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

• Label: 2.11.ac_c
• 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:	$1 - 2 x + 2 x^{2} - 22 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.181611174087$, $\pm0.681611174087$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{21})\)
Galois group:	$C_2^2$
Jacobians:	$16$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$100$	$14800$	$1690900$	$219040000$	$26098652500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$122$	$1270$	$14958$	$162050$	$1771562$	$19499630$	$214367518$	$2357819290$	$25937424602$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{4}}$.

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{21})\).

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

The base change of $A$ to $\F_{11^{4}}$ is 1.14641.gc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{11^{2}}$ is the simple isogeny class 2.121.a_gc and its endomorphism algebra is \(\Q(i, \sqrt{21})\).

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.c_c	$2$	2.121.a_gc
2.11.c_c	$4$	(not in LMFDB)
2.11.a_au	$8$	(not in LMFDB)