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

• Label: 2.71.ar_if
• Base field: $\F_{71}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$1 - 17 x + 213 x^{2} - 1207 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.306662112421$, $\pm0.355673142481$
Angle rank:	$2$ (numerical)
Number field:	4.0.1098725.1
Galois group:	$D_{4}$
Jacobians:	$18$

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4031$	$26116849$	$128935932821$	$646009165860509$	$3255109948404749776$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$55$	$5179$	$360241$	$25421739$	$1804155300$	$128099003383$	$9095114560615$	$645753568760259$	$45848501376207541$	$3255243553899868254$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

• $y^2=46 x^6+42 x^5+x^4+18 x^3+22 x^2+55 x+40$
• $y^2=36 x^6+67 x^5+12 x^4+54 x^3+5 x^2+66 x+52$
• $y^2=7 x^6+40 x^5+27 x^4+11 x^3+15 x^2+12 x+51$
• $y^2=21 x^6+37 x^5+21 x^4+11 x^3+16 x^2+58 x+31$
• $y^2=17 x^6+57 x^5+52 x^4+27 x^3+60 x^2+10 x+17$
• $y^2=49 x^6+34 x^5+24 x^4+60 x^3+26 x^2+35 x+14$
• $y^2=3 x^6+53 x^5+2 x^4+17 x^3+21 x^2+12 x+4$
• $y^2=70 x^6+49 x^5+41 x^4+66 x^3+23 x^2+67 x+7$
• $y^2=45 x^6+51 x^5+21 x^4+40 x^3+3 x^2+3 x+21$
• $y^2=3 x^6+19 x^4+41 x^3+36 x^2+39 x+50$
• $y^2=53 x^6+54 x^5+68 x^4+41 x^3+50 x^2+18 x+66$
• $y^2=51 x^6+70 x^5+57 x^4+59 x^3+47 x^2+52 x+69$
• $y^2=37 x^6+35 x^5+31 x^4+40 x^3+x^2+3 x+65$
• $y^2=25 x^6+28 x^5+70 x^4+19 x^3+33 x^2+40 x+23$
• $y^2=49 x^6+67 x^5+21 x^4+3 x^3+18 x^2+4 x+31$
• $y^2=30 x^6+32 x^5+65 x^4+21 x^3+62 x^2+65 x+57$
• $y^2=51 x^6+43 x^5+29 x^4+65 x^3+61 x^2+9 x+18$
• $y^2=9 x^6+9 x^5+21 x^4+65 x^3+44 x^2+33 x+60$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is 4.0.1098725.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.71.r_if	$2$	(not in LMFDB)