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

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

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$1 + 10 x^{2} + 121 x^{4}$
Frobenius angles:	$\pm0.325099143859$, $\pm0.674900856141$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{3})\)
Galois group:	$C_2^2$
Jacobians:	$24$
Isomorphism classes:	40
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$132$	$17424$	$1768932$	$218566656$	$25937651652$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$142$	$1332$	$14926$	$161052$	$1766302$	$19487172$	$214377118$	$2357947692$	$25937878702$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

• $y^2=2 x^6+9 x^5+2 x^4+9 x^3+x^2+5 x+3$
• $y^2=x^6+4 x^5+6 x^4+6 x^3+6 x^2+4 x+3$
• $y^2=7 x^6+6 x^5+2 x^4+6 x^3+9 x^2+6 x+4$
• $y^2=9 x^6+3 x^5+8 x^4+9 x^3+5 x^2+2 x+7$
• $y^2=7 x^6+6 x^5+5 x^4+7 x^3+10 x^2+4 x+3$
• $y^2=x^6+x^3+6$
• $y^2=10 x^6+7 x^5+3 x^4+6 x^3+2 x^2+4 x+10$
• $y^2=9 x^6+3 x^5+6 x^4+x^3+4 x^2+8 x+9$
• $y^2=7 x^5+6 x^4+4 x^3+4 x^2+4 x+5$
• $y^2=3 x^5+x^4+8 x^3+8 x^2+8 x+10$
• $y^2=3 x^6+9 x^5+6 x^4+9 x^3+5 x^2+6 x+8$
• $y^2=6 x^6+10 x^5+2 x^4+6 x^3+4 x^2+7 x+4$
• $y^2=x^6+10 x^5+10 x^4+8 x^2+3 x+2$
• $y^2=2 x^6+9 x^5+9 x^4+5 x^2+6 x+4$
• $y^2=7 x^6+x^5+5 x^4+4 x^3+10 x^2+4 x+1$
• $y^2=x^6+x^3+10$
• $y^2=3 x^6+5 x^5+9 x^3+7 x^2+8 x+3$
• $y^2=2 x^6+x^5+10 x^4+6 x^3+6 x^2+3 x+1$
• $y^2=4 x^6+2 x^5+9 x^4+x^3+x^2+6 x+2$
• $y^2=5 x^6+7 x^5+7 x^4+4 x^2+7 x+6$
• $y^2=x^6+x^3+2$
• $y^2=6 x^6+9 x^5+7 x^4+3 x^3+5 x^2+2 x+10$
• $y^2=x^6+7 x^5+3 x^4+6 x^3+10 x^2+4 x+9$
• $y^2=8 x^6+2 x^5+5 x^3+7 x+4$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{2}}$.

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{3})\).

Endomorphism algebra over $\overline{\F}_{11}$

The base change of $A$ to $\F_{11^{2}}$ is 1.121.k^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

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.a_ak	$4$	(not in LMFDB)
2.11.ai_bg	$8$	(not in LMFDB)
2.11.i_bg	$8$	(not in LMFDB)