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

• Label: 2.61.ai_fi
• 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:	$( 1 - 4 x + 61 x^{2} )^{2}$
	$1 - 8 x + 138 x^{2} - 488 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.417571001240$, $\pm0.417571001240$
Angle rank:	$1$ (numerical)
Jacobians:	$42$
Cyclic group of points:	no
Non-cyclic primes:	$2, 29$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3364$	$14653584$	$51824522500$	$191602292834304$	$713248450072382884$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$3934$	$228318$	$13838254$	$844484454$	$51520389838$	$3142749720654$	$191707339591774$	$11694145779249558$	$713342908786280254$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

• $y^2=13 x^6+12 x^5+50 x^4+59 x^3+59 x^2+19 x+2$
• $y^2=59 x^6+20 x^5+21 x^4+26 x^3+26 x^2+57 x+36$
• $y^2=15 x^6+19 x^5+35 x^4+4 x^3+24 x^2+3 x+48$
• $y^2=21 x^6+20 x^5+14 x^4+14 x^3+2 x^2+19 x+57$
• $y^2=2 x^6+30 x^3+40$
• $y^2=26 x^6+43 x^5+17 x^4+7 x^3+17 x^2+43 x+26$
• $y^2=12 x^6+56 x^5+36 x^4+47 x^3+27 x^2+12 x+10$
• $y^2=5 x^6+15 x^5+44 x^4+x^3+44 x^2+x+7$
• $y^2=13 x^6+x^5+53 x^4+5 x^3+6 x^2+36 x+8$
• $y^2=4 x^6+23 x^5+x^4+51 x^3+27 x^2+53 x+42$
• $y^2=48 x^6+30 x^5+6 x^4+42 x^3+41 x^2+5 x+24$
• $y^2=46 x^6+36 x^5+49 x^4+15 x^3+49 x^2+36 x+46$
• $y^2=31 x^6+45 x^5+53 x^4+11 x^3+34 x^2+49 x+19$
• $y^2=44 x^6+43 x^5+60 x^4+23 x^2+53 x+44$
• $y^2=51 x^6+5 x^5+48 x^4+18 x^3+48 x^2+5 x+51$
• $y^2=52 x^6+36 x^5+47 x^4+31 x^3+7 x^2+36 x+56$
• $y^2=33 x^6+26 x^4+26 x^2+33$
• $y^2=33 x^6+47 x^5+5 x^4+13 x^3+22 x^2+17 x+8$
• $y^2=22 x^6+36 x^5+31 x^4+58 x^3+48 x^2+19 x+34$
• $y^2=58 x^6+19 x^5+26 x^4+53 x^3+59 x^2+x+52$
• and

• $y^2=3 x^6+42 x^5+41 x^4+38 x^3+20 x^2+24 x+2$
• $y^2=13 x^6+40 x^5+25 x^4+24 x^3+25 x^2+40 x+13$
• $y^2=37 x^6+32 x^5+41 x^4+43 x^3+52 x^2+26 x+8$
• $y^2=29 x^6+52 x^5+55 x^4+4 x^3+11 x^2+7 x+22$
• $y^2=5 x^6+38 x^5+19 x^4+22 x^3+49 x^2+28 x+46$
• $y^2=53 x^6+7 x^5+14 x^4+52 x^3+43 x^2+38 x+60$
• $y^2=31 x^6+35 x^5+42 x^4+20 x^2+31 x$
• $y^2=37 x^6+20 x^5+41 x^3+40 x^2+17 x+45$
• $y^2=40 x^6+8 x^5+52 x^4+60 x^3+55 x^2+26 x+30$
• $y^2=8 x^6+51 x^5+55 x^4+15 x^3+36 x^2+9 x+43$
• $y^2=57 x^6+37 x^5+15 x^4+4 x^3+11 x^2+39$
• $y^2=46 x^6+31 x^5+50 x^4+32 x^3+10 x^2+59 x+28$
• $y^2=32 x^6+15 x^5+60 x^4+59 x^3+60 x^2+15 x+32$
• $y^2=20 x^6+38 x^5+31 x^4+24 x^3+21 x^2+34 x+48$
• $y^2=21 x^6+29 x^5+32 x^4+13 x^3+21 x^2+11 x+40$
• $y^2=2 x^6+2 x^3+18$
• $y^2=x^6+14 x^5+51 x^4+46 x^3+51 x^2+14 x+1$
• $y^2=16 x^6+50 x^4+50 x^2+16$
• $y^2=34 x^6+25 x^5+41 x^4+48 x^3+52 x^2+18 x+30$
• $y^2=50 x^6+49 x^5+2 x^4+12 x^3+43 x^2+34 x+6$
• $y^2=30 x^6+10 x^5+49 x^4+50 x^3+6 x^2+39 x+21$
• $y^2=8 x^6+3 x^5+53 x^4+13 x^2+27 x+50$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-57}) \)$)$

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_ec	$2$	(not in LMFDB)
2.61.i_fi	$2$	(not in LMFDB)
2.61.e_abt	$3$	(not in LMFDB)