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

• Label: 2.71.ak_bd
• 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:	$1 - 10 x + 29 x^{2} - 710 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.0355444598470$, $\pm0.631122206820$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-46})\)
Galois group:	$C_2^2$
Jacobians:	$40$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4351$	$25196641$	$127293395524$	$645542200652329$	$3255248871690942751$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$5000$	$355652$	$25403364$	$1804232302$	$128099161766$	$9095114338162$	$645753549611524$	$45848500259291612$	$3255243547410125000$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=28 x^6+30 x^5+61 x^4+9 x^3+4 x^2+22 x+51$
• $y^2=12 x^6+13 x^5+64 x^4+55 x^3+49 x^2+x+13$
• $y^2=25 x^6+70 x^5+54 x^4+6 x^3+2 x^2+45 x+62$
• $y^2=51 x^6+65 x^5+32 x^4+44 x^3+30 x^2+4 x+56$
• $y^2=59 x^6+30 x^5+13 x^4+3 x^3+47 x^2+50 x+58$
• $y^2=69 x^6+19 x^5+38 x^4+61 x^3+22 x^2+45 x+32$
• $y^2=30 x^6+52 x^5+24 x^4+69 x^3+45 x^2+11$
• $y^2=55 x^6+28 x^5+55 x^4+27 x^3+19 x^2+36$
• $y^2=31 x^6+67 x^5+14 x^4+58 x^3+28 x^2+6 x+12$
• $y^2=13 x^6+5 x^5+69 x^4+24 x^3+19 x^2+55 x+7$
• $y^2=33 x^6+28 x^5+60 x^4+18 x^3+5 x^2+19 x+66$
• $y^2=34 x^6+46 x^5+7 x^4+42 x^3+55 x^2+52 x+20$
• $y^2=52 x^6+46 x^5+50 x^4+21 x^3+17 x^2+38 x+66$
• $y^2=10 x^6+64 x^5+35 x^4+14 x^3+4 x^2+55 x+27$
• $y^2=x^6+42 x^5+8 x^4+21 x^3+15 x^2+53 x+16$
• $y^2=60 x^6+67 x^5+68 x^4+12 x^3+28 x^2+24 x+4$
• $y^2=14 x^6+42 x^5+24 x^4+50 x^3+28 x^2+x+32$
• $y^2=69 x^6+13 x^5+38 x^4+40 x^3+9 x^2+38 x+62$
• $y^2=23 x^6+41 x^5+55 x^4+26 x^3+39 x^2+59 x+3$
• $y^2=7 x^6+49 x^5+61 x^4+14 x^3+61 x^2+20 x+61$
• and

• $y^2=54 x^6+12 x^5+65 x^4+16 x^3+21 x^2+30 x+20$
• $y^2=43 x^6+14 x^5+29 x^4+63 x^3+49 x^2+52 x+10$
• $y^2=47 x^6+20 x^5+39 x^4+40 x^3+27 x^2+60 x+62$
• $y^2=25 x^6+4 x^5+28 x^4+45 x^3+43 x^2+9 x+44$
• $y^2=55 x^6+20 x^5+51 x^4+22 x^3+68 x^2+3 x+12$
• $y^2=33 x^6+25 x^5+23 x^4+12 x^3+66 x^2+19 x+49$
• $y^2=19 x^6+61 x^5+27 x^4+25 x^3+49 x^2+53 x+23$
• $y^2=67 x^6+46 x^5+40 x^4+60 x^3+13 x^2+2 x+43$
• $y^2=68 x^6+34 x^5+13 x^4+44 x^3+36 x^2+61 x+48$
• $y^2=26 x^6+28 x^5+2 x^4+17 x^3+36 x^2+38 x+14$
• $y^2=69 x^6+29 x^5+26 x^4+13 x^3+2 x^2+22 x+32$
• $y^2=34 x^6+57 x^5+3 x^4+17 x^3+34 x^2+6 x+20$
• $y^2=30 x^6+39 x^5+67 x^4+63 x^3+24 x^2+20 x+31$
• $y^2=34 x^6+38 x^5+59 x^4+40 x^3+4 x^2+66 x+1$
• $y^2=66 x^6+11 x^5+41 x^4+20 x^3+x^2+50 x+19$
• $y^2=59 x^6+69 x^5+30 x^4+45 x^3+31 x^2+43 x+54$
• $y^2=47 x^6+58 x^5+57 x^4+9 x^3+46 x^2+24 x+13$
• $y^2=15 x^6+69 x^5+11 x^4+51 x^3+8 x^2+17 x+63$
• $y^2=16 x^6+18 x^5+29 x^4+15 x^3+5 x^2+50 x+62$
• $y^2=67 x^6+57 x^5+51 x^4+6 x^3+46 x^2+35 x+34$

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 \(\Q(\sqrt{-3}, \sqrt{-46})\).

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

The base change of $A$ to $\F_{71^{3}}$ is 1.357911.abrm^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.k_bd	$2$	(not in LMFDB)
2.71.u_ji	$3$	(not in LMFDB)
2.71.au_ji	$6$	(not in LMFDB)