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

• Label: 2.11.a_u
• 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:	$1 + 20 x^{2} + 121 x^{4}$
Frobenius angles:	$\pm0.431611174087$, $\pm0.568388825913$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{2}, \sqrt{-21})\)
Galois group:	$C_2^2$
Jacobians:	$6$
Isomorphism classes:	8
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$142$	$20164$	$1772302$	$209786256$	$25937248702$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$162$	$1332$	$14326$	$161052$	$1773042$	$19487172$	$214367518$	$2357947692$	$25937072802$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{2}, \sqrt{-21})\).

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

The base change of $A$ to $\F_{11^{2}}$ is 1.121.u^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$

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.a_au	$4$	(not in LMFDB)
2.11.ac_c	$8$	(not in LMFDB)
2.11.c_c	$8$	(not in LMFDB)