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

• Label: 2.61.q_he
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$( 1 + 8 x + 61 x^{2} )^{2}$
	$1 + 16 x + 186 x^{2} + 976 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.671149895095$, $\pm0.671149895095$
Angle rank:	$1$ (numerical)
Jacobians:	$53$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4900$	$14288400$	$51089560900$	$191820284006400$	$713385900573062500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$78$	$3838$	$225078$	$13853998$	$844647198$	$51519469678$	$3142746968838$	$191707335120478$	$11694145663746798$	$713342913746066398$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 53 curves (of which all are hyperelliptic):

• $y^2=39 x^6+4 x^5+48 x^4+60 x^3+48 x^2+4 x+39$
• $y^2=8 x^6+29 x^4+29 x^2+8$
• $y^2=43 x^6+39 x^5+12 x^4+33 x^3+59 x^2+39 x+37$
• $y^2=49 x^6+32 x^5+48 x^4+12 x^3+18 x^2+22 x+25$
• $y^2=14 x^6+23 x^5+41 x^4+49 x^3+52 x^2+53 x+36$
• $y^2=55 x^6+27 x^5+5 x^4+17 x^3+34 x^2+37 x+49$
• $y^2=26 x^6+43 x^5+20 x^4+13 x^3+34 x^2+59 x+51$
• $y^2=45 x^6+5 x^5+44 x^4+17 x^3+44 x^2+5 x+45$
• $y^2=54 x^6+2 x^5+18 x^4+9 x^3+18 x^2+2 x+54$
• $y^2=50 x^6+30 x^5+38 x^4+5 x^3+x^2+28 x+34$
• $y^2=2 x^6+32 x^3+18$
• $y^2=15 x^6+57 x^5+23 x^4+4 x^3+23 x^2+57 x+15$
• $y^2=2 x^6+2 x^3+40$
• $y^2=16 x^6+22 x^5+40 x^4+19 x^3+11 x^2+46 x+33$
• $y^2=30 x^6+12 x^5+35 x^4+32 x^3+35 x^2+12 x+30$
• $y^2=29 x^6+5 x^5+28 x^4+32 x^3+28 x^2+5 x+29$
• $y^2=35 x^6+32 x^5+23 x^4+2 x^3+23 x^2+32 x+35$
• $y^2=23 x^6+39 x^5+21 x^4+26 x^3+3 x^2+46 x+22$
• $y^2=36 x^6+38 x^5+54 x^4+15 x^3+58 x^2+30 x+53$
• $y^2=55 x^6+46 x^4+46 x^2+55$
• and

• $y^2=19 x^6+53 x^5+45 x^4+32 x^3+45 x^2+53 x+19$
• $y^2=12 x^6+35 x^5+54 x^4+5 x^3+44 x^2+38 x+52$
• $y^2=34 x^6+6 x^5+33 x^4+21 x^3+50 x^2+33 x+45$
• $y^2=8 x^6+51 x^5+56 x^4+45 x^3+40 x^2+47 x+45$
• $y^2=54 x^6+45 x^5+33 x^4+4 x^3+33 x^2+45 x+54$
• $y^2=48 x^6+9 x^5+60 x^4+19 x^3+10 x^2+25 x+12$
• $y^2=x^5+59 x^4+4 x^3+8 x^2+16 x$
• $y^2=49 x^6+56 x^5+23 x^4+47 x^3+57 x^2+22 x+46$
• $y^2=6 x^6+40 x^5+22 x^4+41 x^3+37 x^2+10 x+41$
• $y^2=2 x^6+17 x^3+54$
• $y^2=56 x^6+35 x^5+24 x^4+46 x^3+32 x^2+7 x+42$
• $y^2=48 x^5+40 x^4+29 x^3+50 x^2+14 x$
• $y^2=15 x^6+41 x^5+56 x^4+19 x^3+33 x^2+46 x+51$
• $y^2=5 x^6+x^5+4 x^4+37 x^3+4 x^2+x+5$
• $y^2=22 x^6+31 x^5+33 x^4+30 x^3+37 x^2+29 x+48$
• $y^2=58 x^6+28 x^5+20 x^4+46 x^3+34 x^2+37 x+34$
• $y^2=34 x^6+28 x^5+28 x^4+4 x^3+33 x^2+28 x+27$
• $y^2=52 x^6+50 x^5+49 x^4+4 x^3+12 x^2+50 x+9$
• $y^2=25 x^6+12 x^5+19 x^4+36 x^3+56 x^2+20 x+40$
• $y^2=48 x^6+31 x^5+3 x^4+19 x^3+33 x^2+34 x+9$
• $y^2=48 x^6+12 x^4+12 x^2+48$
• $y^2=5 x^6+34 x^5+54 x^4+28 x^3+16 x^2+11 x+47$
• $y^2=2 x^6+11 x^3+43$
• $y^2=42 x^6+48 x^5+54 x^4+7 x^3+6 x^2+39 x+47$
• $y^2=16 x^6+50 x^5+44 x^4+37 x^3+44 x^2+50 x+16$
• $y^2=12 x^6+39 x^5+7 x^4+11 x^3+7 x^2+39 x+12$
• $y^2=32 x^6+10 x^5+54 x^4+39 x^3+6 x^2+31 x+30$
• $y^2=46 x^6+28 x^5+23 x^4+9 x^3+11 x^2+38 x+56$
• $y^2=36 x^6+2 x^5+10 x^4+35 x^3+10 x^2+2 x+36$
• $y^2=2 x^6+50 x^3+18$
• $y^2=55 x^6+53 x^5+59 x^4+4 x^3+46 x^2+20 x+12$
• $y^2=2 x^6+51 x^3+55$
• $y^2=59 x^6+42 x^5+26 x^4+4 x^3+8 x^2+10 x+8$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.i^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.61.aq_he	$2$	(not in LMFDB)
2.61.a_cg	$2$	(not in LMFDB)
2.61.ai_d	$3$	(not in LMFDB)