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

• Label: 2.41.a_abn
• 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:	$( 1 - 11 x + 41 x^{2} )( 1 + 11 x + 41 x^{2} )$
	$1 - 39 x^{2} + 1681 x^{4}$
Frobenius angles:	$\pm0.171113726078$, $\pm0.828886273922$
Angle rank:	$1$ (numerical)
Jacobians:	$17$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1643$	$2699449$	$4750241600$	$7995338725609$	$13422659167481003$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$1604$	$68922$	$2829444$	$115856202$	$4750378958$	$194754273882$	$7984929753604$	$327381934393962$	$13422659024809604$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 17 curves (of which all are hyperelliptic):

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

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.al $\times$ 1.41.l 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.al : \(\Q(\sqrt{-43}) \).
• 1.41.l : \(\Q(\sqrt{-43}) \).

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

The base change of $A$ to $\F_{41^{2}}$ is 1.1681.abn^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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.aw_hv	$2$	(not in LMFDB)
2.41.w_hv	$2$	(not in LMFDB)
2.41.a_bn	$4$	(not in LMFDB)