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

• Label: 2.71.c_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:	$\pm0.518899318962$, $\pm0.518899318962$
Angle rank:	$1$ (numerical)
Jacobians:	$39$
Cyclic group of points:	no
Non-cyclic primes:	$73$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5329$	$26863489$	$127949290000$	$645255659945689$	$3255333229043165209$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$5324$	$357488$	$25392084$	$1804279054$	$128101625678$	$9095115287794$	$645753440851684$	$45848501154655568$	$3255243556991654204$

Jacobians and polarizations

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

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.b^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.ac_fn	$2$	(not in LMFDB)
2.71.a_fl	$2$	(not in LMFDB)
2.71.ab_acs	$3$	(not in LMFDB)