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

• Label: 2.23.f_bk
• Base field: $\F_{23}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{23}$
Dimension:	$2$
L-polynomial:	$1 + 5 x + 36 x^{2} + 115 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.448969491287$, $\pm0.738418679133$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-278 +10 \sqrt{65}})\)
Galois group:	$D_{4}$
Jacobians:	$12$
Cyclic group of points:	yes

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$686$	$305956$	$147190904$	$78366346016$	$41384537438986$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$29$	$577$	$12098$	$280041$	$6429819$	$148044538$	$3405031253$	$78310290033$	$1801151185934$	$41426515377897$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-278 +10 \sqrt{65}})\).

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.af_bk	$2$	(not in LMFDB)