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

• Label: 2.61.h_fc
• 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:	$( 1 + 2 x + 61 x^{2} )( 1 + 5 x + 61 x^{2} )$
	$1 + 7 x + 132 x^{2} + 427 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.540867587811$, $\pm0.603713893500$
Angle rank:	$2$ (numerical)
Jacobians:	$48$
Isomorphism classes:	160

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4288$	$14664960$	$51260535808$	$191590369920000$	$713421316712387008$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$69$	$3937$	$225834$	$13837393$	$844689129$	$51520530022$	$3142737372669$	$191707322495233$	$11694146335634994$	$713342910463187977$

Jacobians and polarizations

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

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.c $\times$ 1.61.f 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.c : \(\Q(\sqrt{-15}) \).
• 1.61.f : \(\Q(\sqrt{-219}) \).

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.ah_fc	$2$	(not in LMFDB)
2.61.ad_ei	$2$	(not in LMFDB)
2.61.d_ei	$2$	(not in LMFDB)