Invariants
| Base field: | $\F_{97}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 19 x + 97 x^{2} )( 1 - 16 x + 97 x^{2} )$ |
| $1 - 35 x + 498 x^{2} - 3395 x^{3} + 9409 x^{4}$ | |
| Frobenius angles: | $\pm0.0849741350078$, $\pm0.198227810371$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6478$ | $86403564$ | $832270326496$ | $7837956223958016$ | $73743631456904466718$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $63$ | $9181$ | $911904$ | $88535185$ | $8587482183$ | $832973573122$ | $80798295971799$ | $7837433641990369$ | $760231058585818848$ | $73742412687641168461$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=35 x^6+75 x^5+72 x^4+21 x^3+47 x^2+74 x+23$
- $y^2=15 x^6+38 x^5+56 x^4+51 x^3+x^2+21 x+29$
- $y^2=68 x^6+63 x^5+81 x^4+8 x^3+x^2+61 x+72$
- $y^2=53 x^6+15 x^5+20 x^4+43 x^3+65 x^2+18 x+72$
- $y^2=85 x^6+15 x^5+62 x^4+61 x^3+90 x^2+62 x+80$
- $y^2=68 x^6+91 x^5+80 x^4+33 x^3+54 x^2+76 x+39$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{97}$.
Endomorphism algebra over $\F_{97}$| The isogeny class factors as 1.97.at $\times$ 1.97.aq and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.