Properties

Label 2.2.ac_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:  $( 1 - 2 x + 2 x^{2} )( 1 + 2 x^{2} )$
  $1 - 2x + 4x^{2} - 4x^{3} + 4x^{4}$
Frobenius angles:  $\pm0.250000000000$, $\pm0.5$
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 Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $3$ $45$ $117$ $225$ $1353$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $9$ $13$ $17$ $41$ $81$ $113$ $193$ $481$ $1089$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac $\times$ 1.2.a 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.c_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.c_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)