Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 8 x + 32 x^{2} - 136 x^{3} + 289 x^{4}$ |
Frobenius angles: | $\pm0.00936746300954$, $\pm0.509367463010$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\zeta_{8})\) |
Galois group: | $C_2^2$ |
Jacobians: | $2$ |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $178$ | $82948$ | $23402194$ | $6880370704$ | $2013144841618$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $290$ | $4762$ | $82374$ | $1417850$ | $24137570$ | $410290730$ | $6975432574$ | $118587392074$ | $2015993900450$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^5+8x$
- $y^2=6x^6+x^5+12x^4+12x^2+16x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{4}}$.
Endomorphism algebra over $\F_{17}$The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\). |
The base change of $A$ to $\F_{17^{4}}$ is 1.83521.awc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
- Endomorphism algebra over $\F_{17^{2}}$
The base change of $A$ to $\F_{17^{2}}$ is the simple isogeny class 2.289.a_awc and its endomorphism algebra is \(\Q(\zeta_{8})\).
Base change
This is a primitive isogeny class.