Properties

Label 2.64.abb_lv
Base Field $\F_{2^{6}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
L-polynomial:  $1 - 27 x + 307 x^{2} - 1728 x^{3} + 4096 x^{4}$
Frobenius angles:  $\pm0.0943151045543$, $\pm0.239018228779$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{13})\)
Galois group:  $C_2^2$
Jacobians:  14

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 14 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2649 16315191 68719584492 281559424997691 1152972791571810999 4722381292753126898064 19342812465359708165783589 79228160199598844333193068979 324518553658426705819712766907092 1329227997384074407249099923104502951

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 38 3982 262145 16782250 1073789588 68719692247 4398046363658 281474968487314 18014398509481985 1152921505993895902

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{13})\).
Endomorphism algebra over $\overline{\F}_{2^{6}}$
The base change of $A$ to $\F_{2^{36}}$ is 1.68719476736.gdkl 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{36}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.bb_lv$2$(not in LMFDB)
2.64.a_el$3$(not in LMFDB)
2.64.bb_lv$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.bb_lv$2$(not in LMFDB)
2.64.a_el$3$(not in LMFDB)
2.64.bb_lv$3$(not in LMFDB)
2.64.a_ael$12$(not in LMFDB)