Properties

Label 2.2.a_e
Base field $\F_{2}$
Dimension $2$
$p$-rank $0$
Ordinary No
Supersingular Yes
Simple No
Geometrically simple No
Primitive Yes
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2}$
Dimension:  $2$
L-polynomial:  $( 1 + 2 x^{2} )^{2}$
Frobenius angles:  $\pm0.5$, $\pm0.5$
Angle rank:  $0$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

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
$A(\F_{q^r})$ 9 81 81 81 1089 6561 16641 50625 263169 1185921

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 13 9 1 33 97 129 193 513 1153

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.e 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
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.a_ac$3$2.8.a_q
2.2.a_ae$4$2.16.aq_ds
2.2.ae_i$8$2.256.acm_chc
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.2.a_ac$3$2.8.a_q
2.2.a_ae$4$2.16.aq_ds
2.2.ae_i$8$2.256.acm_chc
2.2.ac_e$8$2.256.acm_chc
2.2.a_a$8$2.256.acm_chc
2.2.c_e$8$2.256.acm_chc
2.2.e_i$8$2.256.acm_chc
2.2.a_c$12$(not in LMFDB)
2.2.ac_c$24$(not in LMFDB)
2.2.c_c$24$(not in LMFDB)