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

• Label: 2.61.aq_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:	$\pm0.328850104905$, $\pm0.328850104905$
Angle rank:	$1$ (numerical)
Jacobians:	$53$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2916$	$14288400$	$51953908356$	$191820284006400$	$713299927426289316$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$3838$	$228886$	$13853998$	$844545406$	$51519469678$	$3142738703206$	$191707335120478$	$11694146521921486$	$713342913746066398$

Jacobians and polarizations

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

• $y^2=50 x^6+2 x^5+24 x^4+30 x^3+24 x^2+2 x+50$
• $y^2=16 x^6+58 x^4+58 x^2+16$
• $y^2=52 x^6+50 x^5+6 x^4+47 x^3+60 x^2+50 x+49$
• $y^2=37 x^6+3 x^5+35 x^4+24 x^3+36 x^2+44 x+50$
• $y^2=7 x^6+42 x^5+51 x^4+55 x^3+26 x^2+57 x+18$
• $y^2=49 x^6+54 x^5+10 x^4+34 x^3+7 x^2+13 x+37$
• $y^2=52 x^6+25 x^5+40 x^4+26 x^3+7 x^2+57 x+41$
• $y^2=29 x^6+10 x^5+27 x^4+34 x^3+27 x^2+10 x+29$
• $y^2=27 x^6+x^5+9 x^4+35 x^3+9 x^2+x+27$
• $y^2=39 x^6+60 x^5+15 x^4+10 x^3+2 x^2+56 x+7$
• $y^2=x^6+16 x^3+9$
• $y^2=38 x^6+59 x^5+42 x^4+2 x^3+42 x^2+59 x+38$
• $y^2=x^6+x^3+20$
• $y^2=32 x^6+44 x^5+19 x^4+38 x^3+22 x^2+31 x+5$
• $y^2=60 x^6+24 x^5+9 x^4+3 x^3+9 x^2+24 x+60$
• $y^2=58 x^6+10 x^5+56 x^4+3 x^3+56 x^2+10 x+58$
• $y^2=9 x^6+3 x^5+46 x^4+4 x^3+46 x^2+3 x+9$
• $y^2=42 x^6+50 x^5+41 x^4+13 x^3+32 x^2+23 x+11$
• $y^2=11 x^6+15 x^5+47 x^4+30 x^3+55 x^2+60 x+45$
• $y^2=58 x^6+23 x^4+23 x^2+58$
• and

• $y^2=38 x^6+45 x^5+29 x^4+3 x^3+29 x^2+45 x+38$
• $y^2=6 x^6+48 x^5+27 x^4+33 x^3+22 x^2+19 x+26$
• $y^2=7 x^6+12 x^5+5 x^4+42 x^3+39 x^2+5 x+29$
• $y^2=4 x^6+56 x^5+28 x^4+53 x^3+20 x^2+54 x+53$
• $y^2=27 x^6+53 x^5+47 x^4+2 x^3+47 x^2+53 x+27$
• $y^2=35 x^6+18 x^5+59 x^4+38 x^3+20 x^2+50 x+24$
• $y^2=31 x^5+60 x^4+2 x^3+4 x^2+8 x$
• $y^2=37 x^6+51 x^5+46 x^4+33 x^3+53 x^2+44 x+31$
• $y^2=12 x^6+19 x^5+44 x^4+21 x^3+13 x^2+20 x+21$
• $y^2=x^6+39 x^3+27$
• $y^2=28 x^6+48 x^5+12 x^4+23 x^3+16 x^2+34 x+21$
• $y^2=24 x^5+20 x^4+45 x^3+25 x^2+7 x$
• $y^2=38 x^6+51 x^5+28 x^4+40 x^3+47 x^2+23 x+56$
• $y^2=10 x^6+2 x^5+8 x^4+13 x^3+8 x^2+2 x+10$
• $y^2=11 x^6+46 x^5+47 x^4+15 x^3+49 x^2+45 x+24$
• $y^2=55 x^6+56 x^5+40 x^4+31 x^3+7 x^2+13 x+7$
• $y^2=17 x^6+14 x^5+14 x^4+2 x^3+47 x^2+14 x+44$
• $y^2=43 x^6+39 x^5+37 x^4+8 x^3+24 x^2+39 x+18$
• $y^2=50 x^6+24 x^5+38 x^4+11 x^3+51 x^2+40 x+19$
• $y^2=35 x^6+x^5+6 x^4+38 x^3+5 x^2+7 x+18$
• $y^2=35 x^6+24 x^4+24 x^2+35$
• $y^2=10 x^6+7 x^5+47 x^4+56 x^3+32 x^2+22 x+33$
• $y^2=x^6+36 x^3+52$
• $y^2=21 x^6+24 x^5+27 x^4+34 x^3+3 x^2+50 x+54$
• $y^2=32 x^6+39 x^5+27 x^4+13 x^3+27 x^2+39 x+32$
• $y^2=24 x^6+17 x^5+14 x^4+22 x^3+14 x^2+17 x+24$
• $y^2=3 x^6+20 x^5+47 x^4+17 x^3+12 x^2+x+60$
• $y^2=31 x^6+56 x^5+46 x^4+18 x^3+22 x^2+15 x+51$
• $y^2=11 x^6+4 x^5+20 x^4+9 x^3+20 x^2+4 x+11$
• $y^2=x^6+25 x^3+9$
• $y^2=49 x^6+45 x^5+57 x^4+8 x^3+31 x^2+40 x+24$
• $y^2=x^6+56 x^3+58$
• $y^2=60 x^6+21 x^5+13 x^4+2 x^3+4 x^2+5 x+4$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ai^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.a_cg	$2$	(not in LMFDB)
2.61.q_he	$2$	(not in LMFDB)
2.61.i_d	$3$	(not in LMFDB)