Invariants
| Base field: | $\F_{97}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 181 x^{2} + 9409 x^{4}$ |
| Frobenius angles: | $\pm0.0585951244730$, $\pm0.941404875527$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-13}, \sqrt{15})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $12$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $9229$ | $85174441$ | $832971184276$ | $7834965238284921$ | $73742412694074247189$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $98$ | $9048$ | $912674$ | $88501396$ | $8587340258$ | $832970363622$ | $80798284478114$ | $7837433559679588$ | $760231058654565218$ | $73742412698655668328$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=23 x^6+34 x^5+12 x^4+26 x^3+82 x^2+41 x+93$
- $y^2=49 x^6+49 x^5+22 x^4+27 x^3+29 x^2+16 x+74$
- $y^2=42 x^6+64 x^5+73 x^4+60 x^3+92 x^2+89 x+8$
- $y^2=81 x^6+45 x^5+64 x^4+28 x^3+20 x^2+51 x+59$
- $y^2=73 x^6+12 x^5+87 x^4+28 x^3+43 x^2+24 x+90$
- $y^2=26 x^6+20 x^5+91 x^4+26 x^3+23 x^2+46 x+65$
- $y^2=70 x^6+19 x^5+10 x^4+43 x^3+36 x^2+60 x+30$
- $y^2=33 x^6+23 x^5+75 x^4+81 x^3+83 x^2+51 x+55$
- $y^2=52 x^6+47 x^5+76 x^4+88 x^3+25 x^2+18 x+50$
- $y^2=29 x^6+60 x^5+93 x^4+55 x^3+29 x^2+74 x+81$
- $y^2=5 x^6+6 x^5+59 x^4+93 x^3+3 x^2+48 x+53$
- $y^2=42 x^6+22 x^5+93 x^4+68 x^2+53 x+70$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{97^{2}}$.
Endomorphism algebra over $\F_{97}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-13}, \sqrt{15})\). |
| The base change of $A$ to $\F_{97^{2}}$ is 1.9409.agz 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-195}) \)$)$ |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.97.a_gz | $4$ | (not in LMFDB) |