Invariants
Base field: | $\F_{2}$ |
Dimension: | $2$ |
L-polynomial: | $1 + x^{2} + 4 x^{4}$ |
Frobenius angles: | $\pm0.290215311628$, $\pm0.709784688372$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{3}, \sqrt{-5})\) |
Galois group: | $C_2^2$ |
Jacobians: | $1$ |
Isomorphism classes: | 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $6$ | $36$ | $54$ | $576$ | $1086$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $7$ | $9$ | $31$ | $33$ | $43$ | $129$ | $223$ | $513$ | $1147$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+1)y=x^5+x^3+x^2+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{3}, \sqrt{-5})\). |
The base change of $A$ to $\F_{2^{2}}$ is 1.4.b 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
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 |
---|---|---|
2.2.a_ab | $4$ | 2.16.o_dd |
2.2.ad_f | $12$ | (not in LMFDB) |
2.2.d_f | $12$ | (not in LMFDB) |