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

• Label: 2.61.i_fi
• 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:	$( 1 + 4 x + 61 x^{2} )^{2}$
	$1 + 8 x + 138 x^{2} + 488 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.582428998760$, $\pm0.582428998760$
Angle rank:	$1$ (numerical)
Jacobians:	$42$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4356$	$14653584$	$51218026596$	$191602292834304$	$713437382886755076$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$70$	$3934$	$225646$	$13838254$	$844708150$	$51520389838$	$3142735951390$	$191707339591774$	$11694146406418726$	$713342908786280254$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

• $y^2=37 x^6+6 x^5+25 x^4+60 x^3+60 x^2+40 x+1$
• $y^2=60 x^6+10 x^5+41 x^4+13 x^3+13 x^2+59 x+18$
• $y^2=30 x^6+38 x^5+9 x^4+8 x^3+48 x^2+6 x+35$
• $y^2=41 x^6+10 x^5+7 x^4+7 x^3+x^2+40 x+59$
• $y^2=x^6+15 x^3+20$
• $y^2=52 x^6+25 x^5+34 x^4+14 x^3+34 x^2+25 x+52$
• $y^2=24 x^6+51 x^5+11 x^4+33 x^3+54 x^2+24 x+20$
• $y^2=33 x^6+38 x^5+22 x^4+31 x^3+22 x^2+31 x+34$
• $y^2=37 x^6+31 x^5+57 x^4+33 x^3+3 x^2+18 x+4$
• $y^2=8 x^6+46 x^5+2 x^4+41 x^3+54 x^2+45 x+23$
• $y^2=24 x^6+15 x^5+3 x^4+21 x^3+51 x^2+33 x+12$
• $y^2=23 x^6+18 x^5+55 x^4+38 x^3+55 x^2+18 x+23$
• $y^2=46 x^6+53 x^5+57 x^4+36 x^3+17 x^2+55 x+40$
• $y^2=27 x^6+25 x^5+59 x^4+46 x^2+45 x+27$
• $y^2=56 x^6+33 x^5+24 x^4+9 x^3+24 x^2+33 x+56$
• $y^2=26 x^6+18 x^5+54 x^4+46 x^3+34 x^2+18 x+28$
• $y^2=5 x^6+52 x^4+52 x^2+5$
• $y^2=47 x^6+54 x^5+33 x^4+37 x^3+11 x^2+39 x+4$
• $y^2=44 x^6+11 x^5+x^4+55 x^3+35 x^2+38 x+7$
• $y^2=55 x^6+38 x^5+52 x^4+45 x^3+57 x^2+2 x+43$
• and

• $y^2=32 x^6+21 x^5+51 x^4+19 x^3+10 x^2+12 x+1$
• $y^2=26 x^6+19 x^5+50 x^4+48 x^3+50 x^2+19 x+26$
• $y^2=49 x^6+16 x^5+51 x^4+52 x^3+26 x^2+13 x+4$
• $y^2=45 x^6+26 x^5+58 x^4+2 x^3+36 x^2+34 x+11$
• $y^2=33 x^6+19 x^5+40 x^4+11 x^3+55 x^2+14 x+23$
• $y^2=45 x^6+14 x^5+28 x^4+43 x^3+25 x^2+15 x+59$
• $y^2=x^6+9 x^5+23 x^4+40 x^2+x$
• $y^2=49 x^6+10 x^5+51 x^3+20 x^2+39 x+53$
• $y^2=19 x^6+16 x^5+43 x^4+59 x^3+49 x^2+52 x+60$
• $y^2=4 x^6+56 x^5+58 x^4+38 x^3+18 x^2+35 x+52$
• $y^2=53 x^6+13 x^5+30 x^4+8 x^3+22 x^2+17$
• $y^2=31 x^6+x^5+39 x^4+3 x^3+20 x^2+57 x+56$
• $y^2=16 x^6+38 x^5+30 x^4+60 x^3+30 x^2+38 x+16$
• $y^2=40 x^6+15 x^5+x^4+48 x^3+42 x^2+7 x+35$
• $y^2=41 x^6+45 x^5+16 x^4+37 x^3+41 x^2+36 x+20$
• $y^2=x^6+x^3+9$
• $y^2=2 x^6+28 x^5+41 x^4+31 x^3+41 x^2+28 x+2$
• $y^2=32 x^6+39 x^4+39 x^2+32$
• $y^2=17 x^6+43 x^5+51 x^4+24 x^3+26 x^2+9 x+15$
• $y^2=39 x^6+37 x^5+4 x^4+24 x^3+25 x^2+7 x+12$
• $y^2=60 x^6+20 x^5+37 x^4+39 x^3+12 x^2+17 x+42$
• $y^2=4 x^6+32 x^5+57 x^4+37 x^2+44 x+25$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.e^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-57}) \)$)$

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.ai_fi	$2$	(not in LMFDB)
2.61.a_ec	$2$	(not in LMFDB)
2.61.ae_abt	$3$	(not in LMFDB)