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

• Label: 2.61.k_fr
• 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:	$( 1 + 5 x + 61 x^{2} )^{2}$
	$1 + 10 x + 147 x^{2} + 610 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.603713893500$, $\pm0.603713893500$
Angle rank:	$1$ (numerical)
Jacobians:	$44$
Cyclic group of points:	no
Non-cyclic primes:	$67$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4489$	$14584761$	$51162820864$	$191652875015625$	$713440932119822929$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$3916$	$225402$	$13841908$	$844712352$	$51520034086$	$3142737458352$	$191707360642468$	$11694146182646082$	$713342908307466556$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 44 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.f^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-219}) \)$)$

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.ak_fr	$2$	(not in LMFDB)
2.61.a_dt	$2$	(not in LMFDB)
2.61.af_abk	$3$	(not in LMFDB)