Abelian variety isogeny class 4.2.ae_i_aq_bc over $\F_{2}$

• Label: 4.2.ae_i_aq_bc
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )^{2}$
	$1 - 4 x + 8 x^{2} - 16 x^{3} + 28 x^{4} - 32 x^{5} + 32 x^{6} - 32 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.0833333333333$, $\pm0.0833333333333$, $\pm0.583333333333$, $\pm0.583333333333$
Angle rank:	$0$ (numerical)
Jacobians:	$0$

This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable.

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$169$	$625$	$28561$	$1745041$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$5$	$-7$	$1$	$49$	$65$	$97$	$321$	$641$	$1025$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 2.2.ac_c^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{12})\)$)$

Endomorphism algebra over $\overline{\F}_{2}$

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey^4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$

  The base change of $A$ to $\F_{2^{2}}$ is 2.4.a_ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{12})\)$)$

• Endomorphism algebra over $\F_{2^{3}}$

  The base change of $A$ to $\F_{2^{3}}$ is 1.8.ae^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$

• Endomorphism algebra over $\F_{2^{4}}$

  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{2^{6}}$

  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
4.2.a_a_a_ae	$2$	4.4.a_ai_a_bw
4.2.e_i_q_bc	$2$	4.4.a_ai_a_bw
4.2.c_c_ae_ai	$3$	(not in LMFDB)
4.2.i_bg_dc_fg	$3$	(not in LMFDB)