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

• Label: 2.71.s_ip
• 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:	$( 1 + 9 x + 71 x^{2} )^{2}$
	$1 + 18 x + 223 x^{2} + 1278 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.679331255589$, $\pm0.679331255589$
Angle rank:	$1$ (numerical)
Jacobians:	$42$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6561$	$26040609$	$127252012176$	$646076909949849$	$3255341340975115401$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$5164$	$355536$	$25424404$	$1804283550$	$128098892878$	$9095128829730$	$645753551967844$	$45848499916285296$	$3255243556758087004$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.j^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-203}) \)$)$

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.as_ip	$2$	(not in LMFDB)
2.71.a_cj	$2$	(not in LMFDB)
2.71.aj_k	$3$	(not in LMFDB)