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

• Label: 2.61.ao_go
• 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:	$( 1 - 8 x + 61 x^{2} )( 1 - 6 x + 61 x^{2} )$
	$1 - 14 x + 170 x^{2} - 854 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.328850104905$, $\pm0.374508117845$
Angle rank:	$2$ (numerical)
Jacobians:	$48$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3024$	$14394240$	$51937952976$	$191764441128960$	$713276213565850704$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$3866$	$228816$	$13849966$	$844517328$	$51519598058$	$3142742090928$	$191707351748446$	$11694146392915056$	$713342911528634426$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ai $\times$ 1.61.ag 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.ai : \(\Q(\sqrt{-5}) \).
• 1.61.ag : \(\Q(\sqrt{-13}) \).

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.ac_cw	$2$	(not in LMFDB)
2.61.c_cw	$2$	(not in LMFDB)
2.61.o_go	$2$	(not in LMFDB)