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

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

Invariants

Base field:	$\F_{23}$
Dimension:	$2$
L-polynomial:	$1 - 23 x^{2} + 529 x^{4}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.833333333333$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-23})\)
Galois group:	$C_2^2$
Jacobians:	$5$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$507$	$257049$	$148060224$	$78607897641$	$41426504777307$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$484$	$12168$	$280900$	$6436344$	$148084558$	$3404825448$	$78311544964$	$1801152661464$	$41426498340964$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{23^{6}}$.

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-23})\).

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

The base change of $A$ to $\F_{23^{6}}$ is 1.148035889.bjzy^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $23$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{23^{2}}$

  The base change of $A$ to $\F_{23^{2}}$ is 1.529.ax^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{23^{3}}$

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

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_bu	$3$	(not in LMFDB)
2.23.a_x	$4$	(not in LMFDB)
2.23.a_bu	$6$	(not in LMFDB)