Properties

Label 2.2.a_ac
Base field $\F_{2}$
Dimension $2$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple yes
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:  $1 - 2 x^{2} + 4 x^{4}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.833333333333$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-2}, \sqrt{-3})\)
Galois group:  $C_2^2$
Jacobians:  0

This isogeny class is simple but not geometrically 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $3$ $9$ $81$ $441$ $993$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $1$ $9$ $25$ $33$ $97$ $129$ $289$ $513$ $961$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-3})\).
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.q 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 $\F_{2^{6}}$.
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.a_e$3$2.8.a_q
2.2.a_c$4$2.16.i_bw
2.2.ac_c$8$2.256.bg_bdo
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.a_e$3$2.8.a_q
2.2.a_c$4$2.16.i_bw
2.2.ac_c$8$2.256.bg_bdo
2.2.c_c$8$2.256.bg_bdo
2.2.a_ae$12$(not in LMFDB)
2.2.ae_i$24$(not in LMFDB)
2.2.ac_e$24$(not in LMFDB)
2.2.a_a$24$(not in LMFDB)
2.2.c_e$24$(not in LMFDB)
2.2.e_i$24$(not in LMFDB)