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

• Label: 2.23.j_cg
• 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:	$1 + 9 x + 58 x^{2} + 207 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.554280324776$, $\pm0.779053008558$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-11})\)
Galois group:	$C_2^2$
Jacobians:	$31$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$804$	$299088$	$145443600$	$78357466944$	$41420191686204$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$33$	$565$	$11952$	$280009$	$6435363$	$148061230$	$3404733981$	$78310453489$	$1801158026256$	$41426499303325$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 31 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{23^{3}}$.

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\).

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

The base change of $A$ to $\F_{23^{3}}$ is 1.12167.aee^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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.aj_cg	$2$	(not in LMFDB)
2.23.as_ex	$3$	(not in LMFDB)
2.23.a_abj	$6$	(not in LMFDB)