Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 26 x^{2} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.111442130273$, $\pm0.888557869727$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{15})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $8$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $264$ | $69696$ | $24142536$ | $6959563776$ | $2015996558664$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $238$ | $4914$ | $83326$ | $1419858$ | $24147502$ | $410338674$ | $6976072318$ | $118587876498$ | $2015999216878$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=4 x^6+10 x^5+6 x^3+x^2+x+11$
- $y^2=16 x^6+16 x^5+14 x^3+3 x^2+1$
- $y^2=14 x^6+9 x^4+6 x^3+10 x^2+11 x+4$
- $y^2=x^6+16 x^5+13 x^4+4 x^3+7 x^2+3 x+13$
- $y^2=13 x^6+2 x^5+2 x^4+3 x^3+3 x^2+9 x+10$
- $y^2=11 x^6+x^5+6 x^4+8 x^2+4 x+2$
- $y^2=5 x^6+6 x^5+2 x^4+3 x^3+10 x^2+14 x+13$
- $y^2=14 x^6+6 x^5+2 x^4+15 x^3+11 x^2+3 x+13$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{2}}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{15})\). |
| The base change of $A$ to $\F_{17^{2}}$ is 1.289.aba 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-30}) \)$)$ |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.17.a_ba | $4$ | (not in LMFDB) |
| 2.17.ae_i | $8$ | (not in LMFDB) |
| 2.17.e_i | $8$ | (not in LMFDB) |