Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 9 x + 29 x^{2} )^{2}$ |
| $1 - 18 x + 139 x^{2} - 522 x^{3} + 841 x^{4}$ | |
| Frobenius angles: | $\pm0.185103371333$, $\pm0.185103371333$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $5$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3, 7$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $441$ | $670761$ | $597509136$ | $501880149225$ | $421068799441521$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $796$ | $24498$ | $709588$ | $20528772$ | $594915046$ | $17250190788$ | $500246583268$ | $14507138388762$ | $420707160077356$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=16 x^6+16 x^5+16 x^4+13 x^3+23 x^2+24 x+20$
- $y^2=18 x^6+23 x^5+3 x^4+16 x^3+10 x^2+x+19$
- $y^2=x^6+23 x^5+28 x^4+10 x^3+28 x^2+23 x+1$
- $y^2=19 x^6+20 x^5+11 x^4+26 x^3+8 x^2+24 x+15$
- $y^2=17 x^6+17 x^5+6 x^4+4 x^3+6 x^2+17 x+17$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The isogeny class factors as 1.29.aj 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-35}) \)$)$ |
Base change
This is a primitive isogeny class.