Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - x + 29 x^{2} )^{2}$ |
| $1 - 2 x + 59 x^{2} - 58 x^{3} + 841 x^{4}$ | |
| Frobenius angles: | $\pm0.470403040323$, $\pm0.470403040323$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $11$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $841$ | $808201$ | $599074576$ | $498033661225$ | $420540699253921$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $956$ | $24562$ | $704148$ | $20503028$ | $594906086$ | $17250194612$ | $500244331108$ | $14507134663258$ | $420707282361356$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):
- $y^2=3 x^6+22 x^5+22 x^4+3 x^3+5 x^2+13 x+17$
- $y^2=3 x^6+8 x^5+6 x^4+2 x^3+4 x^2+5 x+10$
- $y^2=15 x^6+26 x^5+26 x^4+25 x^3+15 x^2+12 x+10$
- $y^2=22 x^6+16 x^5+22 x^4+x^3+x^2+12 x+23$
- $y^2=5 x^6+19 x^5+14 x^4+7 x^3+17 x^2+11 x+13$
- $y^2=22 x^6+23 x^5+18 x^4+12 x^3+10 x^2+13 x+15$
- $y^2=12 x^6+11 x^5+10 x^4+14 x^3+5 x^2+5 x+9$
- $y^2=20 x^6+2 x^5+16 x^4+13 x^2+2 x+9$
- $y^2=10 x^6+21 x^5+5 x^4+22 x^3+6 x^2+5 x+20$
- $y^2=13 x^6+23 x^5+16 x^4+21 x^3+23 x^2+20 x+9$
- $y^2=22 x^6+9 x^5+15 x^4+10 x^3+x^2+2 x+22$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The isogeny class factors as 1.29.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-115}) \)$)$ |
Base change
This is a primitive isogeny class.