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

• Label: 2.71.a_aem
• 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:	$1 - 116 x^{2} + 5041 x^{4}$
Frobenius angles:	$\pm0.0978449860573$, $\pm0.902155013943$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-26}, \sqrt{258})\)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4926$	$24265476$	$128100477294$	$645582115422864$	$3255243554610073326$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$4810$	$357912$	$25404934$	$1804229352$	$128100670666$	$9095120158392$	$645753610124734$	$45848500718449032$	$3255243558210265450$

Jacobians and polarizations

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

• $y^2=6 x^6+30 x^5+8 x^4+15 x^3+12 x^2+48 x+53$
• $y^2=42 x^6+68 x^5+56 x^4+34 x^3+13 x^2+52 x+16$
• $y^2=x^6+56 x^5+6 x^4+48 x^3+11 x^2+28 x+35$
• $y^2=7 x^6+37 x^5+42 x^4+52 x^3+6 x^2+54 x+32$
• $y^2=15 x^6+27 x^5+19 x^4+70 x^3+34 x^2+67 x+66$
• $y^2=34 x^6+47 x^5+62 x^4+64 x^3+25 x^2+43 x+36$
• $y^2=16 x^6+32 x^5+52 x^4+20 x^3+45 x^2+3 x+40$
• $y^2=41 x^6+11 x^5+9 x^4+69 x^3+31 x^2+21 x+67$
• $y^2=36 x^6+38 x^5+40 x^4+28 x^3+62 x^2+6 x+62$
• $y^2=39 x^6+53 x^5+67 x^4+54 x^3+8 x^2+42 x+8$
• $y^2=52 x^6+8 x^5+4 x^4+6 x^3+8 x^2+57 x+33$
• $y^2=9 x^6+56 x^5+28 x^4+42 x^3+56 x^2+44 x+18$
• $y^2=33 x^6+28 x^5+30 x^4+8 x^3+64 x^2+2 x+14$
• $y^2=18 x^6+54 x^5+68 x^4+56 x^3+22 x^2+14 x+27$
• $y^2=18 x^6+36 x^5+13 x^3+3 x^2+43 x+8$
• $y^2=55 x^6+39 x^5+20 x^3+21 x^2+17 x+56$
• $y^2=18 x^6+58 x^5+34 x^4+51 x^3+15 x^2+16 x+41$
• $y^2=55 x^6+51 x^5+25 x^4+2 x^3+34 x^2+41 x+3$
• $y^2=31 x^6+9 x^5+38 x^4+18 x^3+17 x^2+42$
• $y^2=4 x^6+63 x^5+53 x^4+55 x^3+48 x^2+10$
• and

• $y^2=49 x^6+2 x^5+21 x^4+15 x^3+12 x^2+45 x+69$
• $y^2=59 x^6+14 x^5+5 x^4+34 x^3+13 x^2+31 x+57$
• $y^2=17 x^6+32 x^5+52 x^4+3 x^3+63 x^2+9 x+47$
• $y^2=48 x^6+11 x^5+9 x^4+21 x^3+15 x^2+63 x+45$
• $y^2=57 x^6+41 x^5+34 x^4+32 x^3+4 x^2+63 x+25$
• $y^2=44 x^6+3 x^5+25 x^4+11 x^3+28 x^2+15 x+33$
• $y^2=27 x^6+17 x^5+60 x^4+4 x^3+4 x^2+48 x+7$
• $y^2=47 x^6+48 x^5+65 x^4+28 x^3+28 x^2+52 x+49$
• $y^2=65 x^6+59 x^5+45 x^4+50 x^3+34 x^2+51 x+18$
• $y^2=29 x^6+58 x^5+31 x^4+66 x^3+25 x^2+2 x+55$
• $y^2=43 x^6+27 x^5+62 x^4+46 x^3+43 x^2+48 x+13$
• $y^2=17 x^6+47 x^5+8 x^4+38 x^3+17 x^2+52 x+20$
• $y^2=69 x^6+8 x^5+21 x^4+33 x^3+47 x^2+24 x+25$
• $y^2=57 x^6+56 x^5+5 x^4+18 x^3+45 x^2+26 x+33$
• $y^2=34 x^6+20 x^5+53 x^4+31 x^3+23 x^2+29 x+55$
• $y^2=25 x^6+69 x^5+16 x^4+4 x^3+19 x^2+61 x+30$
• $y^2=68 x^6+3 x^5+10 x^4+20 x^3+53 x^2+4 x+23$
• $y^2=50 x^6+21 x^5+70 x^4+69 x^3+16 x^2+28 x+19$
• $y^2=x^6+41 x^5+22 x^4+45 x^3+7 x^2+18 x+13$
• $y^2=7 x^6+3 x^5+12 x^4+31 x^3+49 x^2+55 x+20$

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 \(\Q(\sqrt{-26}, \sqrt{258})\).

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

The base change of $A$ to $\F_{71^{2}}$ is 1.5041.aem^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1677}) \)$)$

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.a_em	$4$	(not in LMFDB)