Abelian variety isogeny class 2.49.abb_ku over $\F_{7^{2}}$

• Label: 2.49.abb_ku
• Base field: $\F_{7^{2}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{7^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 7 x )^{2}( 1 - 13 x + 49 x^{2} )$
	$1 - 27 x + 280 x^{2} - 1323 x^{3} + 2401 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.121037718324$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1332$	$5370624$	$13727362896$	$33203882880000$	$79785852330432852$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$23$	$2233$	$116678$	$5759761$	$282452543$	$13841205406$	$678222886847$	$33232930512481$	$1628413594752422$	$79792266178649353$

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

Endomorphism algebra over $\F_{7^{2}}$

The isogeny class factors as 1.49.ao $\times$ 1.49.an and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.49.ao : the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$.
• 1.49.an : \(\Q(\sqrt{-3}) \).

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
2.49.ab_adg	$2$	(not in LMFDB)
2.49.b_adg	$2$	(not in LMFDB)
2.49.bb_ku	$2$	(not in LMFDB)
2.49.am_cs	$3$	(not in LMFDB)
2.49.ad_ace	$3$	(not in LMFDB)