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

• Label: 2.61.a_adt
• 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:	$1 - 97 x^{2} + 3721 x^{4}$
Frobenius angles:	$\pm0.103713893500$, $\pm0.896286106500$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{219})\)
Galois group:	$C_2^2$
Jacobians:	$40$
Cyclic group of points:	no
Non-cyclic primes:	$5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3625$	$13140625$	$51520544500$	$191652875015625$	$713342913340590625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$3528$	$226982$	$13841908$	$844596302$	$51520714638$	$3142742836022$	$191707360642468$	$11694146092834142$	$713342915018298648$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61^{2}}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{219})\).

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

The base change of $A$ to $\F_{61^{2}}$ is 1.3721.adt^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	$4$	(not in LMFDB)
2.61.a_dt	$4$	(not in LMFDB)
2.61.k_fr	$4$	(not in LMFDB)