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

• Label: 2.41.a_acd
• 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:	$1 - 55 x^{2} + 1681 x^{4}$
Frobenius angles:	$\pm0.132990101477$, $\pm0.867009898523$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{137})\)
Galois group:	$C_2^2$
Jacobians:	$30$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1627$	$2647129$	$4750215232$	$7986835557801$	$13422659428165627$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$1572$	$68922$	$2826436$	$115856202$	$4750326222$	$194754273882$	$7984936305028$	$327381934393962$	$13422659546178852$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

• $y^2=15 x^6+4 x^5+12 x^4+39 x^3+18 x^2+x+6$
• $y^2=8 x^6+24 x^5+31 x^4+29 x^3+26 x^2+6 x+36$
• $y^2=12 x^6+9 x^5+37 x^3+33 x^2+6 x+27$
• $y^2=37 x^6+11 x^5+8 x^4+21 x^3+20 x^2+3 x+6$
• $y^2=8 x^6+3 x^5+34 x^4+2 x^3+14 x^2+35 x+10$
• $y^2=12 x^6+38 x^5+22 x^4+31 x^3+33 x^2+24 x+21$
• $y^2=31 x^6+23 x^5+9 x^4+22 x^3+34 x^2+21 x+3$
• $y^2=14 x^6+12 x^5+40 x^4+6 x^3+18 x^2+37 x+21$
• $y^2=2 x^6+31 x^5+35 x^4+36 x^3+26 x^2+17 x+3$
• $y^2=11 x^6+37 x^5+30 x^4+34 x^3+37 x^2+32 x+9$
• $y^2=6 x^6+7 x^5+2 x^4+25 x^3+33 x^2+13 x+18$
• $y^2=13 x^6+11 x^5+30 x^4+30 x^3+37 x^2+8 x+5$
• $y^2=37 x^6+25 x^5+16 x^4+16 x^3+17 x^2+7 x+30$
• $y^2=40 x^6+4 x^5+9 x^4+15 x^3+38 x^2+5 x+38$
• $y^2=16 x^6+20 x^5+7 x^4+10 x^3+6 x^2+5 x+9$
• $y^2=14 x^6+38 x^5+x^4+19 x^3+36 x^2+30 x+13$
• $y^2=25 x^6+34 x^5+26 x^4+26 x^3+2 x^2+16 x+2$
• $y^2=22 x^6+33 x^5+33 x^4+31 x^3+33 x^2+11 x+14$
• $y^2=10 x^6+36 x^5+40 x^4+19 x^3+30 x^2+35 x+9$
• $y^2=3 x^6+14 x^5+4 x^4+38 x^3+24 x^2+37 x+9$
• $y^2=18 x^6+2 x^5+24 x^4+23 x^3+21 x^2+17 x+13$
• $y^2=40 x^6+9 x^5+39 x^4+37 x^3+19 x^2+28 x+1$
• $y^2=35 x^6+13 x^5+29 x^4+17 x^3+32 x^2+4 x+6$
• $y^2=15 x^6+24 x^5+5 x^4+8 x^3+23 x^2+34 x+36$
• $y^2=8 x^6+21 x^5+30 x^4+7 x^3+15 x^2+40 x+11$
• $y^2=24 x^6+11 x^5+23 x^4+14 x^3+27 x^2+2 x+9$
• $y^2=21 x^6+25 x^5+15 x^4+2 x^3+39 x^2+12 x+13$
• $y^2=29 x^6+15 x^5+6 x^3+20 x^2+2 x+17$
• $y^2=20 x^6+17 x^5+33 x^4+14 x^3+19 x^2+3 x+29$
• $y^2=28 x^6+35 x^5+14 x^4+26 x^3+5 x^2+25 x+17$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41^{2}}$.

Endomorphism algebra over $\F_{41}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{137})\).

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

The base change of $A$ to $\F_{41^{2}}$ is 1.1681.acd^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-411}) \)$)$

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.aj_cq	$3$	(not in LMFDB)
2.41.j_cq	$3$	(not in LMFDB)
2.41.a_cd	$4$	(not in LMFDB)