Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 2 x + 29 x^{2} )( 1 + 4 x + 29 x^{2} )$ |
| $1 + 6 x + 66 x^{2} + 174 x^{3} + 841 x^{4}$ | |
| Frobenius angles: | $\pm0.559453748998$, $\pm0.621118941591$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $10$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1088$ | $792064$ | $583943744$ | $499317145600$ | $421032356368448$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $36$ | $938$ | $23940$ | $705966$ | $20526996$ | $594812666$ | $17249501844$ | $500247712606$ | $14507151420420$ | $420707188975178$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=23 x^6+12 x^5+25 x^4+7 x^3+25 x^2+12 x+23$
- $y^2=26 x^6+11 x^5+5 x^4+8 x^3+5 x^2+11 x+26$
- $y^2=25 x^6+23 x^5+2 x^4+19 x^3+26 x^2+x+28$
- $y^2=2 x^6+2 x^5+17 x^4+6 x^3+2 x^2+21 x+11$
- $y^2=7 x^6+26 x^5+x^4+x^2+26 x+7$
- $y^2=12 x^5+15 x^4+7 x^3+15 x^2+12 x$
- $y^2=23 x^6+2 x^5+9 x^4+18 x^3+9 x^2+2 x+23$
- $y^2=9 x^6+26 x^5+x^4+27 x^3+25 x^2+10 x+4$
- $y^2=21 x^6+11 x^5+5 x^4+27 x^3+23 x^2+17 x+18$
- $y^2=9 x^6+16 x^5+11 x^4+6 x^3+11 x^2+16 x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The isogeny class factors as 1.29.c $\times$ 1.29.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.