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

• Label: 2.61.at_id
• 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:	$1 - 19 x + 211 x^{2} - 1159 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.262088288773$, $\pm0.319708272189$
Angle rank:	$2$ (numerical)
Number field:	4.0.570125.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2755$	$14080805$	$51905359855$	$191875201190405$	$713355493881250000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$43$	$3783$	$228673$	$13857963$	$844611198$	$51519832743$	$3142737158233$	$191707291563123$	$11694146200434163$	$713342913664697398$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=48 x^6+18 x^5+32 x^4+29 x^3+11 x^2+17 x+21$
• $y^2=10 x^6+55 x^5+27 x^4+10 x^3+6 x^2+46$
• $y^2=38 x^6+35 x^5+7 x^4+8 x^3+58 x^2+18 x+21$
• $y^2=28 x^6+13 x^5+29 x^4+24 x^3+4 x^2+45 x+3$
• $y^2=51 x^6+43 x^5+3 x^4+24 x^3+56 x^2+44 x+37$
• $y^2=23 x^6+18 x^5+19 x^4+19 x^3+17 x^2+12 x+16$
• $y^2=32 x^6+44 x^5+48 x^4+7 x^3+34 x^2+11 x+4$
• $y^2=50 x^6+7 x^5+24 x^4+56 x^3+29 x^2+37 x+58$
• $y^2=16 x^6+27 x^5+8 x^4+48 x^3+38 x^2+40 x+18$
• $y^2=52 x^6+6 x^5+x^4+24 x^3+43 x^2+59 x+34$
• $y^2=6 x^6+32 x^5+35 x^4+34 x^3+23 x^2+34 x+7$
• $y^2=45 x^6+14 x^5+33 x^4+32 x^3+44 x^2+12 x+53$

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 4.0.570125.1.

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.t_id	$2$	(not in LMFDB)