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

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

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$( 1 + 2 x + 11 x^{2} )^{2}$
	$1 + 4 x + 26 x^{2} + 44 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.597491114521$, $\pm0.597491114521$
Angle rank:	$1$ (numerical)
Jacobians:	$4$
Cyclic group of points:	no
Non-cyclic primes:	$2, 7$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$196$	$19600$	$1623076$	$211993600$	$26196717316$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$158$	$1216$	$14478$	$162656$	$1770158$	$19472336$	$214403998$	$2358020656$	$25936782398$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=5 x^6+4 x^5+9 x^4+x^3+9 x^2+4 x+5$
• $y^2=3 x^6+9 x^5+7 x^4+4 x^3+7 x^2+9 x+3$
• $y^2=2 x^5+9 x^4+6 x^3+6 x^2+5 x+10$
• $y^2=8 x^6+x^4+x^2+8$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{11}$

The isogeny class factors as 1.11.c^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$

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
2.11.ae_ba	$2$	2.121.bk_vu
2.11.a_s	$2$	2.121.bk_vu
2.11.ac_ah	$3$	(not in LMFDB)