Abelian variety isogeny class 2.11.ah_bh over $\F_{11}$

• Label: 2.11.ah_bh
• Base field: $\F_{11}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$1 - 7 x + 33 x^{2} - 77 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.254874969775$, $\pm0.383085442404$
Angle rank:	$2$ (numerical)
Number field:	4.0.21725.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$71$	$16969$	$1936241$	$217559549$	$25896756976$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$139$	$1451$	$14859$	$160800$	$1769743$	$19485065$	$214358499$	$2357904971$	$25937196054$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=4x^6+x^5+5x^4+8x^3+9x^2+8x+6$
• $y^2=2x^6+7x^3+x^2+8x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11}$.

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is 4.0.21725.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.11.h_bh	$2$	2.121.r_jt