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

• Label: 2.71.au_ji
• 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:	$( 1 - 10 x + 71 x^{2} )^{2}$
	$1 - 20 x + 242 x^{2} - 1420 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.297788873486$, $\pm0.297788873486$
Angle rank:	$1$ (numerical)
Jacobians:	$34$
Cyclic group of points:	no
Non-cyclic primes:	$2, 31$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3844$	$25847056$	$128911157764$	$646176400000000$	$3255254199580219204$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$5126$	$360172$	$25428318$	$1804235252$	$128099161766$	$9095108517932$	$645753494514238$	$45848501177606452$	$3255243558209393606$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 34 curves (of which all are hyperelliptic):

• $y^2=51 x^6+66 x^5+65 x^4+45 x^3+39 x^2+70 x+58$
• $y^2=43 x^6+2 x^5+13 x^4+38 x^3+13 x^2+2 x+43$
• $y^2=57 x^6+10 x^5+53 x^4+20 x^3+64 x^2+65 x+4$
• $y^2=69 x^6+19 x^5+12 x^4+15 x^3+37 x+33$
• $y^2=16 x^6+37 x^5+45 x^3+37 x+16$
• $y^2=31 x^6+34 x^5+61 x^4+10 x^3+61 x^2+34 x+31$
• $y^2=47 x^6+21 x^5+27 x^4+59 x^3+52 x^2+45 x+56$
• $y^2=63 x^6+4 x^5+24 x^4+43 x^3+24 x^2+4 x+63$
• $y^2=65 x^6+30 x^5+50 x^4+51 x^3+42 x^2+43 x+47$
• $y^2=49 x^6+18 x^5+60 x^4+62 x^3+70 x^2+50 x+42$
• $y^2=28 x^6+7 x^5+4 x^4+45 x^3+25 x^2+62 x+67$
• $y^2=54 x^6+67 x^5+8 x^4+48 x^3+8 x^2+67 x+54$
• $y^2=58 x^6+35 x^5+7 x^4+8 x^3+7 x^2+24 x+49$
• $y^2=17 x^6+63 x^5+14 x^3+46 x^2+36 x+8$
• $y^2=35 x^6+37 x^5+68 x^4+x^3+68 x^2+37 x+35$
• $y^2=26 x^6+66 x^5+20 x^4+65 x^3+43 x^2+47 x+61$
• $y^2=54 x^6+50 x^5+7 x^4+19 x^3+51 x^2+x+3$
• $y^2=22 x^6+52 x^5+47 x^4+61 x^3+44 x^2+17 x+61$
• $y^2=44 x^6+57 x^5+38 x^4+27 x^3+58 x^2+34 x+46$
• $y^2=15 x^6+67 x^5+46 x^4+27 x^3+4 x^2+68 x+27$
• and

• $y^2=11 x^6+33 x^5+38 x^4+15 x^3+38 x^2+33 x+11$
• $y^2=14 x^6+25 x^5+6 x^4+13 x^3+62 x^2+14 x+32$
• $y^2=42 x^6+51 x^5+69 x^4+39 x^3+56 x^2+45 x+56$
• $y^2=55 x^6+44 x^5+39 x^4+20 x^3+57 x^2+57 x+21$
• $y^2=61 x^6+59 x^5+20 x^4+4 x^3+30 x^2+44 x+55$
• $y^2=47 x^6+61 x^5+23 x^4+12 x^3+60 x^2+7 x+14$
• $y^2=2 x^6+6 x^5+50 x^4+27 x^3+20 x^2+18 x+45$
• $y^2=51 x^6+62 x^5+70 x^4+48 x^3+62 x^2+45 x+22$
• $y^2=29 x^6+53 x^4+53 x^2+29$
• $y^2=28 x^6+34 x^5+31 x^4+25 x^3+47 x^2+9 x+68$
• $y^2=63 x^6+20 x^5+30 x^4+58 x^3+64 x^2+13 x+33$
• $y^2=13 x^6+14 x^4+14 x^2+13$
• $y^2=60 x^6+33 x^5+30 x^4+63 x^3+69 x^2+47 x+66$
• $y^2=59 x^6+67 x^5+9 x^4+60 x^3+46 x^2+51 x+14$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-46}) \)$)$

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_bq	$2$	(not in LMFDB)
2.71.u_ji	$2$	(not in LMFDB)
2.71.k_bd	$3$	(not in LMFDB)