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: | $\pm0.529596959677$, $\pm0.529596959677$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $961$ | $808201$ | $590684416$ | $498033661225$ | $420873882374521$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $956$ | $24218$ | $704148$ | $20519272$ | $594906086$ | $17249558008$ | $500244331108$ | $14507157288482$ | $420707282361356$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):
- $y^2=6 x^6+15 x^5+15 x^4+6 x^3+10 x^2+26 x+5$
- $y^2=6 x^6+16 x^5+12 x^4+4 x^3+8 x^2+10 x+20$
- $y^2=x^6+23 x^5+23 x^4+21 x^3+x^2+24 x+20$
- $y^2=15 x^6+3 x^5+15 x^4+2 x^3+2 x^2+24 x+17$
- $y^2=10 x^6+9 x^5+28 x^4+14 x^3+5 x^2+22 x+26$
- $y^2=11 x^6+26 x^5+9 x^4+6 x^3+5 x^2+21 x+22$
- $y^2=6 x^6+20 x^5+5 x^4+7 x^3+17 x^2+17 x+19$
- $y^2=10 x^6+x^5+8 x^4+21 x^2+x+19$
- $y^2=20 x^6+13 x^5+10 x^4+15 x^3+12 x^2+10 x+11$
- $y^2=26 x^6+17 x^5+3 x^4+13 x^3+17 x^2+11 x+18$
- $y^2=11 x^6+19 x^5+22 x^4+5 x^3+15 x^2+x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The isogeny class factors as 1.29.b 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-115}) \)$)$ |
Base change
This is a primitive isogeny class.