Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 2 x - 25 x^{2} - 58 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.107212917668$, $\pm0.773879584335$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $8$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $757$ | $663889$ | $586802176$ | $501120014425$ | $420557903622277$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $788$ | $24058$ | $708516$ | $20503868$ | $594865766$ | $17250129932$ | $500246521156$ | $14507161118722$ | $420707245305428$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=6 x^6+2 x^5+27 x^4+26 x^3+11 x^2+24 x+22$
- $y^2=14 x^6+11 x^5+3 x^4+6 x^3+x^2+10 x+1$
- $y^2=6 x^6+x^5+23 x^4+11 x^3+25 x^2+10 x+10$
- $y^2=27 x^6+14 x^5+27 x^4+15 x^3+22 x^2+16 x+14$
- $y^2=27 x^6+15 x^5+x^4+5 x^3+10 x^2+2 x+18$
- $y^2=4 x^6+23 x^5+12 x^4+27 x^3+26 x^2+18 x+13$
- $y^2=21 x^6+3 x^5+26 x^4+11 x^3+27 x^2+9 x+1$
- $y^2=20 x^6+16 x^5+18 x^4+13 x^3+10 x^2+23 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29^{3}}$.
Endomorphism algebra over $\F_{29}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\). |
| The base change of $A$ to $\F_{29^{3}}$ is 1.24389.agk 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This is a primitive isogeny class.