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

• Label: 2.61.au_ih
• 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:	$1 - 20 x + 215 x^{2} - 1220 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.199705063640$, $\pm0.343962623824$
Angle rank:	$2$ (numerical)
Number field:	4.0.12519696.1
Galois group:	$D_{4}$
Jacobians:	$18$
Isomorphism classes:	18
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2697$	$13962369$	$51802562772$	$191829671847081$	$713360732182135377$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$3752$	$228222$	$13854676$	$844617402$	$51520298222$	$3142742940162$	$191707324320868$	$11694146125511862$	$713342910292233752$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

• $y^2=23 x^6+25 x^5+24 x^4+5 x^3+47 x^2+59 x+30$
• $y^2=31 x^6+57 x^5+27 x^4+54 x^3+36 x^2+45 x+56$
• $y^2=24 x^6+36 x^5+12 x^4+5 x^3+8 x^2+3 x+54$
• $y^2=13 x^6+49 x^5+48 x^4+54 x^3+21 x^2+22 x+41$
• $y^2=35 x^6+19 x^5+40 x^4+23 x^3+36 x^2+2 x+42$
• $y^2=21 x^6+24 x^5+60 x^4+13 x^3+53 x^2+50 x+4$
• $y^2=53 x^6+9 x^5+54 x^4+40 x^3+16 x^2+52 x+9$
• $y^2=10 x^6+45 x^5+20 x^4+22 x^3+41 x^2+17 x+30$
• $y^2=7 x^6+32 x^5+48 x^4+51 x^3+9 x^2+13 x+30$
• $y^2=7 x^6+9 x^5+19 x^4+50 x^3+52 x^2+13 x+26$
• $y^2=40 x^6+60 x^5+21 x^4+29 x^3+9 x^2+45 x+32$
• $y^2=38 x^6+19 x^5+13 x^4+45 x^3+6 x^2+7 x+38$
• $y^2=43 x^6+24 x^5+14 x^4+7 x^3+28 x^2+41 x+30$
• $y^2=54 x^6+36 x^5+46 x^4+46 x^3+59 x^2+46 x+1$
• $y^2=51 x^6+31 x^5+39 x^4+57 x^3+28 x^2+34 x+23$
• $y^2=13 x^6+39 x^5+27 x^4+57 x^3+20 x^2+34 x+50$
• $y^2=37 x^6+7 x^5+50 x^3+40 x^2+29 x+15$
• $y^2=24 x^6+5 x^5+7 x^4+15 x^3+19 x^2+x+27$

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 4.0.12519696.1.

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.u_ih	$2$	(not in LMFDB)