Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 6 x + 29 x^{2} )^{2}$ |
| $1 + 12 x + 94 x^{2} + 348 x^{3} + 841 x^{4}$ | |
| Frobenius angles: | $\pm0.688080637848$, $\pm0.688080637848$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $26$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1296$ | $746496$ | $580039056$ | $501943910400$ | $420776440762896$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $886$ | $23778$ | $709678$ | $20514522$ | $594733606$ | $17250316818$ | $500246371678$ | $14507133448842$ | $420707309659606$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):
- $y^2=23 x^6+15 x^5+13 x^4+12 x^3+22 x^2+27 x+25$
- $y^2=17 x^6+10 x^4+10 x^2+17$
- $y^2=17 x^6+24 x^5+26 x^4+18 x^3+11 x^2+23 x+18$
- $y^2=23 x^6+4 x^5+9 x^4+13 x^3+16 x^2+13 x+4$
- $y^2=6 x^6+26 x^5+7 x^4+x^3+25 x^2+15 x+13$
- $y^2=8 x^5+7 x^4+16 x^3+24 x^2+10 x$
- $y^2=22 x^6+10 x^5+24 x^4+3 x^3+25 x^2+25 x+5$
- $y^2=22 x^6+9 x^5+22 x^4+25 x^3+22 x^2+9 x+22$
- $y^2=11 x^6+19 x^4+19 x^2+11$
- $y^2=26 x^6+23 x^4+23 x^2+26$
- $y^2=28 x^6+20 x^5+26 x^4+16 x^3+26 x^2+20 x+28$
- $y^2=4 x^6+17 x^4+17 x^2+4$
- $y^2=9 x^6+19 x^5+2 x^4+18 x^3+15 x^2+3 x+16$
- $y^2=27 x^5+22 x^4+17 x^3+7 x^2+27 x$
- $y^2=27 x^6+28 x^5+2 x^4+5 x^3+11 x^2+6 x+8$
- $y^2=5 x^6+13 x^5+7 x^4+10 x^3+7 x^2+13 x+5$
- $y^2=22 x^6+11 x^5+23 x^4+15 x^3+23 x^2+11 x+22$
- $y^2=3 x^6+7 x^5+3 x^3+23 x+8$
- $y^2=25 x^5+12 x^4+22 x^3+3 x^2+7 x$
- $y^2=19 x^6+14 x^5+24 x^4+5 x^3+5 x^2+14 x+10$
- $y^2=7 x^6+13 x^5+24 x^4+27 x^3+28 x^2+4 x+4$
- $y^2=19 x^6+12 x^5+18 x^4+26 x^3+27 x^2+27 x+17$
- $y^2=7 x^6+27 x^5+19 x^4+13 x^3+26 x^2+12 x+28$
- $y^2=6 x^6+17 x^5+20 x^4+7 x^3+24 x^2+21 x+28$
- $y^2=x^6+11 x^4+11 x^2+1$
- $y^2=26 x^6+3 x^5+19 x^4+16 x^3+19 x^2+3 x+26$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The isogeny class factors as 1.29.g 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$ |
Base change
This is a primitive isogeny class.