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

• Label: 2.61.a_ec
• 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:	$( 1 - 4 x + 61 x^{2} )( 1 + 4 x + 61 x^{2} )$
	$1 + 106 x^{2} + 3721 x^{4}$
Frobenius angles:	$\pm0.417571001240$, $\pm0.582428998760$
Angle rank:	$1$ (numerical)
Jacobians:	$112$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3828$	$14653584$	$51520382100$	$191602292834304$	$713342910224581428$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$3934$	$226982$	$13838254$	$844596302$	$51520389838$	$3142742836022$	$191707339591774$	$11694146092834142$	$713342908786280254$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 112 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ae $\times$ 1.61.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.61.ae : \(\Q(\sqrt{-57}) \).
• 1.61.e : \(\Q(\sqrt{-57}) \).

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

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