Abelian variety isogeny class 2.61.ax_jr over $\F_{61}$

• Label: 2.61.ax_jr
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 - 23 x + 251 x^{2} - 1403 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.175618717362$, $\pm0.286807421120$
Angle rank:	$2$ (numerical)
Number field:	4.0.188773.1
Galois group:	$D_{4}$
Jacobians:	$15$
Isomorphism classes:	15

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2547$	$13751253$	$51734941623$	$191860795907973$	$713398549244422032$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$39$	$3695$	$227925$	$13856923$	$844662174$	$51520531583$	$3142741955973$	$191707304479699$	$11694146093293383$	$713342911991630390$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):

• $y^2=44 x^6+24 x^5+31 x^4+2 x^3+42 x^2+12 x+42$
• $y^2=50 x^6+19 x^5+15 x^4+28 x^3+40 x^2+19 x+33$
• $y^2=6 x^6+7 x^5+23 x^4+46 x^3+45 x^2+39 x+51$
• $y^2=51 x^6+45 x^5+26 x^4+26 x^3+42 x^2+44 x+39$
• $y^2=28 x^6+25 x^5+53 x^4+6 x^3+36 x^2+44 x+45$
• $y^2=55 x^6+2 x^5+2 x^4+40 x^3+17 x^2+46 x+46$
• $y^2=54 x^6+28 x^5+2 x^4+7 x^3+35 x^2+x+32$
• $y^2=29 x^6+28 x^5+12 x^4+40 x^3+11 x^2+57 x+50$
• $y^2=55 x^6+60 x^5+45 x^4+33 x^3+22 x^2+36 x+30$
• $y^2=31 x^6+23 x^5+46 x^4+37 x^3+21 x^2+47 x+10$
• $y^2=21 x^6+51 x^5+50 x^4+21 x^3+25 x^2+18 x+47$
• $y^2=44 x^6+42 x^5+32 x^4+18 x^3+35 x^2+20 x+43$
• $y^2=43 x^6+24 x^5+56 x^4+27 x^3+35 x^2+58 x+35$
• $y^2=29 x^6+3 x^5+25 x^4+27 x^3+22 x^2+50 x+35$
• $y^2=12 x^6+31 x^5+2 x^4+2 x^3+58 x^2+52 x+9$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is 4.0.188773.1.

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
2.61.x_jr	$2$	(not in LMFDB)