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

• Label: 2.71.ac_fn
• 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:	$( 1 - x + 71 x^{2} )^{2}$
	$1 - 2 x + 143 x^{2} - 142 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.481100681038$, $\pm0.481100681038$
Angle rank:	$1$ (numerical)
Jacobians:	$39$
Cyclic group of points:	no
Non-cyclic primes:	$71$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5041$	$26863489$	$128252799376$	$645255659945689$	$3255153881428659001$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$70$	$5324$	$358336$	$25392084$	$1804179650$	$128101625678$	$9095125028990$	$645753440851684$	$45848500282242496$	$3255243556991654204$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 39 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-283}) \)$)$

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_fl	$2$	(not in LMFDB)
2.71.c_fn	$2$	(not in LMFDB)
2.71.b_acs	$3$	(not in LMFDB)