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

• Label: 2.71.a_aen
• 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:	$1 - 117 x^{2} + 5041 x^{4}$
Frobenius angles:	$\pm0.0958846096257$, $\pm0.904115390374$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{259})\)
Galois group:	$C_2^2$
Jacobians:	$40$
Isomorphism classes:	52
Cyclic group of points:	no
Non-cyclic primes:	$5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4925$	$24255625$	$128100451700$	$645570275205625$	$3255243554588223125$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$4808$	$357912$	$25404468$	$1804229352$	$128100619478$	$9095120158392$	$645753606871588$	$45848500718449032$	$3255243558166565048$

Jacobians and polarizations

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

• $y^2=10 x^6+3 x^5+49 x^4+28 x^3+39 x^2+6 x+18$
• $y^2=70 x^6+21 x^5+59 x^4+54 x^3+60 x^2+42 x+55$
• $y^2=47 x^6+20 x^5+37 x^4+21 x^3+42 x^2+50 x+61$
• $y^2=65 x^6+21 x^5+45 x^4+21 x^3+5 x^2+12 x+22$
• $y^2=29 x^6+5 x^5+31 x^4+5 x^3+35 x^2+13 x+12$
• $y^2=29 x^6+31 x^5+9 x^4+42 x^3+5 x^2+2 x+19$
• $y^2=27 x^6+48 x^5+61 x^4+7 x^3+40 x^2+28 x+55$
• $y^2=47 x^6+52 x^5+x^4+49 x^3+67 x^2+54 x+30$
• $y^2=31 x^6+43 x^5+24 x^4+70 x^3+55 x^2+48 x+5$
• $y^2=27 x^6+62 x^5+51 x^4+44 x^3+10 x^2+46 x+5$
• $y^2=60 x^6+52 x^5+70 x^4+50 x^3+49 x^2+60 x+33$
• $y^2=20 x^6+12 x^5+22 x^4+8 x^3+11 x^2+25 x+67$
• $y^2=69 x^6+13 x^5+12 x^4+56 x^3+6 x^2+33 x+43$
• $y^2=27 x^6+29 x^5+60 x^4+46 x^3+61 x^2+49 x+11$
• $y^2=47 x^6+61 x^5+65 x^4+38 x^3+x^2+59 x+6$
• $y^2=53 x^6+67 x^5+24 x^4+25 x^3+42 x^2+53 x+63$
• $y^2=16 x^6+43 x^5+26 x^4+33 x^3+10 x^2+16 x+15$
• $y^2=6 x^6+30 x^5+62 x^4+4 x^3+48 x^2+25 x+22$
• $y^2=42 x^6+68 x^5+8 x^4+28 x^3+52 x^2+33 x+12$
• $y^2=44 x^6+50 x^5+34 x^4+33 x^3+62 x^2+60 x+21$
• and

• $y^2=24 x^6+66 x^5+25 x^4+18 x^3+8 x^2+65 x+5$
• $y^2=22 x^6+62 x^5+70 x^4+4 x^3+9 x^2+52 x+7$
• $y^2=12 x^6+8 x^5+64 x^4+28 x^3+63 x^2+9 x+49$
• $y^2=5 x^6+5 x^5+34 x^4+41 x^3+48 x^2+43 x+44$
• $y^2=35 x^6+35 x^5+25 x^4+3 x^3+52 x^2+17 x+24$
• $y^2=12 x^6+56 x^5+57 x^4+35 x^3+31 x^2+11 x+14$
• $y^2=13 x^6+37 x^5+44 x^4+32 x^3+4 x^2+6 x+27$
• $y^2=11 x^6+51 x^5+24 x^4+36 x^3+57 x^2+25 x+21$
• $y^2=6 x^6+2 x^5+26 x^4+39 x^3+44 x^2+33 x+5$
• $y^2=46 x^6+23 x^4+22 x^3+42 x^2+44 x+56$
• $y^2=29 x^6+55 x^5+32 x^4+19 x^3+28 x^2+55 x+62$
• $y^2=32 x^6+9 x^5+58 x^4+19 x^3+23 x^2+50 x+41$
• $y^2=11 x^6+63 x^5+51 x^4+62 x^3+19 x^2+66 x+3$
• $y^2=4 x^6+24 x^5+22 x^4+64 x^3+38 x^2+15 x+27$
• $y^2=52 x^6+16 x^5+67 x^4+8 x^3+64 x^2+12 x+39$
• $y^2=9 x^6+41 x^5+43 x^4+56 x^3+22 x^2+13 x+60$
• $y^2=67 x^6+24 x^5+54 x^4+61 x^3+23 x^2+29 x+27$
• $y^2=43 x^6+26 x^5+23 x^4+x^3+19 x^2+61 x+47$
• $y^2=24 x^6+27 x^5+28 x^4+48 x^3+30 x^2+11 x+41$
• $y^2=26 x^6+47 x^5+54 x^4+52 x^3+68 x^2+6 x+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71^{2}}$.

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{259})\).

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

The base change of $A$ to $\F_{71^{2}}$ is 1.5041.aen^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-259}) \)$)$

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.ak_gl	$4$	(not in LMFDB)
2.71.a_en	$4$	(not in LMFDB)
2.71.k_gl	$4$	(not in LMFDB)