Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 8 x + 29 x^{2} )^{2}$ |
| $1 + 16 x + 122 x^{2} + 464 x^{3} + 841 x^{4}$ | |
| Frobenius angles: | $\pm0.766493812366$, $\pm0.766493812366$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1444$ | $698896$ | $585930436$ | $502578909184$ | $420386049009124$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $46$ | $830$ | $24022$ | $710574$ | $20495486$ | $594853166$ | $17250091814$ | $500243823454$ | $14507160442318$ | $420707192664350$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=21 x^6+27 x^4+27 x^2+21$
- $y^2=7 x^6+27 x^5+12 x^4+22 x^3+12 x^2+27 x+7$
- $y^2=16 x^6+2 x^5+2 x^4+13 x^3+2 x^2+2 x+16$
- $y^2=16 x^6+21 x^5+24 x^4+23 x^3+22 x^2+23 x$
- $y^2=21 x^6+19 x^5+28 x^4+9 x^3+5 x^2+4 x+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29}$.
Endomorphism algebra over $\F_{29}$| The isogeny class factors as 1.29.i 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$ |
Base change
This is a primitive isogeny class.