Abelian variety isogeny class 2.121.abp_zk over $\F_{11^{2}}$

• Label: 2.121.abp_zk
• Base field: $\F_{11^{2}}$
• 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:	$\F_{11^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 11 x )^{2}( 1 - 19 x + 121 x^{2} )$
	$1 - 41 x + 660 x^{2} - 4961 x^{3} + 14641 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.168181340661$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10300$	$209131200$	$3133783240000$	$45946694313388800$	$672748972932909557500$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$81$	$14281$	$1768938$	$214344721$	$25937385201$	$3138428375278$	$379749827693961$	$45949729634927521$	$5559917308574433018$	$672749994852640153801$

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 $\F_{11^{2}}$.

Endomorphism algebra over $\F_{11^{2}}$

The isogeny class factors as 1.121.aw $\times$ 1.121.at and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.121.aw : the quaternion algebra over \(\Q\) ramified at $11$ and $\infty$.
• 1.121.at : \(\Q(\sqrt{-123}) \).

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
2.121.ad_agu	$2$	(not in LMFDB)
2.121.d_agu	$2$	(not in LMFDB)
2.121.bp_zk	$2$	(not in LMFDB)
2.121.ai_bh	$3$	(not in LMFDB)