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

• Label: 2.11.a_al
• Base field: $\F_{11}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$1 - 11 x^{2} + 121 x^{4}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.833333333333$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-11})\)
Galois group:	$C_2^2$
Jacobians:	$3$
Cyclic group of points:	yes

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$111$	$12321$	$1774224$	$217946169$	$25937263551$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$100$	$1332$	$14884$	$161052$	$1776886$	$19487172$	$214388164$	$2357947692$	$25937102500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{6}}$.

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\).

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

The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.dyk^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $11$ and $\infty$.

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{11^{2}}$ is 1.121.al^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{11^{3}}$

  The base change of $A$ to $\F_{11^{3}}$ is 1.1331.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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_w	$3$	(not in LMFDB)
2.11.a_l	$4$	(not in LMFDB)
2.11.a_aw	$12$	(not in LMFDB)