Abelian variety isogeny class 2.23.ab_aw over $\F_{23}$

• Label: 2.23.ab_aw
• Base field: $\F_{23}$
• 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:	$\F_{23}$
Dimension:	$2$
L-polynomial:	$1 - x - 22 x^{2} - 23 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.133420151237$, $\pm0.800086817904$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-91})\)
Galois group:	$C_2^2$
Jacobians:	$10$
Isomorphism classes:	40
Cyclic group of points:	no
Non-cyclic primes:	$2, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$484$	$257488$	$146410000$	$78582247744$	$41410220796604$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$23$	$485$	$12032$	$280809$	$6433813$	$148075310$	$3404903371$	$78311360689$	$1801156996736$	$41426504746925$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

• $y^2=6 x^5+9 x^4+5 x^3+8 x+18$
• $y^2=7 x^6+3 x^5+16 x^4+21 x^3+18 x^2+3 x$
• $y^2=4 x^6+2 x^5+x^4+10 x^2+4 x+14$
• $y^2=6 x^6+2 x^5+19 x^4+8 x^3+7 x^2+13 x+14$
• $y^2=3 x^6+20 x^5+3 x^4+6 x^3+4 x^2+12 x+16$
• $y^2=14 x^6+12 x^5+22 x^4+18 x^3+18 x^2+9 x+22$
• $y^2=3 x^6+18 x^4+12 x^3+6 x^2+8 x$
• $y^2=11 x^6+3 x^5+17 x^4+21 x^3+17 x^2+9$
• $y^2=17 x^6+15 x^5+7 x^4+x^3+15 x^2+18 x+20$
• $y^2=18 x^6+2 x^5+9 x^4+9 x^3+14 x^2+21 x+17$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{23^{3}}$.

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-91})\).

Endomorphism algebra over $\overline{\F}_{23}$

The base change of $A$ to $\F_{23^{3}}$ is 1.12167.acq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-91}) \)$)$

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
2.23.b_aw	$2$	(not in LMFDB)
2.23.c_bv	$3$	(not in LMFDB)
2.23.ac_bv	$6$	(not in LMFDB)