Properties

Label 2.2.a_ad
Base Field $\F_{2}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2}$
Dimension:  $2$
L-polynomial:  $1 - 3 x^{2} + 4 x^{4}$
Frobenius angles:  $\pm0.115026728081$, $\pm0.884973271919$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{7})\)
Galois group:  $C_2^2$
Jacobians:  0

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 is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2 4 74 256 1082 5476 16298 82944 261146 1170724

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 -1 9 15 33 83 129 319 513 1139

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{7})\).
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.2.ac_f$4$2.16.ac_bh
2.2.a_d$4$2.16.ac_bh
2.2.c_f$4$2.16.ac_bh
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.2.ac_f$4$2.16.ac_bh
2.2.a_d$4$2.16.ac_bh
2.2.c_f$4$2.16.ac_bh
2.2.ab_ab$12$(not in LMFDB)
2.2.b_ab$12$(not in LMFDB)