Properties

Label 2.2.c_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 Yes

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Invariants

Base field:  $\F_{2}$
Dimension:  $2$
L-polynomial:  $( 1 + 2 x^{2} )( 1 + 2 x + 2 x^{2} )$
Frobenius angles:  $\pm0.5$, $\pm0.750000000000$
Angle rank:  $0$ (numerical)
Jacobians:  1

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 contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 15 45 45 225 825 5265 18705 50625 279585 1116225

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 9 5 17 25 81 145 193 545 1089

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.a $\times$ 1.2.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg 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 $\F_{2^{8}}$.
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
2.2.ac_e$2$2.4.e_i
2.2.ae_i$8$2.256.acm_chc
2.2.a_ae$8$2.256.acm_chc
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.2.ac_e$2$2.4.e_i
2.2.ae_i$8$2.256.acm_chc
2.2.a_ae$8$2.256.acm_chc
2.2.a_a$8$2.256.acm_chc
2.2.a_e$8$2.256.acm_chc
2.2.e_i$8$2.256.acm_chc
2.2.ac_c$24$(not in LMFDB)
2.2.a_ac$24$(not in LMFDB)
2.2.a_c$24$(not in LMFDB)
2.2.c_c$24$(not in LMFDB)