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

• Label: 2.61.r_gy
• Base field: $\F_{61}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 + 17 x + 180 x^{2} + 1037 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.597818699027$, $\pm0.787759329610$
Angle rank:	$2$ (numerical)
Number field:	4.0.4200957.1
Galois group:	$D_{4}$
Jacobians:	$80$
Isomorphism classes:	80
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4956$	$14114688$	$51258738384$	$191764465976832$	$713343785884258956$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$79$	$3793$	$225826$	$13849969$	$844597339$	$51520548454$	$3142739711575$	$191707318344385$	$11694146379941770$	$713342908411829593$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 80 curves (of which all are hyperelliptic):

• $y^2=5 x^5+40 x^4+18 x^3+29 x^2+5 x+47$
• $y^2=19 x^6+42 x^5+58 x^4+19 x^3+60 x^2+24 x+50$
• $y^2=17 x^6+44 x^5+28 x^4+44 x^3+51 x^2+59 x+4$
• $y^2=16 x^6+26 x^5+8 x^4+31 x^3+31 x^2+49 x+5$
• $y^2=17 x^6+45 x^5+35 x^4+41 x^3+8 x^2+53 x+59$
• $y^2=9 x^6+38 x^5+14 x^4+51 x^3+44 x^2+37 x+37$
• $y^2=44 x^6+9 x^5+29 x^4+43 x^3+37 x^2+43 x+2$
• $y^2=60 x^6+56 x^5+52 x^3+20 x^2+23 x+15$
• $y^2=38 x^6+8 x^5+56 x^4+32 x^3+26 x^2+5 x+43$
• $y^2=56 x^6+44 x^5+26 x^4+19 x^3+28 x^2+9 x+48$
• $y^2=60 x^6+59 x^5+48 x^4+55 x^3+49 x^2+57 x+13$
• $y^2=22 x^6+14 x^5+40 x^4+12 x^3+3 x^2+30 x+11$
• $y^2=51 x^6+35 x^5+31 x^4+60 x^3+43 x^2+50 x+20$
• $y^2=23 x^6+59 x^5+48 x^4+4 x^3+14 x^2+47 x$
• $y^2=40 x^6+44 x^5+7 x^4+16 x^3+56 x^2+4 x+3$
• $y^2=22 x^6+33 x^5+10 x^4+36 x^3+41 x^2+30 x+33$
• $y^2=25 x^6+x^5+23 x^4+3 x^3+37 x^2+12 x+2$
• $y^2=45 x^6+16 x^5+59 x^4+40 x^3+43 x^2+47 x+56$
• $y^2=25 x^6+57 x^5+39 x^4+27 x^3+26 x^2+31 x+12$
• $y^2=58 x^5+30 x^4+25 x^3+41 x^2+47 x+57$
• and

• $y^2=35 x^6+14 x^5+15 x^4+3 x^3+47 x^2+16 x+25$
• $y^2=43 x^6+46 x^5+44 x^4+5 x^3+30 x^2+2 x+31$
• $y^2=x^6+27 x^5+10 x^4+40 x^3+60 x^2+49 x+44$
• $y^2=36 x^6+47 x^5+41 x^4+30 x^3+59 x^2+39 x+58$
• $y^2=40 x^6+56 x^5+36 x^4+50 x^3+29 x^2+2 x+51$
• $y^2=25 x^6+40 x^5+35 x^4+31 x^3+10 x^2+6 x+50$
• $y^2=4 x^6+7 x^5+32 x^4+25 x^3+44 x^2+19 x+31$
• $y^2=3 x^6+40 x^5+40 x^4+13 x^3+49 x^2+33 x+4$
• $y^2=11 x^6+55 x^5+55 x^4+38 x^3+5 x^2+11 x+11$
• $y^2=3 x^6+51 x^5+59 x^4+19 x^3+17 x^2+25 x+17$
• $y^2=39 x^6+34 x^5+34 x^4+15 x^3+34 x^2+33 x+3$
• $y^2=4 x^6+22 x^5+12 x^4+44 x^3+5 x^2+20 x+46$
• $y^2=41 x^6+50 x^5+60 x^4+x^3+26 x^2+45 x+21$
• $y^2=8 x^6+14 x^5+27 x^4+48 x^3+53 x^2+50 x$
• $y^2=33 x^6+19 x^5+4 x^4+36 x^3+42 x^2+60 x+36$
• $y^2=x^6+48 x^5+24 x^4+40 x^3+4 x^2+9 x+33$
• $y^2=50 x^6+x^5+57 x^4+41 x^3+51 x^2+43 x+48$
• $y^2=52 x^6+35 x^5+23 x^4+8 x^2+40 x$
• $y^2=21 x^6+43 x^5+51 x^4+38 x^3+54 x^2+18 x+16$
• $y^2=3 x^6+28 x^5+38 x^4+30 x^3+37 x^2+13 x+47$
• $y^2=43 x^6+15 x^5+56 x^4+6 x^3+46 x^2+13 x+26$
• $y^2=57 x^6+48 x^5+54 x^4+11 x^3+60 x^2+33 x+47$
• $y^2=49 x^6+39 x^5+4 x^4+60 x^3+47 x^2+20 x+4$
• $y^2=3 x^6+10 x^5+36 x^4+27 x^3+60 x^2+27 x+12$
• $y^2=35 x^6+26 x^5+7 x^4+54 x^3+52 x^2+20 x+37$
• $y^2=19 x^6+56 x^5+47 x^4+52 x^3+37 x^2+16 x+30$
• $y^2=13 x^6+34 x^5+33 x^4+20 x^3+31 x^2+57 x+34$
• $y^2=11 x^6+52 x^5+21 x^4+58 x^3+23 x^2+34 x+27$
• $y^2=14 x^6+55 x^5+56 x^4+5 x^3+51 x^2+37 x+42$
• $y^2=38 x^6+52 x^5+54 x^4+5 x^3+16 x^2+31 x+6$
• $y^2=33 x^6+59 x^5+14 x^4+11 x^3+56 x^2+38 x+57$
• $y^2=27 x^6+47 x^5+43 x^4+24 x^3+28 x^2+46 x+3$
• $y^2=60 x^6+20 x^5+45 x^4+5 x^3+16 x^2+55 x+27$
• $y^2=20 x^6+56 x^5+18 x^4+37 x^3+35 x^2+30 x+28$
• $y^2=45 x^6+39 x^5+4 x^4+40 x^3+36 x^2+3 x$
• $y^2=52 x^6+28 x^5+29 x^4+37 x^3+48 x^2+57 x+4$
• $y^2=48 x^6+5 x^5+51 x^4+3 x^3+2 x^2+21 x+51$
• $y^2=3 x^6+28 x^5+36 x^4+17 x^2+11 x+1$
• $y^2=57 x^6+55 x^5+5 x^4+31 x^3+3 x^2+59 x+23$
• $y^2=40 x^6+48 x^5+11 x^4+21 x^3+13 x^2+53 x+22$
• $y^2=6 x^6+35 x^5+55 x^4+53 x^3+36 x^2+3 x+43$
• $y^2=58 x^6+30 x^5+2 x^4+5 x^3+34 x^2+60$
• $y^2=49 x^6+56 x^5+60 x^4+43 x^3+23 x^2+30 x+20$
• $y^2=27 x^6+8 x^5+34 x^4+44 x^3+56 x^2+53 x+44$
• $y^2=26 x^6+21 x^5+24 x^4+29 x^3+53 x^2+44 x+39$
• $y^2=4 x^6+50 x^5+11 x^4+50 x^3+45 x^2+x+27$
• $y^2=6 x^6+30 x^5+20 x^4+49 x^3+7 x^2+48 x+31$
• $y^2=17 x^6+26 x^5+9 x^4+3 x^3+53 x^2+29 x+39$
• $y^2=5 x^6+2 x^5+14 x^4+45 x^3+26 x^2+34 x+23$
• $y^2=x^6+24 x^5+7 x^4+19 x^3+17 x^2+36 x+9$
• $y^2=46 x^6+48 x^5+58 x^4+37 x^3+47 x^2+18 x+2$
• $y^2=57 x^6+40 x^5+5 x^4+32 x^3+15 x^2+41 x+23$
• $y^2=19 x^6+42 x^5+48 x^4+56 x^3+23 x^2+26 x+29$
• $y^2=25 x^6+10 x^5+27 x^4+51 x^3+55 x^2+32 x+52$
• $y^2=8 x^6+58 x^5+17 x^4+13 x^3+16 x^2+9 x+52$
• $y^2=39 x^6+9 x^5+23 x^4+26 x^3+33 x^2+15 x+35$
• $y^2=49 x^6+55 x^5+38 x^4+22 x^3+4 x^2+13 x+48$
• $y^2=34 x^6+10 x^5+5 x^4+49 x^3+42 x^2+19 x+40$
• $y^2=19 x^6+3 x^5+10 x^4+53 x^3+51 x^2+40 x+3$
• $y^2=12 x^6+52 x^5+44 x^4+22 x^3+12 x^2+47 x+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is 4.0.4200957.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.61.ar_gy	$2$	(not in LMFDB)