Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 4 x + 29 x^{2} )^{2}$ |
| $1 + 8 x + 74 x^{2} + 232 x^{3} + 841 x^{4}$ | |
| Frobenius angles: | $\pm0.621118941591$, $\pm0.621118941591$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $9$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1156$ | $781456$ | $581099236$ | $500131840000$ | $421058662641796$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $38$ | $926$ | $23822$ | $707118$ | $20528278$ | $594759566$ | $17249634622$ | $500249228638$ | $14507141722118$ | $420707168660606$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=22 x^6+27 x^5+25 x^4+9 x^3+x^2+18 x+6$
- $y^2=24 x^6+4 x^4+4 x^2+24$
- $y^2=19 x^6+15 x^5+18 x^4+2 x^3+12 x^2+26 x+11$
- $y^2=14 x^6+19 x^5+16 x^4+24 x^3+3 x^2+27 x+3$
- $y^2=23 x^6+2 x^5+15 x^4+6 x^3+9 x^2+25 x+22$
- $y^2=21 x^6+18 x^5+23 x^4+16 x^3+7 x^2+10 x+17$
- $y^2=13 x^6+15 x^5+13 x^4+5 x^3+11 x^2+17 x+6$
- $y^2=10 x^6+17 x^5+13 x^4+16 x^3+28 x^2+14 x+18$
- $y^2=13 x^6+27 x^4+9 x^3+5 x^2+18 x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The isogeny class factors as 1.29.e 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.