Abelian variety isogeny class 2.41.ah_i over $\F_{41}$

• Label: 2.41.ah_i
• Base field: $\F_{41}$
• 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:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$1 - 7 x + 8 x^{2} - 287 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0174146042242$, $\pm0.649252062442$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-115})\)
Galois group:	$C_2^2$
Jacobians:	$10$
Isomorphism classes:	16
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1396$	$2769664$	$4679107216$	$7978504612864$	$13422042056685076$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$1649$	$67886$	$2823489$	$115850875$	$4749843278$	$194753578915$	$7984924744129$	$327381870616766$	$13422659106816929$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

• $y^2=25 x^6+23 x^5+19 x^4+9 x^3+14 x^2+31 x+22$
• $y^2=18 x^6+39 x^5+31 x^4+17 x^3+6 x^2+39 x+31$
• $y^2=23 x^6+7 x^5+35 x^4+36 x^3+5 x^2+25 x+4$
• $y^2=2 x^6+9 x^5+12 x^4+28 x^3+35 x^2+37 x+9$
• $y^2=x^6+29 x^5+39 x^4+30 x^3+14 x^2+38 x+24$
• $y^2=19 x^6+4 x^5+3 x^4+35 x^3+2 x^2+17 x+11$
• $y^2=22 x^6+34 x^5+31 x^4+14 x^3+38 x^2+39 x+1$
• $y^2=34 x^5+37 x^4+16 x^3+30 x^2+8 x+5$
• $y^2=8 x^6+x^5+18 x^4+3 x^2+2 x+6$
• $y^2=35 x^6+17 x^5+29 x^4+16 x^3+2 x^2+15 x+12$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41^{3}}$.

Endomorphism algebra over $\F_{41}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-115})\).

Endomorphism algebra over $\overline{\F}_{41}$

The base change of $A$ to $\F_{41^{3}}$ is 1.68921.aty^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-115}) \)$)$

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
2.41.h_i	$2$	(not in LMFDB)
2.41.o_fb	$3$	(not in LMFDB)
2.41.ao_fb	$6$	(not in LMFDB)