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

• Label: 2.11.a_at
• 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:	$1 - 19 x^{2} + 121 x^{4}$
Frobenius angles:	$\pm0.0840906703303$, $\pm0.915909329670$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{41})\)
Galois group:	$C_2^2$
Jacobians:	$1$
Isomorphism classes:	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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$103$	$10609$	$1771600$	$210917529$	$25937707303$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$84$	$1332$	$14404$	$161052$	$1771638$	$19487172$	$214389124$	$2357947692$	$25937990004$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{41})\).

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

The base change of $A$ to $\F_{11^{2}}$ is 1.121.at^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$

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.ad_o	$3$	(not in LMFDB)
2.11.d_o	$3$	(not in LMFDB)
2.11.a_t	$4$	(not in LMFDB)