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

• Label: 2.71.j_k
• 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:	$1 + 9 x + 10 x^{2} + 639 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.345997922256$, $\pm0.987335411078$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-203})\)
Galois group:	$C_2^2$
Jacobians:	$28$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5700$	$25102800$	$128952810000$	$645591902587200$	$3255292442751217500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$81$	$4981$	$360288$	$25405321$	$1804256451$	$128098892878$	$9095124494061$	$645753520884721$	$45848501520612768$	$3255243548135778301$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

• $y^2=49 x^6+31 x^5+61 x^4+8 x^3+x^2+60 x+63$
• $y^2=63 x^6+21 x^5+7 x^4+42 x^3+66 x^2+2 x+63$
• $y^2=57 x^6+19 x^5+42 x^4+43 x^3+22 x^2+2 x+41$
• $y^2=36 x^6+12 x^5+9 x^4+34 x^3+63 x^2+62 x+36$
• $y^2=18 x^6+59 x^5+67 x^4+69 x^3+68 x^2+22 x+40$
• $y^2=64 x^6+57 x^5+43 x^4+33 x^3+52 x^2+46 x+13$
• $y^2=44 x^6+33 x^5+3 x^4+53 x^3+47 x^2+29 x+67$
• $y^2=4 x^6+18 x^5+51 x^4+55 x^3+42 x^2+8 x+63$
• $y^2=25 x^6+32 x^5+13 x^4+35 x^3+40 x^2+38 x+29$
• $y^2=33 x^6+21 x^5+52 x^4+25 x^3+51 x^2+12 x+20$
• $y^2=36 x^6+44 x^5+67 x^4+41 x^3+38 x^2+70 x+69$
• $y^2=11 x^6+23 x^5+63 x^4+42 x^3+55 x^2+65 x+33$
• $y^2=54 x^6+4 x^5+56 x^4+25 x^3+20 x^2+35 x+38$
• $y^2=55 x^6+46 x^5+62 x^4+24 x^3+36 x^2+x+8$
• $y^2=25 x^6+34 x^5+14 x^4+9 x^3+8 x^2+54 x+10$
• $y^2=60 x^6+21 x^5+38 x^4+38 x^3+17 x^2+20 x+18$
• $y^2=41 x^6+57 x^5+35 x^4+41 x^3+49 x^2+53 x+1$
• $y^2=34 x^6+4 x^5+61 x^4+60 x^3+47 x^2+62 x+58$
• $y^2=32 x^6+29 x^5+69 x^4+26 x^3+6 x^2+5 x+49$
• $y^2=22 x^6+35 x^5+50 x^4+52 x^3+34 x^2+31 x+37$
• $y^2=16 x^6+33 x^5+42 x^4+8 x^3+46 x^2+63 x+16$
• $y^2=61 x^6+30 x^5+11 x^4+5 x^3+68 x^2+62$
• $y^2=57 x^6+30 x^5+23 x^4+40 x^3+18 x^2+28 x+57$
• $y^2=54 x^6+24 x^5+47 x^4+4 x^3+63 x^2+50 x+5$
• $y^2=3 x^6+16 x^5+58 x^4+26 x^3+5 x^2+51 x$
• $y^2=36 x^6+12 x^5+40 x^4+41 x^3+9 x^2+26 x+13$
• $y^2=6 x^6+38 x^5+x^4+15 x^3+46 x^2+64 x+43$
• $y^2=12 x^6+47 x^5+13 x^4+6 x^3+62 x^2+23 x+9$

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 \(\Q(\sqrt{-3}, \sqrt{-203})\).

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

The base change of $A$ to $\F_{71^{3}}$ is 1.357911.bts^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.aj_k	$2$	(not in LMFDB)
2.71.as_ip	$3$	(not in LMFDB)
2.71.aj_k	$6$	(not in LMFDB)