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

• Label: 2.71.o_he
• Base field: $\F_{71}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$1 + 14 x + 186 x^{2} + 994 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.591225890391$, $\pm0.684633382989$
Angle rank:	$2$ (numerical)
Number field:	4.0.1298000.1
Galois group:	$D_{4}$
Jacobians:	$72$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6236$	$26315920$	$127355464556$	$645822571685120$	$3255432015299855116$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$86$	$5218$	$355826$	$25414398$	$1804333806$	$128099502658$	$9095119483466$	$645753568491198$	$45848500572572966$	$3255243550733723298$

Jacobians and polarizations

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

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is 4.0.1298000.1.

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.ao_he	$2$	(not in LMFDB)