Abelian variety isogeny class 2.71.aq_hy over $\F_{71}$

• Label: 2.71.aq_hy
• Base field: $\F_{71}$
• 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:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$( 1 - 8 x + 71 x^{2} )^{2}$
	$1 - 16 x + 206 x^{2} - 1136 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.342551982147$, $\pm0.342551982147$
Angle rank:	$1$ (numerical)
Jacobians:	$88$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4096$	$26214400$	$128955682816$	$645956789862400$	$3255053579256303616$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$5198$	$360296$	$25419678$	$1804124056$	$128098873838$	$9095116353736$	$645753600924478$	$45848501546009336$	$3255243552683174798$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 88 curves (of which all are hyperelliptic):

• $y^2=21 x^6+69 x^5+54 x^4+26 x^3+38 x^2+27 x+51$
• $y^2=12 x^6+53 x^5+19 x^4+62 x^3+28 x^2+66 x+42$
• $y^2=29 x^5+7 x^4+66 x^3+5 x^2+23 x+10$
• $y^2=9 x^6+57 x^5+43 x^4+12 x^3+43 x^2+57 x+9$
• $y^2=21 x^6+11 x^5+68 x^4+25 x^3+35 x^2+22 x+7$
• $y^2=56 x^6+66 x^5+52 x^4+59 x^3+31 x^2+63 x+13$
• $y^2=36 x^6+23 x^5+10 x^4+8 x^3+35 x^2+40 x+46$
• $y^2=22 x^6+20 x^5+30 x^4+2 x^3+30 x^2+20 x+22$
• $y^2=28 x^6+29 x^5+39 x^4+43 x^3+39 x^2+29 x+28$
• $y^2=33 x^6+19 x^5+48 x^4+7 x^3+27 x^2+21 x+63$
• $y^2=36 x^6+44 x^5+4 x^4+8 x^3+21 x^2+23 x+13$
• $y^2=9 x^6+24 x^5+2 x^4+8 x^3+2 x^2+24 x+9$
• $y^2=5 x^6+54 x^4+54 x^2+5$
• $y^2=23 x^6+31 x^5+60 x^4+58 x^3+60 x^2+31 x+23$
• $y^2=51 x^6+47 x^5+25 x^4+54 x^3+30 x^2+62 x+62$
• $y^2=20 x^6+61 x^5+64 x^4+47 x^3+5 x^2+63 x+19$
• $y^2=20 x^6+20 x^5+59 x^4+51 x^3+29 x^2+16 x+21$
• $y^2=21 x^6+51 x^5+36 x^4+4 x^3+61 x^2+31 x+61$
• $y^2=21 x^6+45 x^4+45 x^2+21$
• $y^2=17 x^6+37 x^5+30 x^4+19 x^3+63 x^2+19 x+16$
• and

• $y^2=65 x^6+38 x^5+68 x^4+46 x^3+68 x^2+38 x+65$
• $y^2=69 x^6+2 x^5+49 x^4+53 x^3+31 x^2+28 x+5$
• $y^2=56 x^6+35 x^5+3 x^4+41 x^3+40 x^2+17 x+39$
• $y^2=7 x^6+59 x^5+30 x^4+62 x^3+30 x^2+59 x+7$
• $y^2=6 x^6+26 x^4+26 x^2+6$
• $y^2=63 x^6+15 x^5+48 x^4+29 x^3+63 x^2+68 x+56$
• $y^2=35 x^6+51 x^5+27 x^4+57 x^3+37 x^2+35 x+39$
• $y^2=52 x^6+31 x^5+59 x^4+65 x^3+63 x^2+69 x+68$
• $y^2=28 x^6+50 x^5+30 x^4+7 x^3+61 x^2+59 x+41$
• $y^2=5 x^6+2 x^5+63 x^4+38 x^3+63 x^2+2 x+5$
• $y^2=56 x^6+43 x^5+67 x^4+46 x^3+26 x^2+24 x+28$
• $y^2=24 x^6+41 x^5+57 x^4+61 x^3+57 x^2+41 x+24$
• $y^2=65 x^6+60 x^5+52 x^4+65 x^3+46 x^2+39 x+7$
• $y^2=69 x^6+42 x^5+56 x^4+46 x^3+31 x^2+62 x+62$
• $y^2=53 x^6+30 x^5+58 x^4+40 x^3+23 x^2+6 x+24$
• $y^2=63 x^5+29 x^4+12 x^3+20 x^2+45 x+36$
• $y^2=38 x^6+35 x^5+33 x^4+49 x^3+33 x^2+35 x+38$
• $y^2=46 x^6+32 x^5+2 x^4+18 x^3+49 x^2+38 x+47$
• $y^2=19 x^6+31 x^5+31 x^4+24 x^3+41 x^2+13 x+48$
• $y^2=3 x^6+5 x^5+51 x^4+28 x^3+44 x^2+14 x+43$
• $y^2=38 x^5+20 x^4+63 x^3+46 x^2+5 x+42$
• $y^2=31 x^6+40 x^5+51 x^4+29 x^3+51 x^2+40 x+31$
• $y^2=37 x^6+20 x^5+31 x^4+46 x^3+23 x^2+43 x+6$
• $y^2=6 x^6+41 x^5+27 x^4+46 x^3+27 x^2+41 x+6$
• $y^2=41 x^6+45 x^5+29 x^4+28 x^3+29 x^2+45 x+41$
• $y^2=34 x^6+58 x^5+12 x^4+70 x^3+12 x^2+58 x+34$
• $y^2=66 x^6+45 x^5+15 x^4+27 x^3+15 x^2+45 x+66$
• $y^2=32 x^6+18 x^5+48 x^4+39 x^3+19 x^2+30 x+29$
• $y^2=26 x^6+21 x^5+16 x^4+38 x^3+7 x^2+4 x+2$
• $y^2=33 x^6+41 x^5+10 x^4+6 x^3+x^2+44 x+4$
• $y^2=38 x^6+31 x^4+31 x^2+38$
• $y^2=37 x^6+63 x^5+10 x^4+25 x^3+15 x^2+53 x+45$
• $y^2=45 x^6+32 x^5+49 x^4+7 x^3+64 x^2+5 x+57$
• $y^2=56 x^6+63 x^5+55 x^4+61 x^3+55 x^2+63 x+56$
• $y^2=40 x^6+22 x^5+21 x^4+57 x^3+19 x^2+8 x+46$
• $y^2=26 x^6+31 x^5+22 x^4+26 x^3+22 x^2+31 x+26$
• $y^2=6 x^6+49 x^4+49 x^2+6$
• $y^2=70 x^6+55 x^5+17 x^4+43 x^3+14 x^2+33 x+69$
• $y^2=50 x^6+42 x^4+42 x^2+50$
• $y^2=16 x^6+70 x^4+70 x^2+16$
• $y^2=5 x^6+7 x^5+51 x^4+28 x^3+51 x^2+7 x+5$
• $y^2=55 x^6+36 x^5+69 x^4+32 x^3+65 x^2+40 x+65$
• $y^2=58 x^6+50 x^4+50 x^2+58$
• $y^2=68 x^6+12 x^5+23 x^4+7 x^3+33 x^2+42 x+65$
• $y^2=26 x^6+20 x^5+6 x^4+19 x^3+54 x^2+58 x+68$
• $y^2=14 x^6+32 x^5+36 x^4+55 x^3+36 x^2+32 x+14$
• $y^2=35 x^6+35 x^5+50 x^4+56 x^3+55 x^2+15 x+37$
• $y^2=35 x^6+18 x^5+32 x^4+32 x^2+18 x+35$
• $y^2=3 x^6+16 x^5+23 x^4+35 x^3+23 x^2+16 x+3$
• $y^2=56 x^6+25 x^5+32 x^4+10 x^3+8 x^2+6 x+63$
• $y^2=65 x^6+38 x^5+69 x^4+46 x^3+69 x^2+38 x+65$
• $y^2=22 x^6+67 x^5+64 x^4+26 x^3+25 x^2+14 x+31$
• $y^2=49 x^6+27 x^5+37 x^4+5 x^3+30 x^2+8 x+9$
• $y^2=46 x^6+61 x^5+70 x^4+43 x^3+11 x^2+68 x+47$
• $y^2=45 x^6+70 x^5+54 x^4+34 x^3+54 x^2+70 x+45$
• $y^2=15 x^6+20 x^5+39 x^4+39 x^2+20 x+15$
• $y^2=42 x^6+49 x^5+36 x^4+29 x^3+15 x^2+9 x+59$
• $y^2=57 x^6+17 x^5+64 x^4+64 x^3+64 x^2+17 x+57$
• $y^2=62 x^6+47 x^5+37 x^4+6 x^3+37 x^2+47 x+62$
• $y^2=70 x^5+35 x^4+15 x^3+62 x^2+31 x$
• $y^2=12 x^6+64 x^5+30 x^4+69 x^3+30 x^2+64 x+12$
• $y^2=27 x^6+10 x^5+40 x^4+34 x^3+40 x^2+10 x+27$
• $y^2=8 x^6+36 x^5+49 x^4+29 x^3+2 x^2+49 x+32$
• $y^2=69 x^6+34 x^5+50 x^4+47 x^3+56 x^2+60 x+33$
• $y^2=x^5+49 x^4+16 x^3+24 x^2+54 x$
• $y^2=5 x^6+37 x^5+15 x^4+48 x^3+25 x^2+16 x+10$
• $y^2=34 x^6+39 x^5+29 x^4+60 x^3+41 x^2+45 x+61$
• $y^2=23 x^6+29 x^5+63 x^4+63 x^3+63 x^2+29 x+23$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71}$.

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.ai^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-55}) \)$)$

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
2.71.a_da	$2$	(not in LMFDB)
2.71.q_hy	$2$	(not in LMFDB)
2.71.i_ah	$3$	(not in LMFDB)