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

• Label: 2.23.a_bu
• Base field: $\F_{23}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{23}$
Dimension:	$2$
L-polynomial:	$( 1 + 23 x^{2} )^{2}$
	$1 + 46 x^{2} + 529 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.5$
Angle rank:	$0$ (numerical)
Jacobians:	$30$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$576$	$331776$	$148060224$	$77720518656$	$41426524086336$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$622$	$12168$	$277726$	$6436344$	$148084558$	$3404825448$	$78309865918$	$1801152661464$	$41426536959022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-23}) \)$)$

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

The base change of $A$ to $\F_{23^{2}}$ is 1.529.bu^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $23$ and $\infty$.

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.a_ax	$3$	(not in LMFDB)
2.23.a_abu	$4$	(not in LMFDB)
2.23.a_ax	$6$	(not in LMFDB)