Abelian variety isogeny class 3.4.ag_p_abc over $\F_{2^{2}}$

• Label: 3.4.ag_p_abc
• Base field: $\F_{2^{2}}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x )^{2}( 1 - 2 x + 3 x^{2} - 8 x^{3} + 16 x^{4} )$
	$1 - 6 x + 15 x^{2} - 28 x^{3} + 60 x^{4} - 96 x^{5} + 64 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.168977707736$, $\pm0.618033150523$
Angle rank:	$2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10$	$2700$	$162190$	$15660000$	$1129223050$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$11$	$35$	$239$	$1079$	$4019$	$16211$	$65759$	$260855$	$1043651$

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

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

The isogeny class factors as 1.4.ae $\times$ 2.4.ac_d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.4.ae : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.4.ac_d : 4.0.32832.1.

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
3.4.ac_ab_e	$2$	3.16.ag_j_ai
3.4.c_ab_ae	$2$	3.16.ag_j_ai
3.4.g_p_bc	$2$	3.16.ag_j_ai
3.4.a_d_ak	$3$	(not in LMFDB)