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

• Label: 2.71.ao_hf
• 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:	$( 1 - 9 x + 71 x^{2} )( 1 - 5 x + 71 x^{2} )$
	$1 - 14 x + 187 x^{2} - 994 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.320668744411$, $\pm0.404115390374$
Angle rank:	$2$ (numerical)
Jacobians:	$96$
Isomorphism classes:	160

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4221$	$26326377$	$128863753200$	$645823542897225$	$3255041710561131981$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$58$	$5220$	$360040$	$25414436$	$1804117478$	$128099420622$	$9095121002858$	$645753579419716$	$45848500940805880$	$3255243550305642180$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 96 curves (of which all are hyperelliptic):

• $y^2=53 x^6+19 x^5+34 x^4+2 x^3+34 x^2+19 x+53$
• $y^2=6 x^6+15 x^5+65 x^4+65 x^3+65 x^2+15 x+6$
• $y^2=6 x^6+62 x^5+3 x^4+70 x^3+3 x^2+62 x+6$
• $y^2=35 x^6+41 x^5+34 x^4+25 x^3+x^2+11 x+51$
• $y^2=59 x^6+63 x^5+15 x^4+70 x^3+15 x^2+63 x+59$
• $y^2=63 x^6+59 x^5+34 x^4+57 x^3+34 x^2+59 x+63$
• $y^2=5 x^6+34 x^5+11 x^3+28 x^2+23 x+46$
• $y^2=2 x^6+69 x^5+51 x^4+69 x^3+45 x^2+8 x+20$
• $y^2=31 x^6+52 x^5+54 x^4+70 x^3+56 x^2+60 x+36$
• $y^2=68 x^6+8 x^5+3 x^4+31 x^3+3 x^2+8 x+68$
• $y^2=41 x^6+37 x^5+16 x^4+31 x^3+18 x^2+7 x+60$
• $y^2=26 x^6+20 x^5+42 x^4+41 x^3+42 x^2+20 x+26$
• $y^2=56 x^6+53 x^5+49 x^4+27 x^3+49 x^2+53 x+56$
• $y^2=54 x^6+18 x^5+10 x^4+7 x^3+10 x^2+18 x+54$
• $y^2=64 x^6+18 x^5+32 x^4+25 x^3+22 x^2+46 x+49$
• $y^2=44 x^6+57 x^5+64 x^4+35 x^3+64 x^2+57 x+44$
• $y^2=7 x^6+21 x^5+25 x^4+69 x^3+25 x^2+21 x+7$
• $y^2=70 x^6+55 x^5+28 x^4+30 x^3+64 x^2+26 x+22$
• $y^2=13 x^6+46 x^5+31 x^4+9 x^3+28 x^2+56 x+10$
• $y^2=3 x^6+59 x^5+6 x^4+31 x^3+64 x^2+23 x+26$
• and

• $y^2=37 x^6+14 x^5+30 x^4+42 x^3+14 x^2+61 x+11$
• $y^2=33 x^6+32 x^5+42 x^4+27 x^3+42 x^2+17 x+22$
• $y^2=20 x^6+45 x^5+62 x^4+60 x^3+62 x^2+45 x+20$
• $y^2=53 x^6+7 x^5+6 x^4+21 x^3+6 x^2+7 x+53$
• $y^2=26 x^6+46 x^5+45 x^4+62 x^3+5 x^2+55 x+50$
• $y^2=25 x^6+48 x^5+31 x^4+47 x^3+14 x^2+13 x+64$
• $y^2=35 x^6+22 x^5+62 x^4+27 x^3+3 x^2+7 x+29$
• $y^2=6 x^6+49 x^5+32 x^4+x^3+32 x^2+49 x+6$
• $y^2=45 x^6+17 x^5+12 x^4+x^3+40 x^2+65 x+39$
• $y^2=9 x^6+28 x^5+12 x^4+5 x^3+12 x^2+28 x+9$
• $y^2=50 x^6+20 x^5+12 x^4+60 x^3+5 x^2+63 x+47$
• $y^2=33 x^6+8 x^5+64 x^4+49 x^3+64 x^2+8 x+33$
• $y^2=33 x^6+23 x^5+33 x^4+20 x^3+16 x^2+5 x+44$
• $y^2=65 x^6+33 x^5+64 x^4+13 x^3+64 x^2+33 x+65$
• $y^2=27 x^6+13 x^5+55 x^4+35 x^3+14 x^2+22 x+1$
• $y^2=25 x^6+45 x^5+11 x^4+36 x^3+28 x^2+52 x+49$
• $y^2=21 x^6+63 x^5+45 x^4+15 x^3+37 x^2+18 x+60$
• $y^2=30 x^6+3 x^5+41 x^4+51 x^3+53 x^2+17 x+52$
• $y^2=44 x^6+39 x^5+67 x^4+38 x^3+67 x^2+39 x+44$
• $y^2=31 x^6+27 x^5+63 x^4+15 x^3+63 x^2+27 x+31$
• $y^2=17 x^6+51 x^5+x^4+29 x^3+26 x^2+26 x+18$
• $y^2=38 x^6+13 x^5+53 x^4+32 x^3+53 x^2+13 x+38$
• $y^2=66 x^6+49 x^5+25 x^4+31 x^3+48 x^2+24 x+49$
• $y^2=67 x^6+40 x^5+39 x^4+51 x^3+39 x^2+40 x+67$
• $y^2=63 x^6+33 x^5+30 x^4+43 x^3+30 x^2+33 x+63$
• $y^2=14 x^6+22 x^5+29 x^4+37 x^3+10 x^2+25 x+4$
• $y^2=54 x^6+65 x^5+29 x^4+56 x^3+50 x^2+9 x+66$
• $y^2=65 x^6+39 x^5+47 x^4+30 x^3+21 x^2+15 x+29$
• $y^2=64 x^6+22 x^5+23 x^4+50 x^3+49 x^2+5 x+21$
• $y^2=16 x^6+55 x^5+21 x^4+11 x^3+22 x^2+58 x+46$
• $y^2=48 x^6+46 x^5+56 x^4+18 x^3+56 x^2+46 x+48$
• $y^2=62 x^6+x^5+26 x^4+37 x^3+26 x^2+x+62$
• $y^2=22 x^6+35 x^5+69 x^4+40 x^3+42 x^2+66 x+29$
• $y^2=56 x^6+13 x^5+4 x^4+65 x^3+4 x^2+13 x+56$
• $y^2=31 x^6+x^5+7 x^4+22 x^3+62 x^2+20 x+53$
• $y^2=67 x^6+64 x^5+41 x^4+64 x^3+41 x^2+64 x+67$
• $y^2=32 x^6+65 x^4+62 x^3+65 x^2+32$
• $y^2=59 x^6+24 x^5+37 x^4+31 x^3+37 x^2+24 x+59$
• $y^2=37 x^6+26 x^5+57 x^4+70 x^3+30 x^2+14 x+46$
• $y^2=35 x^6+44 x^5+49 x^4+61 x^3+49 x^2+44 x+35$
• $y^2=20 x^6+55 x^5+48 x^4+13 x^3+48 x^2+7 x+61$
• $y^2=28 x^6+11 x^5+38 x^4+42 x^3+24 x^2+60 x+62$
• $y^2=59 x^6+64 x^5+60 x^4+36 x^3+60 x^2+64 x+59$
• $y^2=38 x^6+66 x^5+25 x^4+30 x^3+25 x^2+66 x+38$
• $y^2=52 x^6+62 x^5+49 x^4+39 x^3+49 x^2+62 x+52$
• $y^2=61 x^6+34 x^5+5 x^4+46 x^3+5 x^2+34 x+61$
• $y^2=53 x^6+10 x^5+57 x^4+8 x^3+56 x^2+46 x+4$
• $y^2=x^6+39 x^5+9 x^4+52 x^3+60 x^2+2 x+44$
• $y^2=27 x^6+35 x^5+30 x^4+56 x^3+40 x^2+40 x+41$
• $y^2=57 x^6+14 x^5+61 x^4+8 x^3+22 x^2+65 x+61$
• $y^2=12 x^6+2 x^5+13 x^4+68 x^3+3 x^2+26 x+66$
• $y^2=43 x^6+26 x^5+2 x^4+23 x^3+2 x^2+26 x+43$
• $y^2=38 x^6+37 x^5+34 x^4+34 x^3+34 x^2+37 x+38$
• $y^2=26 x^6+7 x^5+59 x^4+13 x^3+59 x^2+7 x+26$
• $y^2=63 x^6+51 x^5+55 x^4+49 x^3+55 x^2+51 x+63$
• $y^2=7 x^6+68 x^5+8 x^4+11 x^3+8 x^2+68 x+7$
• $y^2=67 x^6+5 x^5+55 x^4+36 x^3+27 x^2+30 x+61$
• $y^2=4 x^6+16 x^5+7 x^4+7 x^2+16 x+4$
• $y^2=49 x^6+28 x^5+37 x^4+69 x^3+70 x^2+59 x+39$
• $y^2=63 x^6+30 x^5+69 x^4+38 x^3+69 x^2+30 x+63$
• $y^2=31 x^6+52 x^5+52 x^4+69 x^3+62 x^2+69 x+39$
• $y^2=57 x^6+3 x^5+12 x^4+25 x^3+38 x^2+31 x+38$
• $y^2=15 x^6+30 x^5+4 x^4+24 x^3+4 x^2+30 x+15$
• $y^2=51 x^6+34 x^5+39 x^4+9 x^3+39 x^2+34 x+51$
• $y^2=49 x^6+6 x^5+68 x^4+38 x^2+19 x+44$
• $y^2=32 x^6+10 x^5+70 x^4+59 x^3+70 x^2+10 x+32$
• $y^2=63 x^6+65 x^5+51 x^4+70 x^3+39 x^2+69 x+63$
• $y^2=67 x^6+27 x^5+53 x^4+30 x^3+x^2+25 x+55$
• $y^2=66 x^6+69 x^5+33 x^4+10 x^3+33 x^2+69 x+66$
• $y^2=x^6+62 x^5+60 x^4+20 x^3+60 x^2+62 x+1$
• $y^2=12 x^6+61 x^5+42 x^4+69 x^3+42 x^2+61 x+12$
• $y^2=17 x^6+28 x^5+29 x^4+10 x^3+16 x^2+39 x+62$
• $y^2=7 x^6+30 x^5+32 x^4+2 x^3+32 x^2+30 x+7$
• $y^2=17 x^6+49 x^5+67 x^4+67 x^3+38 x^2+8 x+69$
• $y^2=65 x^6+28 x^5+27 x^4+42 x^3+27 x^2+28 x+65$
• $y^2=60 x^6+39 x^5+11 x^4+51 x^3+5 x^2+19 x+45$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.aj $\times$ 1.71.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.71.aj : \(\Q(\sqrt{-203}) \).
• 1.71.af : \(\Q(\sqrt{-259}) \).

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
2.71.ae_dt	$2$	(not in LMFDB)
2.71.e_dt	$2$	(not in LMFDB)
2.71.o_hf	$2$	(not in LMFDB)