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

• Label: 2.41.o_fa
• Base field: $\F_{41}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$( 1 + 6 x + 41 x^{2} )( 1 + 8 x + 41 x^{2} )$
	$1 + 14 x + 130 x^{2} + 574 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.655213070720$, $\pm0.714776712523$
Angle rank:	$2$ (numerical)
Jacobians:	$20$
Isomorphism classes:	80

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2400$	$2937600$	$4681980000$	$7997040230400$	$13423699166460000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$1746$	$67928$	$2830046$	$115865176$	$4749884658$	$194755393336$	$7984925750206$	$327381892525688$	$13422659555121426$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.g $\times$ 1.41.i and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.41.g : \(\Q(\sqrt{-2}) \).
• 1.41.i : \(\Q(\sqrt{-1}) \).

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
2.41.ao_fa	$2$	(not in LMFDB)
2.41.ac_bi	$2$	(not in LMFDB)
2.41.c_bi	$2$	(not in LMFDB)