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

• Label: 2.23.a_e
• 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:	$1 + 4 x^{2} + 529 x^{4}$
Frobenius angles:	$\pm0.263857061043$, $\pm0.736142938957$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{-21})\)
Galois group:	$C_2^2$
Jacobians:	$40$
Isomorphism classes:	100
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$534$	$285156$	$148029606$	$78895821456$	$41426516642214$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$538$	$12168$	$281926$	$6436344$	$148023322$	$3404825448$	$78309933118$	$1801152661464$	$41426522070778$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=x^6+x^5+8 x^4+17 x^3+17 x^2+x+15$
• $y^2=5 x^6+5 x^5+17 x^4+16 x^3+16 x^2+5 x+6$
• $y^2=7 x^6+7 x^5+3 x^4+2 x^3+4 x^2+x+8$
• $y^2=12 x^6+12 x^5+15 x^4+10 x^3+20 x^2+5 x+17$
• $y^2=17 x^6+20 x^5+14 x^4+7 x^3+2 x^2+3 x+8$
• $y^2=16 x^6+8 x^5+x^4+12 x^3+10 x^2+15 x+17$
• $y^2=21 x^6+12 x^5+9 x^4+20 x^3+9 x^2+14 x+3$
• $y^2=13 x^6+14 x^5+22 x^4+8 x^3+22 x^2+x+15$
• $y^2=14 x^6+4 x^5+8 x^4+5 x^3+15 x^2+13 x+11$
• $y^2=x^6+20 x^5+17 x^4+2 x^3+6 x^2+19 x+9$
• $y^2=x^6+18 x^5+6 x^4+6 x^3+3 x^2+15 x+14$
• $y^2=5 x^6+21 x^5+7 x^4+7 x^3+15 x^2+6 x+1$
• $y^2=13 x^6+15 x^5+17 x^4+14 x^3+3 x^2+10 x+11$
• $y^2=19 x^6+6 x^5+16 x^4+x^3+15 x^2+4 x+9$
• $y^2=x^6+4 x^5+9 x^4+12 x^3+14 x^2+6 x+22$
• $y^2=5 x^6+20 x^5+22 x^4+14 x^3+x^2+7 x+18$
• $y^2=18 x^6+17 x^5+21 x^3+8 x^2+14 x+21$
• $y^2=21 x^6+16 x^5+13 x^3+17 x^2+x+13$
• $y^2=21 x^6+7 x^5+15 x^4+12 x^3+16 x^2+11 x+13$
• $y^2=13 x^6+12 x^5+6 x^4+14 x^3+11 x^2+9 x+19$
• and

• $y^2=4 x^6+2 x^5+15 x^4+9 x^3+2 x^2+19$
• $y^2=20 x^6+10 x^5+6 x^4+22 x^3+10 x^2+3$
• $y^2=6 x^6+14 x^5+20 x^4+18 x^3+7 x^2+7 x+17$
• $y^2=7 x^6+x^5+8 x^4+21 x^3+12 x^2+12 x+16$
• $y^2=19 x^6+21 x^5+5 x^4+11 x^3+12 x^2+6 x+12$
• $y^2=3 x^6+13 x^5+2 x^4+9 x^3+14 x^2+7 x+14$
• $y^2=15 x^6+7 x^4+16 x^3+7 x^2+17 x+17$
• $y^2=6 x^6+12 x^4+11 x^3+12 x^2+16 x+16$
• $y^2=17 x^6+6 x^5+19 x^3+20 x^2+x+9$
• $y^2=16 x^6+7 x^5+3 x^3+8 x^2+5 x+22$
• $y^2=10 x^6+19 x^5+22 x^4+4 x^3+16 x^2+6 x+21$
• $y^2=4 x^6+3 x^5+18 x^4+20 x^3+11 x^2+7 x+13$
• $y^2=21 x^6+15 x^5+13 x^4+20 x^3+7 x^2+17 x+5$
• $y^2=13 x^6+6 x^5+19 x^4+8 x^3+12 x^2+16 x+2$
• $y^2=16 x^6+18 x^5+22 x^4+8 x^3+17 x^2+4 x+12$
• $y^2=11 x^6+21 x^5+18 x^4+17 x^3+16 x^2+20 x+14$
• $y^2=7 x^6+4 x^5+8 x^4+x^3+18 x^2+20 x+14$
• $y^2=12 x^6+20 x^5+17 x^4+5 x^3+21 x^2+8 x+1$
• $y^2=14 x^6+12 x^5+15 x^4+9 x^3+11 x^2+21 x+9$
• $y^2=x^6+14 x^5+6 x^4+22 x^3+9 x^2+13 x+22$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{23^{2}}$.

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-21})\).

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

The base change of $A$ to $\F_{23^{2}}$ is 1.529.e^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$

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
2.23.a_ae	$4$	(not in LMFDB)
2.23.ak_by	$8$	(not in LMFDB)
2.23.k_by	$8$	(not in LMFDB)