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

• Label: 2.61.ay_kc
• 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:	$( 1 - 14 x + 61 x^{2} )( 1 - 10 x + 61 x^{2} )$
	$1 - 24 x + 262 x^{2} - 1464 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.146275019398$, $\pm0.278857938376$
Angle rank:	$2$ (numerical)
Jacobians:	$60$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2496$	$13658112$	$51667761600$	$191830914662400$	$713391542560721856$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$3670$	$227630$	$13854766$	$844653878$	$51520560262$	$3142742866238$	$191707316101726$	$11694146217341510$	$713342913190197430$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 60 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ao $\times$ 1.61.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.61.ao : \(\Q(\sqrt{-3}) \).
• 1.61.ak : \(\Q(\sqrt{-1}) \).

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.ae_as	$2$	(not in LMFDB)
2.61.e_as	$2$	(not in LMFDB)
2.61.y_kc	$2$	(not in LMFDB)
2.61.aj_ei	$3$	(not in LMFDB)
2.61.d_ai	$3$	(not in LMFDB)