Properties

Label 3.2.ae_k_aq
Base field $\F_{2}$
Dimension $3$
$p$-rank $0$
Ordinary No
Supersingular Yes
Simple No
Geometrically simple No
Primitive Yes
Principally polarizable Yes
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $( 1 + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}$
Frobenius angles:  $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.5$
Angle rank:  $0$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

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
$A(\F_{q^r})$ 3 225 1521 5625 55473 342225 1647201 11390625 118688193 1144130625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 9 17 25 49 81 97 161 449 1089

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac 2 $\times$ 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 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, 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
3.2.a_c_a$2$3.4.e_m_bg
3.2.e_k_q$2$3.4.e_m_bg
3.2.c_e_i$3$3.8.i_bo_ey
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.a_c_a$2$3.4.e_m_bg
3.2.e_k_q$2$3.4.e_m_bg
3.2.c_e_i$3$3.8.i_bo_ey
3.2.ac_e_ai$6$(not in LMFDB)
3.2.ag_s_abg$8$(not in LMFDB)
3.2.ac_ac_i$8$(not in LMFDB)
3.2.ac_c_a$8$(not in LMFDB)
3.2.ac_g_ai$8$(not in LMFDB)
3.2.a_ac_a$8$(not in LMFDB)
3.2.a_g_a$8$(not in LMFDB)
3.2.c_ac_ai$8$(not in LMFDB)
3.2.c_c_a$8$(not in LMFDB)
3.2.c_g_i$8$(not in LMFDB)
3.2.g_s_bg$8$(not in LMFDB)
3.2.ae_i_am$24$(not in LMFDB)
3.2.ac_a_e$24$(not in LMFDB)
3.2.ac_e_ae$24$(not in LMFDB)
3.2.a_a_ae$24$(not in LMFDB)
3.2.a_a_a$24$(not in LMFDB)
3.2.a_a_e$24$(not in LMFDB)
3.2.a_e_a$24$(not in LMFDB)
3.2.c_a_ae$24$(not in LMFDB)
3.2.c_e_e$24$(not in LMFDB)
3.2.e_i_m$24$(not in LMFDB)