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

• Label: 2.23.ai_bp
• 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:	$1 - 8 x + 41 x^{2} - 184 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.147321344739$, $\pm0.519345321928$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-7})\)
Galois group:	$C_2^2$
Jacobians:	$28$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$379$	$289177$	$147088384$	$78105840169$	$41458402075099$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$548$	$12088$	$279108$	$6441296$	$148081358$	$3404893424$	$78310964356$	$1801155453544$	$41426522863268$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{-7})\).

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

The base change of $A$ to $\F_{23^{3}}$ is 1.12167.abo^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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.i_bp	$2$	(not in LMFDB)
2.23.q_eg	$3$	(not in LMFDB)
2.23.aq_eg	$6$	(not in LMFDB)