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

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

Invariants

Base field:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$1 - 20 x + 180 x^{2} - 820 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.149793866772$, $\pm0.266106536017$
Angle rank:	$2$ (numerical)
Number field:	4.0.194816.2
Galois group:	$D_{4}$
Jacobians:	$6$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1022$	$2761444$	$4773640382$	$7997152869776$	$13425683554533502$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$1642$	$69262$	$2830086$	$115882302$	$4750193962$	$194754344422$	$7984924623678$	$327381939284182$	$13422659423960202$

Jacobians and polarizations

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

• $y^2=14 x^6+10 x^5+36 x^4+22 x^3+7 x^2+15 x+34$
• $y^2=17 x^6+5 x^5+26 x^4+15 x^3+23 x^2+3 x+19$
• $y^2=38 x^6+33 x^5+4 x^4+5 x^3+4 x+27$
• $y^2=27 x^6+17 x^5+8 x^4+13 x^3+6 x^2+38 x+7$
• $y^2=11 x^6+28 x^5+36 x^4+28 x^3+2 x^2+22 x+13$
• $y^2=11 x^6+24 x^5+30 x^3+31 x^2+27 x+24$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{41}$

The endomorphism algebra of this simple isogeny class is 4.0.194816.2.

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.u_gy	$2$	(not in LMFDB)