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

• Label: 2.71.s_hq
• 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:	$( 1 + 4 x + 71 x^{2} )( 1 + 14 x + 71 x^{2} )$
	$1 + 18 x + 198 x^{2} + 1278 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.576280895962$, $\pm0.812086478560$
Angle rank:	$2$ (numerical)
Jacobians:	$72$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6536$	$25777984$	$127733969576$	$645788405377024$	$3255235786222668776$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$5114$	$356886$	$25413054$	$1804225050$	$128101037978$	$9095110855830$	$645753547970494$	$45848501317313946$	$3255243545006293754$

Jacobians and polarizations

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

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.e $\times$ 1.71.o 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.e : \(\Q(\sqrt{-67}) \).
• 1.71.o : \(\Q(\sqrt{-22}) \).

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.as_hq	$2$	(not in LMFDB)
2.71.ak_di	$2$	(not in LMFDB)
2.71.k_di	$2$	(not in LMFDB)