16 x^4+79 x^3+6 x^2+8 x+62$
- $y^2=92 x^6+83 x^5+52 x^4+18 x^3+34 x^2+79 x+60$
- $y^2=62 x^6+89 x^5+60 x^4+90 x^3+35 x^2+6 x+41$
- $y^2=71 x^6+45 x^5+58 x^4+89 x^3+x^2+30 x+14$
- $y^2=48 x^6+37 x^5+14 x^4+37 x^3+53 x^2+33 x+9$
- $y^2=x^6+x^3+23$
- $y^2=92 x^6+71 x^5+36 x^4+7 x^3+83 x^2+56 x+63$
- $y^2=10 x^6+17 x^5+91 x^4+49 x^3+64 x^2+30 x+23$
- $y^2=5 x^6+44 x^5+45 x^4+61 x^3+75 x^2+47 x+21$
- $y^2=95 x^6+46 x^5+52 x^4+62 x^3+31 x+46$
- $y^2=74 x^6+41 x^5+19 x^4+50 x^3+4 x^2+32 x+28$
- $y^2=48 x^6+42 x^5+36 x^4+70 x^3+24 x^2+33 x+79$
- $y^2=10 x^6+51 x^5+15 x^4+80 x^3+4 x^2+89 x+83$
- $y^2=37 x^6+75 x^5+33 x^4+17 x^3+61 x^2+59 x+50$
- $y^2=10 x^6+25 x^5+11 x^4+83 x^3+19 x^2+12 x$
- $y^2=32 x^6+64 x^5+26 x^4+87 x^3+11 x^2+90 x+40$
- $y^2=78 x^6+46 x^5+4 x^4+37 x^3+65 x^2+92 x+36$
- $y^2=93 x^5+68 x^4+37 x^3+19 x^2+71 x+71$
- $y^2=3 x^6+76 x^5+52 x^4+44 x^3+70 x^2+28 x+13$
- $y^2=35 x^6+5 x^5+51 x^4+52 x^3+36 x^2+32 x+21$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{97^{6}}$.
Endomorphism algebra over $\F_{97}$| The isogeny class factors as 1.97.at $\times$ 1.97.ao and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
| The base change of $A$ to $\F_{97^{6}}$ is 1.832972004929.dfna 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
- Endomorphism algebra over $\F_{97^{2}}$
The base change of $A$ to $\F_{97^{2}}$ is 1.9409.agl $\times$ 1.9409.ac. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{97^{3}}$
The base change of $A$ to $\F_{97^{3}}$ is 1.912673.abze $\times$ 1.912673.bze. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.