Invariants
| Base field: | $\F_{29}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 54 x^{2} + 841 x^{4}$ |
| Frobenius angles: | $\pm0.0594537489984$, $\pm0.940546251002$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{7})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 12 |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $788$ | $620944$ | $594802100$ | $498503778304$ | $420707245305428$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $734$ | $24390$ | $704814$ | $20511150$ | $594780878$ | $17249876310$ | $500246196574$ | $14507145975870$ | $420707257310654$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=8 x^6+27 x^5+15 x^4+13 x^2+11 x+16$
- $y^2=x^6+9 x^5+12 x^4+19 x^2+x+6$
- $y^2=19 x^6+17 x^5+21 x^4+3 x^3+23 x^2+15 x+13$
- $y^2=x^6+13 x^5+12 x^4+19 x^2+24 x+6$
- $y^2=26 x^6+17 x^5+13 x^4+25 x^2+8 x+14$
- $y^2=x^6+2 x^5+11 x^4+21 x^2+15 x+16$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{29^{2}}$.
Endomorphism algebra over $\F_{29}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{7})\). |
| The base change of $A$ to $\F_{29^{2}}$ is 1.841.acc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This is a primitive isogeny class.