Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 8 x^{2} - 36 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.0937471905441$, $\pm0.593747190544$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(i, \sqrt{14})\) |
Galois group: | $C_2^2$ |
Jacobians: | $6$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $50$ | $6500$ | $478850$ | $42250000$ | $3512625250$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $82$ | $654$ | $6438$ | $59486$ | $531442$ | $4781174$ | $43065278$ | $387473766$ | $3486784402$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+2)x^6+(a+2)x^5+(a+1)x^4+(a+1)x^2+(2a+1)x+a+2$
- $y^2=(2a+1)x^6+(a+2)x^5+(a+1)x^4+(a+1)x^2+(2a+1)x+2a+1$
- $y^2=(a+2)x^6+(a+2)x^5+ax^4+(2a+1)x^3+x$
- $y^2=(2a+2)x^6+(a+1)x^5+(2a+1)x^4+(a+2)x^3+x^2+(2a+2)x+2a+2$
- $y^2=(a+2)x^5+(a+1)x^4+ax^3+(a+2)x^2+(a+2)x+a+2$
- $y^2=2ax^6+2ax^5+(2a+1)x^4+ax^3+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{8}}$.
Endomorphism algebra over $\F_{3^{2}}$The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{14})\). |
The base change of $A$ to $\F_{3^{8}}$ is 1.6561.ack 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-14}) \)$)$ |
- Endomorphism algebra over $\F_{3^{4}}$
The base change of $A$ to $\F_{3^{4}}$ is the simple isogeny class 2.81.a_ack and its endomorphism algebra is \(\Q(i, \sqrt{14})\).
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 |
---|---|---|
2.9.e_i | $2$ | 2.81.a_ack |
2.9.a_ak | $8$ | (not in LMFDB) |
2.9.a_k | $8$ | (not in LMFDB) |