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

• Label: 2.61.ao_gp
• 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:	$( 1 - 7 x + 61 x^{2} )^{2}$
	$1 - 14 x + 171 x^{2} - 854 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.352090495177$, $\pm0.352090495177$
Angle rank:	$1$ (numerical)
Jacobians:	$40$
Cyclic group of points:	no
Non-cyclic primes:	$5, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3025$	$14402025$	$51947526400$	$191765857682025$	$713271247777515625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$3868$	$228858$	$13850068$	$844511448$	$51519522598$	$3142742049768$	$191707359451108$	$11694146465972418$	$713342911441167148$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=22 x^6+28 x^5+29 x^4+26 x^3+33 x^2+10 x+45$
• $y^2=10 x^6+38 x^5+21 x^4+44 x^3+46 x^2+6 x+4$
• $y^2=7 x^6+6 x^5+23 x^4+50 x^3+55 x^2+8 x+35$
• $y^2=22 x^6+51 x^5+30 x^4+56 x^3+52 x^2+32 x+20$
• $y^2=60 x^6+35 x^5+21 x^4+56 x^3+18 x^2+17 x+20$
• $y^2=35 x^6+34 x^5+31 x^4+6 x^2+16 x+18$
• $y^2=13 x^6+45 x^5+36 x^4+22 x^3+27 x^2+52 x+56$
• $y^2=2 x^6+2 x^3+7$
• $y^2=44 x^6+50 x^5+22 x^4+34 x^3+14 x^2+28 x+1$
• $y^2=33 x^6+4 x^5+37 x^4+20 x^3+24 x^2+4 x+28$
• $y^2=50 x^6+35 x^5+54 x^4+14 x^3+46 x^2+15 x+59$
• $y^2=5 x^6+28 x^5+43 x^4+44 x^3+59 x^2+17 x+6$
• $y^2=23 x^6+30 x^5+7 x^4+26 x^3+29 x^2+2 x+11$
• $y^2=2 x^6+33 x^5+32 x^4+2 x^2+18 x+7$
• $y^2=26 x^6+34 x^5+28 x^4+8 x^3+11 x^2+58 x+51$
• $y^2=23 x^6+20 x^5+4 x^4+52 x^3+51 x^2+43 x+9$
• $y^2=49 x^6+20 x^5+25 x^4+53 x^3+5 x^2+13 x+57$
• $y^2=48 x^6+29 x^5+56 x^4+37 x^3+22 x^2+10 x+5$
• $y^2=30 x^6+33 x^5+48 x^4+36 x^3+x^2+2 x+31$
• $y^2=48 x^6+54 x^5+10 x^4+3 x^3+52 x^2+28 x+25$
• and

• $y^2=46 x^6+9 x^5+32 x^4+45 x^3+28 x^2+25 x+22$
• $y^2=2 x^6+10 x^3+6$
• $y^2=6 x^6+28 x^5+49 x^4+48 x^3+52 x^2+31 x+54$
• $y^2=54 x^6+20 x^5+13 x^4+44 x^3+53 x^2+11 x+56$
• $y^2=23 x^6+27 x^5+28 x^4+47 x^3+28 x^2+27 x+23$
• $y^2=44 x^6+48 x^5+28 x^4+37 x^3+25 x^2+9 x+40$
• $y^2=24 x^6+7 x^5+51 x^4+14 x^3+18 x^2+30 x+24$
• $y^2=8 x^6+39 x^5+3 x^4+25 x^3+3 x^2+39 x+8$
• $y^2=20 x^6+56 x^5+43 x^4+28 x^3+32 x^2+44 x+18$
• $y^2=53 x^6+19 x^5+7 x^4+7 x^3+53 x^2+41 x+50$
• $y^2=12 x^6+60 x^5+43 x^4+35 x^3+21 x^2+41 x+25$
• $y^2=58 x^6+19 x^5+54 x^4+47 x^3+52 x^2+59 x+2$
• $y^2=20 x^6+10 x^5+x^4+46 x^3+54 x^2+44 x+55$
• $y^2=35 x^6+55 x^5+24 x^4+59 x^3+33 x^2+16 x+32$
• $y^2=24 x^6+10 x^5+54 x^4+3 x^3+45 x^2+50 x+30$
• $y^2=48 x^6+49 x^5+44 x^4+19 x^3+54 x^2+52 x+22$
• $y^2=44 x^6+9 x^5+10 x^4+55 x^3+53 x^2+57 x+17$
• $y^2=54 x^6+29 x^5+25 x^4+4 x^3+12 x^2+10 x+59$
• $y^2=10 x^6+34 x^5+5 x^4+38 x^3+23 x^2+36 x+44$
• $y^2=9 x^6+27 x^5+7 x^4+12 x^3+36 x^2+34 x+34$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ah^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-195}) \)$)$

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_cv	$2$	(not in LMFDB)
2.61.o_gp	$2$	(not in LMFDB)
2.61.h_am	$3$	(not in LMFDB)