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

• Label: 2.41.af_de
• Base field: $\F_{41}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 - 5 x + 41 x^{2} )( 1 + 41 x^{2} )$
	$1 - 5 x + 82 x^{2} - 205 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.372324822061$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$48$
Cyclic group of points:	yes

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1554$	$3067596$	$4784013864$	$7975749600000$	$13420397449520754$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$1821$	$69412$	$2822513$	$115836677$	$4750139826$	$194754563117$	$7984925216353$	$327381950729092$	$13422659392351581$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.af $\times$ 1.41.a 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.af : \(\Q(\sqrt{-139}) \).
• 1.41.a : \(\Q(\sqrt{-41}) \).

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

The base change of $A$ to $\F_{41^{2}}$ is 1.1681.cf $\times$ 1.1681.de. The endomorphism algebra for each factor is:

• 1.1681.cf : \(\Q(\sqrt{-139}) \).
• 1.1681.de : the quaternion algebra over \(\Q\) ramified at $41$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.41.f_de	$2$	(not in LMFDB)