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

• Label: 2.11.ab_ak
• 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:	$1 - x - 10 x^{2} - 11 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.118495992215$, $\pm0.785162658881$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-43})\)
Galois group:	$C_2^2$
Jacobians:	$3$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$100$	$12400$	$1690000$	$217297600$	$25848827500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$101$	$1268$	$14841$	$160501$	$1774838$	$19494871$	$214369201$	$2358137708$	$25937406101$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{3}}$.

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-43})\).

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

The base change of $A$ to $\F_{11^{3}}$ is 1.1331.abg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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.b_ak	$2$	2.121.av_mi
2.11.c_x	$3$	(not in LMFDB)
2.11.ac_x	$6$	(not in LMFDB)