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

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

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$1 - x - 70 x^{2} - 71 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.147767347705$, $\pm0.814434014371$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-283})\)
Galois group:	$C_2^2$
Jacobians:	$63$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5, 7$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4900$	$24715600$	$127949290000$	$646002610926400$	$3255198712919597500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$71$	$4901$	$357488$	$25421481$	$1804204501$	$128101625678$	$9095122593691$	$645753576442801$	$45848501154655568$	$3255243548018994701$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 63 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71^{3}}$.

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-283})\).

Endomorphism algebra over $\overline{\F}_{71}$

The base change of $A$ to $\F_{71^{3}}$ is 1.357911.aie^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.b_acs	$2$	(not in LMFDB)
2.71.c_fn	$3$	(not in LMFDB)
2.71.ac_fn	$6$	(not in LMFDB)