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

• Label: 2.61.v_iu
• 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:	$1 + 21 x + 228 x^{2} + 1281 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.681657552267$, $\pm0.797390396758$
Angle rank:	$2$ (numerical)
Number field:	4.0.1076236.1
Galois group:	$D_{4}$
Jacobians:	$36$
Isomorphism classes:	36

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5252$	$13907296$	$51235129712$	$191859993360256$	$713307811815189812$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$83$	$3737$	$225722$	$13856865$	$844554743$	$51520292102$	$3142744606931$	$191707306783969$	$11694146079562562$	$713342911395776177$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

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

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

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 4.0.1076236.1.

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.av_iu	$2$	(not in LMFDB)