Invariants
| Base field: | $\F_{97}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 97 x^{2} )^{2}$ |
| $1 - 194 x^{2} + 9409 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $1$, $1$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{97}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $19$ |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $9216$ | $84934656$ | $832970179584$ | $7834102237495296$ | $73742412672318145536$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $98$ | $9022$ | $912674$ | $88491646$ | $8587340258$ | $832968354238$ | $80798284478114$ | $7837433240259838$ | $760231058654565218$ | $73742412655143465022$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 19 curves (of which all are hyperelliptic):
- $y^2=7 x^6+74 x^5+10 x^4+24 x^3+96 x^2+44 x+3$
- $y^2=x^6+17 x^3+63$
- $y^2=55 x^6+81 x^5+34 x^4+36 x^3+88 x^2+49 x+19$
- $y^2=65 x^6+62 x^5+87 x^4+3 x^3+3 x^2+89 x+87$
- $y^2=x^6+x^3+63$
- $y^2=87 x^6+89 x^5+38 x^4+55 x^2+31 x+39$
- $y^2=90 x^6+87 x^5+34 x^4+62 x^3+35 x^2+8 x+38$
- $y^2=62 x^6+47 x^5+73 x^4+19 x^3+78 x^2+40 x+93$
- $y^2=34 x^6+48 x^4+95 x^3+37 x^2+70$
- $y^2=47 x^6+54 x^5+67 x^4+28 x^3+88 x^2+25 x+19$
- $y^2=41 x^6+76 x^5+44 x^4+43 x^3+52 x^2+28 x+95$
- $y^2=79 x^6+79 x^5+56 x^4+14 x^3+79 x^2+36 x+30$
- $y^2=x^6+x^3+69$
- $y^2=6 x^6+8 x^5+33 x^4+75 x^3+82 x^2+73 x+17$
- $y^2=37 x^6+88 x^5+64 x^4+31 x^3+83 x^2+8 x+53$
- $y^2=85 x^6+96 x^5+11 x^4+62 x^3+57 x^2+95 x+52$
- $y^2=20 x^6+92 x^5+44 x^4+71 x^3+60 x^2+38 x+22$
- $y^2=20 x^6+90 x^5+81 x^4+20 x^3+52 x^2+17 x+47$
- $y^2=x^6+x^3+34$
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 the quaternion algebra over \(\Q(\sqrt{97}) \) ramified at both real infinite places. |
| The base change of $A$ to $\F_{97^{2}}$ is 1.9409.ahm 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $97$ and $\infty$. |
Base change
This is a primitive isogeny class.