Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 4 x + 29 x^{2} )( 1 - 2 x + 29 x^{2} )$ |
| $1 - 6 x + 66 x^{2} - 174 x^{3} + 841 x^{4}$ | |
| Frobenius angles: | $\pm0.378881058409$, $\pm0.440546251002$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $10$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $728$ | $792064$ | $605894744$ | $499317145600$ | $420382317002648$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $938$ | $24840$ | $705966$ | $20495304$ | $594812666$ | $17250250776$ | $500247712606$ | $14507140531320$ | $420707188975178$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=26 x^6+6 x^5+27 x^4+18 x^3+27 x^2+6 x+26$
- $y^2=23 x^6+22 x^5+10 x^4+16 x^3+10 x^2+22 x+23$
- $y^2=27 x^6+26 x^5+x^4+24 x^3+13 x^2+15 x+14$
- $y^2=4 x^6+4 x^5+5 x^4+12 x^3+4 x^2+13 x+22$
- $y^2=18 x^6+13 x^5+15 x^4+15 x^2+13 x+18$
- $y^2=6 x^5+22 x^4+18 x^3+22 x^2+6 x$
- $y^2=17 x^6+4 x^5+18 x^4+7 x^3+18 x^2+4 x+17$
- $y^2=18 x^6+23 x^5+2 x^4+25 x^3+21 x^2+20 x+8$
- $y^2=13 x^6+22 x^5+10 x^4+25 x^3+17 x^2+5 x+7$
- $y^2=18 x^6+3 x^5+22 x^4+12 x^3+22 x^2+3 x+18$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The isogeny class factors as 1.29.ae $\times$ 1.29.ac 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.