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

• Label: 2.11.ae_w
• Base field: $\F_{11}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$( 1 - 4 x + 11 x^{2} )( 1 + 11 x^{2} )$
	$1 - 4 x + 22 x^{2} - 44 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.293962833700$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$18$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$96$	$18432$	$1864800$	$213811200$	$25949986656$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$150$	$1400$	$14606$	$161128$	$1772262$	$19478488$	$214316446$	$2357990600$	$25938063030$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{2}}$.

Endomorphism algebra over $\F_{11}$

The isogeny class factors as 1.11.ae $\times$ 1.11.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.11.ae : \(\Q(\sqrt{-7}) \).
• 1.11.a : \(\Q(\sqrt{-11}) \).

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

The base change of $A$ to $\F_{11^{2}}$ is 1.121.g $\times$ 1.121.w. The endomorphism algebra for each factor is:

• 1.121.g : \(\Q(\sqrt{-7}) \).
• 1.121.w : the quaternion algebra over \(\Q\) ramified at $11$ and $\infty$.

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.e_w	$2$	2.121.bc_ok