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

• Label: 2.23.a_ak
• 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:	$1 - 10 x^{2} + 529 x^{4}$
Frobenius angles:	$\pm0.215122617226$, $\pm0.784877382774$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{14})\)
Galois group:	$C_2^2$
Jacobians:	$82$
Isomorphism classes:	176
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$520$	$270400$	$148050760$	$78848640000$	$41426499766600$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$510$	$12168$	$281758$	$6436344$	$148065630$	$3404825448$	$78310269118$	$1801152661464$	$41426488319550$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 82 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{14})\).

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

The base change of $A$ to $\F_{23^{2}}$ is 1.529.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-14}) \)$)$

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.am_de	$4$	(not in LMFDB)
2.23.a_k	$4$	(not in LMFDB)
2.23.m_de	$4$	(not in LMFDB)