Abelian variety isogeny class 3.2.ae_j_ap over $\F_{2}$

• Label: 3.2.ae_j_ap
• Base field: $\F_{2}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$3$
L-polynomial:	$1 - 4 x + 9 x^{2} - 15 x^{3} + 18 x^{4} - 16 x^{5} + 8 x^{6}$
Frobenius angles:	$\pm0.0435981566527$, $\pm0.329312442367$, $\pm0.527830414776$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{7})\)
Galois group:	$C_6$
Jacobians:	$0$
Isomorphism classes:	1

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$71$	$421$	$2059$	$25621$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$7$	$8$	$7$	$24$	$52$	$90$	$231$	$575$	$1092$

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 $\F_{2^{7}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{7})\).

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

The base change of $A$ to $\F_{2^{7}}$ is 1.128.an^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

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
3.2.e_j_p	$2$	3.4.c_ad_an
3.2.d_c_ab	$7$	(not in LMFDB)
3.2.d_j_n	$7$	(not in LMFDB)

Additional information

This isogeny class appears as a sporadic example in the classification of abelian varieties with one rational point.