Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 10 x + 29 x^{2} )( 1 - 5 x + 29 x^{2} )$ |
| $1 - 15 x + 108 x^{2} - 435 x^{3} + 841 x^{4}$ | |
| Frobenius angles: | $\pm0.121118941591$, $\pm0.346328109963$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $6$ |
| Isomorphism classes: | 40 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $500$ | $700000$ | $599222000$ | $500609200000$ | $420644184762500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $833$ | $24570$ | $707793$ | $20508075$ | $594807878$ | $17250047655$ | $500248883713$ | $14507160399810$ | $420707270341673$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=17 x^6+10 x^5+x^4+16 x^3+25 x^2+22 x+21$
- $y^2=11 x^6+27 x^5+8 x^4+7 x^3+26 x^2+22 x+26$
- $y^2=13 x^5+16 x^4+10 x^3+27 x^2+13 x+11$
- $y^2=10 x^6+17 x^5+24 x^4+12 x^3+28 x^2+x+26$
- $y^2=13 x^6+2 x^5+27 x^4+27 x^2+6 x+20$
- $y^2=21 x^6+4 x^4+20 x^3+x^2+4 x+27$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The isogeny class factors as 1.29.ak $\times$ 1.29.af 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.