Invariants
| Base field: | $\F_{97}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 122 x^{2} + 9409 x^{4}$ |
| Frobenius angles: | $\pm0.141759705500$, $\pm0.858240294500$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{79})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $216$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $9288$ | $86266944$ | $832973632776$ | $7838130335302656$ | $73742412693889452168$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $98$ | $9166$ | $912674$ | $88537150$ | $8587340258$ | $832975260622$ | $80798284478114$ | $7837433917541374$ | $760231058654565218$ | $73742412698286078286$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 216 curves (of which all are hyperelliptic):
- $y^2=88 x^6+x^5+5 x^4+40 x^3+54 x^2+7 x+10$
- $y^2=52 x^6+5 x^5+25 x^4+6 x^3+76 x^2+35 x+50$
- $y^2=76 x^6+59 x^5+66 x^4+83 x^3+79 x^2+54 x+46$
- $y^2=89 x^6+4 x^5+39 x^4+27 x^3+7 x^2+76 x+36$
- $y^2=62 x^6+16 x^5+71 x^4+53 x^3+66 x^2+x+16$
- $y^2=19 x^6+80 x^5+64 x^4+71 x^3+39 x^2+5 x+80$
- $y^2=61 x^6+24 x^5+47 x^4+38 x^3+42 x^2+36 x+37$
- $y^2=41 x^6+78 x^4+65 x^3+64 x^2+94 x+77$
- $y^2=11 x^6+2 x^4+34 x^3+29 x^2+82 x+94$
- $y^2=65 x^6+8 x^5+24 x^4+19 x^3+51 x^2+24 x+87$
- $y^2=27 x^6+x^5+21 x^4+60 x^3+60 x^2+68 x+51$
- $y^2=38 x^6+5 x^5+8 x^4+9 x^3+9 x^2+49 x+61$
- $y^2=51 x^6+40 x^5+70 x^4+2 x^3+4 x^2+94 x+47$
- $y^2=61 x^6+6 x^5+59 x^4+10 x^3+20 x^2+82 x+41$
- $y^2=65 x^6+58 x^5+7 x^4+13 x^3+71 x^2+15 x+33$
- $y^2=34 x^6+96 x^5+35 x^4+65 x^3+64 x^2+75 x+68$
- $y^2=40 x^6+53 x^5+59 x^4+89 x^3+31 x^2+67 x+35$
- $y^2=6 x^6+71 x^5+4 x^4+57 x^3+58 x^2+44 x+78$
- $y^2=67 x^6+65 x^5+82 x^4+15 x^3+31 x^2+75 x+79$
- $y^2=37 x^6+25 x^5+24 x^4+69 x^3+57 x^2+82 x+22$
- and 196 more
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 \(\Q(\sqrt{-2}, \sqrt{79})\). |
| The base change of $A$ to $\F_{97^{2}}$ is 1.9409.aes 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-158}) \)$)$ |
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_es | $4$ | (not in LMFDB) |
| 2.97.am_cu | $8$ | (not in LMFDB) |
| 2.97.m_cu | $8$ | (not in LMFDB) |