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

• Label: 2.61.as_gg
• Base field: $\F_{61}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 - 18 x + 162 x^{2} - 1098 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0531652068084$, $\pm0.446834793192$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{41})\)
Galois group:	$C_2^2$
Jacobians:	$72$
Isomorphism classes:	124
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2768$	$13840000$	$51434646608$	$191545600000000$	$713273575365721808$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$44$	$3722$	$226604$	$13834158$	$844514204$	$51520374362$	$3142744713404$	$191707300122718$	$11694145862442764$	$713342911662882602$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 72 curves (of which all are hyperelliptic):

• $y^2=52 x^6+12 x^5+12 x^4+33 x^3+49 x^2+32$
• $y^2=47 x^6+43 x^5+20 x^4+55 x^3+46 x^2+15 x+38$
• $y^2=31 x^6+51 x^5+58 x^4+52 x^3+2 x^2+9 x$
• $y^2=19 x^6+33 x^5+50 x^4+23 x^3+35 x^2+48 x+13$
• $y^2=8 x^6+43 x^5+56 x^4+36 x^3+9 x^2+39 x+35$
• $y^2=7 x^6+4 x^5+48 x^4+48 x^3+53 x^2+6 x+8$
• $y^2=57 x^6+59 x^5+45 x^4+7 x^3+13 x^2+20 x+18$
• $y^2=29 x^6+9 x^5+19 x^4+30 x^3+7 x^2+31 x+39$
• $y^2=26 x^6+x^5+4 x^4+x^3+34 x^2+7 x+18$
• $y^2=4 x^6+10 x^5+13 x^4+13 x^3+54 x^2+31 x+11$
• $y^2=55 x^6+38 x^5+27 x^4+57 x^3+25 x^2+7$
• $y^2=8 x^6+57 x^5+25 x^4+46 x^3+16 x^2+26$
• $y^2=19 x^6+26 x^5+31 x^4+34 x^3+3 x^2+24 x+29$
• $y^2=46 x^6+46 x^5+31 x^4+3 x^3+45 x^2+24 x+8$
• $y^2=18 x^6+50 x^5+25 x^4+47 x^3+38 x^2+30$
• $y^2=8 x^6+36 x^5+14 x^4+10 x^3+51 x+55$
• $y^2=2 x^6+57 x^5+57 x^4+60 x^3+19 x^2+14 x+55$
• $y^2=26 x^6+21 x^5+57 x^4+38 x^3+50 x^2+53 x+26$
• $y^2=44 x^6+6 x^5+18 x^4+44 x^3+55 x^2+16 x+59$
• $y^2=9 x^6+46 x^5+28 x^4+46 x^3+10 x^2+43 x+39$
• and

• $y^2=28 x^6+6 x^5+5 x^4+17 x^3+50 x^2+x+14$
• $y^2=36 x^6+43 x^5+21 x^4+41 x^3+3 x^2+6 x+54$
• $y^2=55 x^6+45 x^5+56 x^4+11 x^3+4 x^2+2 x+2$
• $y^2=6 x^6+50 x^5+59 x^4+27 x^3+5 x^2+21 x+33$
• $y^2=48 x^6+13 x^5+28 x^4+3 x^3+42 x^2+27 x+51$
• $y^2=4 x^6+17 x^5+6 x^4+33 x^3+28 x^2+4 x+59$
• $y^2=41 x^6+54 x^5+41 x^4+15 x^2+29 x+19$
• $y^2=13 x^6+44 x^5+51 x^4+39 x^3+32 x^2+55$
• $y^2=x^6+50 x^5+46 x^4+33 x^3+40 x^2+36 x+1$
• $y^2=59 x^6+48 x^5+38 x^4+31 x^3+59 x^2+37 x+29$
• $y^2=44 x^6+48 x^5+40 x^4+8 x^3+53 x^2+7 x+7$
• $y^2=26 x^6+43 x^5+49 x^4+27 x^3+57 x^2+32 x+31$
• $y^2=31 x^6+39 x^5+37 x^4+35 x^3+4 x^2+48 x+26$
• $y^2=59 x^6+44 x^5+12 x^4+38 x^3+55 x^2+14 x+47$
• $y^2=57 x^6+45 x^5+40 x^4+14 x^3+30 x^2+29 x+52$
• $y^2=34 x^6+52 x^5+10 x^4+19 x^3+19 x^2+59 x+57$
• $y^2=21 x^6+57 x^5+22 x^4+28 x^3+40 x^2+10 x+22$
• $y^2=19 x^6+20 x^5+56 x^4+12 x^3+24 x^2+45 x+28$
• $y^2=56 x^6+36 x^5+20 x^4+59 x^3+8 x^2+42 x+59$
• $y^2=44 x^6+46 x^5+8 x^4+21 x^3+39 x^2+57 x+10$
• $y^2=53 x^5+16 x^4+16 x^2+8 x$
• $y^2=9 x^6+25 x^5+24 x^4+24 x^2+36 x+9$
• $y^2=55 x^6+33 x^5+8 x^4+38 x^3+35 x^2+24 x+60$
• $y^2=50 x^5+16 x^4+16 x^2+11 x$
• $y^2=8 x^5+28 x^4+19 x^3+54 x^2+24 x+19$
• $y^2=53 x^6+3 x^5+23 x^4+26 x^3+15 x^2+49 x+31$
• $y^2=60 x^6+5 x^5+x^4+54 x^3+34 x^2+32 x+26$
• $y^2=2 x^6+39 x^5+56 x^4+52 x^3+20 x^2+4 x+56$
• $y^2=8 x^6+18 x^5+22 x^4+58 x^3+21 x^2+19 x+54$
• $y^2=13 x^6+8 x^5+8 x^4+50 x^3+60 x^2+57 x+26$
• $y^2=54 x^6+18 x^5+5 x^4+5 x^3+50 x^2+17 x+56$
• $y^2=37 x^6+10 x^5+29 x^4+37 x^3+51 x^2+59$
• $y^2=10 x^6+18 x^5+16 x^4+22 x^3+20 x^2+57 x+18$
• $y^2=21 x^5+8 x^4+8 x^2+40 x$
• $y^2=43 x^5+55 x^4+52 x^3+3 x^2+10 x+60$
• $y^2=12 x^6+31 x^5+12 x^4+53 x^3+22 x^2+51 x+51$
• $y^2=21 x^6+57 x^5+32 x^4+44 x^3+47 x^2+20 x+31$
• $y^2=21 x^6+27 x^5+7 x^4+40 x^3+36 x^2+4 x+38$
• $y^2=29 x^6+59 x^5+7 x^4+40 x^3+37 x^2+57 x+53$
• $y^2=48 x^6+52 x^5+3 x^4+46 x^3+14 x^2+22 x+48$
• $y^2=41 x^6+40 x^5+58 x^4+7 x^3+16 x^2+38 x+52$
• $y^2=39 x^6+24 x^5+41 x^4+52 x^3+53 x^2+17 x+12$
• $y^2=30 x^6+43 x^5+22 x^4+34 x^3+10 x^2+10 x+20$
• $y^2=21 x^6+x^5+9 x^4+8 x^3+30 x^2+27 x+35$
• $y^2=47 x^6+23 x^5+16 x^4+46 x^3+8 x^2+13 x+54$
• $y^2=12 x^6+48 x^5+23 x^4+11 x^3+44 x^2+49 x+5$
• $y^2=17 x^6+31 x^5+58 x^4+22 x^3+22 x^2+57 x+39$
• $y^2=25 x^6+7 x^5+10 x^4+10 x^3+24 x^2+3 x+32$
• $y^2=31 x^6+9 x^5+8 x^4+28 x^3+53 x^2+35 x+14$
• $y^2=31 x^6+18 x^5+16 x^3+51 x^2+57 x+8$
• $y^2=52 x^6+53 x^5+60 x^4+30 x^3+19 x^2+53 x+14$
• $y^2=28 x^6+26 x^5+54 x^4+45 x^3+13 x^2+29 x+22$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61^{4}}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{41})\).

Endomorphism algebra over $\overline{\F}_{61}$

The base change of $A$ to $\F_{61^{4}}$ is 1.13845841.aiqs^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-41}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{61^{2}}$

  The base change of $A$ to $\F_{61^{2}}$ is the simple isogeny class 2.3721.a_aiqs and its endomorphism algebra is \(\Q(i, \sqrt{41})\).

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.s_gg	$2$	(not in LMFDB)
2.61.a_abo	$8$	(not in LMFDB)
2.61.a_bo	$8$	(not in LMFDB)