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

• Label: 2.41.av_hi
• 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:	$( 1 - 12 x + 41 x^{2} )( 1 - 9 x + 41 x^{2} )$
	$1 - 21 x + 190 x^{2} - 861 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.113551764296$, $\pm0.251940962052$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	30

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$990$	$2726460$	$4758831000$	$7993064629440$	$13424895517794750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$1621$	$69048$	$2828641$	$115875501$	$4750173538$	$194754329301$	$7984925003521$	$327381946348968$	$13422659535300901$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=11 x^6+30 x^5+13 x^4+16 x^3+19 x^2+15 x+29$
• $y^2=35 x^6+40 x^5+36 x^4+9 x^3+15 x^2+22$
• $y^2=13 x^6+34 x^5+30 x^4+36 x^3+14 x^2+35 x+1$
• $y^2=22 x^6+15 x^5+11 x^4+35 x^3+38 x^2+30 x+40$
• $y^2=11 x^6+18 x^5+11 x^4+3 x^3+36 x^2+4 x+14$
• $y^2=35 x^6+31 x^5+4 x^4+35 x^3+3 x^2+2 x+37$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.am $\times$ 1.41.aj 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.am : \(\Q(\sqrt{-5}) \).
• 1.41.aj : \(\Q(\sqrt{-83}) \).

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.ad_aba	$2$	(not in LMFDB)
2.41.d_aba	$2$	(not in LMFDB)
2.41.v_hi	$2$	(not in LMFDB)