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

• Label: 2.61.au_io
• 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:	$( 1 - 10 x + 61 x^{2} )^{2}$
	$1 - 20 x + 222 x^{2} - 1220 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.278857938376$, $\pm0.278857938376$
Angle rank:	$1$ (numerical)
Jacobians:	$49$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2704$	$14017536$	$51898307344$	$191900067840000$	$713374923911023504$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$3766$	$228642$	$13859758$	$844634202$	$51519904486$	$3142735825362$	$191707271553118$	$11694146106042762$	$713342914323062806$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 49 curves (of which all are hyperelliptic):

• $y^2=6 x^6+58 x^5+35 x^3+58 x^2+7 x+21$
• $y^2=32 x^6+23 x^5+46 x^4+39 x^3+13 x^2+33 x+35$
• $y^2=16 x^6+20 x^5+45 x^4+24 x^3+39 x^2+34 x+13$
• $y^2=28 x^6+41 x^5+18 x^4+28 x^3+18 x^2+41 x+28$
• $y^2=47 x^6+25 x^4+25 x^2+47$
• $y^2=52 x^6+53 x^4+53 x^2+52$
• $y^2=24 x^6+10 x^4+10 x^2+24$
• $y^2=59 x^6+18 x^4+18 x^2+59$
• $y^2=51 x^6+24 x^5+3 x^4+49 x^3+15 x^2+51 x+31$
• $y^2=17 x^6+45 x^4+45 x^2+17$
• $y^2=43 x^6+22 x^5+5 x^4+18 x^3+31 x^2+18 x+18$
• $y^2=7 x^6+29 x^5+5 x^4+8 x^3+x^2+28 x+43$
• $y^2=14 x^6+33 x^5+42 x^4+3 x^3+51 x^2+34 x+16$
• $y^2=2 x^6+38 x^3+54$
• $y^2=40 x^6+24 x^5+27 x^4+11 x^3+25 x^2+2 x+43$
• $y^2=27 x^6+39 x^5+19 x^4+32 x^3+19 x^2+39 x+27$
• $y^2=32 x^6+32 x^5+55 x^4+8 x^3+50 x^2+28 x+18$
• $y^2=39 x^6+32 x^5+9 x^4+12 x^3+60 x^2+26 x+13$
• $y^2=17 x^6+53 x^5+36 x^4+38 x^3+36 x^2+53 x+17$
• $y^2=2 x^6+2 x^3+43$
• and

• $y^2=48 x^6+41 x^5+49 x^4+2 x^3+25 x^2+3 x+15$
• $y^2=43 x^6+2 x^5+4 x^4+x^3+57 x^2+51 x+21$
• $y^2=23 x^6+15 x^4+15 x^2+23$
• $y^2=46 x^6+33 x^5+58 x^4+17 x^3+25 x^2+10 x+55$
• $y^2=41 x^6+27 x^5+21 x^4+12 x^3+12 x^2+14 x+46$
• $y^2=6 x^6+56 x^5+47 x^4+54 x^3+34 x^2+34 x+59$
• $y^2=2 x^6+2 x^3+6$
• $y^2=38 x^6+38 x^5+42 x^4+11 x^3+42 x^2+38 x+38$
• $y^2=23 x^6+28 x^5+11 x^4+10 x^3+11 x^2+28 x+23$
• $y^2=21 x^6+5 x^5+55 x^4+26 x^3+32 x^2+27 x+54$
• $y^2=3 x^6+40 x^5+2 x^4+50 x^3+55 x^2+55 x+41$
• $y^2=2 x^6+8 x^3+6$
• $y^2=2 x^6+28 x^3+59$
• $y^2=51 x^6+27 x^5+35 x^4+21 x^3+51 x^2+52 x+31$
• $y^2=18 x^6+42 x^5+15 x^4+9 x^3+57 x^2+x+18$
• $y^2=20 x^6+15 x^5+16 x^4+50 x^3+46 x^2+22 x+3$
• $y^2=33 x^6+26 x^5+23 x^4+4 x^3+37 x^2+32 x+38$
• $y^2=35 x^6+4 x^5+15 x^4+34 x^3+9 x^2+60 x+10$
• $y^2=28 x^6+26 x^5+25 x^4+15 x^3+49 x^2+30 x+53$
• $y^2=46 x^6+55 x^5+6 x^4+45 x^3+6 x^2+55 x+46$
• $y^2=2 x^6+35 x^3+59$
• $y^2=17 x^6+37 x^5+38 x^4+25 x^3+55 x^2+31 x+44$
• $y^2=18 x^6+46 x^5+38 x^4+3 x^3+54 x^2+52 x+7$
• $y^2=32 x^6+34 x^5+11 x^4+50 x^3+24 x^2+58 x+17$
• $y^2=33 x^6+55 x^5+13 x^4+39 x^3+41 x^2+13 x+21$
• $y^2=26 x^5+48 x^4+39 x^3+14 x^2+2 x$
• $y^2=59 x^6+53 x^5+16 x^4+46 x^3+57 x^2+30 x+21$
• $y^2=38 x^6+2 x^5+60 x^4+12 x^3+12 x^2+44 x+33$
• $y^2=32 x^6+17 x^5+48 x^4+18 x^3+9 x^2+28 x+35$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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_w	$2$	(not in LMFDB)
2.61.u_io	$2$	(not in LMFDB)
2.61.k_bn	$3$	(not in LMFDB)