Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 8 x + 35 x^{2} + 232 x^{3} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.433160479032$, $\pm0.900172854301$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-13})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 36 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1117$ | $711529$ | $603881476$ | $499084228825$ | $420546630802477$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $38$ | $848$ | $24758$ | $705636$ | $20503318$ | $594853166$ | $17249984062$ | $500247707716$ | $14507131509422$ | $420707253618128$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=7 x^6+5 x^5+8 x^4+15 x^3+x^2+5 x+20$
- $y^2=5 x^6+23 x^5+18 x^4+6 x^3+9 x^2+13 x+4$
- $y^2=17 x^6+26 x^5+20 x^4+16 x^3+9 x^2+22 x+12$
- $y^2=23 x^6+13 x^5+7 x^4+x^3+27 x^2+5 x+14$
- $y^2=9 x^6+6 x^5+7 x^4+23 x^3+15 x^2+8 x+13$
- $y^2=16 x^6+15 x^5+22 x^4+20 x^3+7 x^2+21 x+28$
- $y^2=6 x^6+23 x^5+15 x^4+13 x^3+2 x+14$
- $y^2=22 x^6+20 x^5+21 x^4+20 x^3+12 x^2+13 x+25$
- $y^2=28 x^6+15 x^5+10 x^4+20 x^3+23 x^2+14 x+22$
- $y^2=16 x^6+26 x^5+13 x^4+11 x^3+21 x^2+2 x+6$
- $y^2=25 x^6+7 x^5+27 x^4+28 x^3+20 x^2+7 x+20$
- $y^2=27 x^6+18 x^5+6 x^4+5 x^3+2 x^2+3 x+17$
- $y^2=24 x^6+16 x^5+11 x^4+18 x^3+21 x^2+4 x+4$
- $y^2=16 x^6+2 x^5+3 x^4+7 x^3+2 x^2+16 x+5$
- $y^2=10 x^6+15 x^5+13 x^4+20 x^3+25 x^2+11 x+5$
- $y^2=22 x^6+12 x^5+9 x^4+21 x^3+9 x+17$
- $y^2=25 x^6+27 x^5+19 x^4+4 x^3+13 x^2+10 x+25$
- $y^2=11 x^6+15 x^5+x^4+24 x^3+10 x+14$
- $y^2=20 x^6+12 x^5+5 x^4+20 x^3+15 x^2+13 x+9$
- $y^2=22 x^6+2 x^5+28 x^4+11 x^3+15 x^2+6 x+20$
- $y^2=28 x^6+23 x^5+14 x^4+3 x^3+x^2+28 x+10$
- $y^2=18 x^6+21 x^5+17 x^4+17 x^3+11 x^2+8 x+6$
- $y^2=12 x^6+8 x^5+x^4+12 x^3+x^2+17 x+28$
- $y^2=22 x^6+2 x^5+18 x^4+22 x^3+x^2+22 x+9$
- $y^2=6 x^6+7 x^5+28 x^4+5 x^2+22 x+7$
- $y^2=3 x^6+9 x^5+8 x^4+4 x^3+7 x^2+17 x+27$
- $y^2=11 x^6+25 x^5+19 x^4+24 x^3+24 x^2+24 x+3$
- $y^2=4 x^6+13 x^5+8 x^3+2 x^2+5 x+1$
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 \(\Q(\sqrt{-3}, \sqrt{-13})\). |
| The base change of $A$ to $\F_{29^{3}}$ is 1.24389.hc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$ |
Base change
This is a primitive isogeny class.