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

• Label: 2.23.a_a
• 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:	$1 + 529 x^{4}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.750000000000$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(i, \sqrt{46})\)
Galois group:	$C_2^2$
Jacobians:	$25$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$530$	$280900$	$148035890$	$78904810000$	$41426511213650$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$530$	$12168$	$281958$	$6436344$	$148035890$	$3404825448$	$78309865918$	$1801152661464$	$41426511213650$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 25 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{23^{4}}$.

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{46})\).

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

The base change of $A$ to $\F_{23^{4}}$ is 1.279841.bos^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.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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_abu	$8$	(not in LMFDB)
2.23.a_bu	$8$	(not in LMFDB)
2.23.a_ax	$24$	(not in LMFDB)