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

• Label: 2.71.a_aek
• 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:	$( 1 - 16 x + 71 x^{2} )( 1 + 16 x + 71 x^{2} )$
	$1 - 114 x^{2} + 5041 x^{4}$
Frobenius angles:	$\pm0.101666819831$, $\pm0.898333180169$
Angle rank:	$1$ (numerical)
Jacobians:	$49$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4928$	$24285184$	$128100526400$	$645605491277824$	$3255243554613393728$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$4814$	$357912$	$25405854$	$1804229352$	$128100768878$	$9095120158392$	$645753615909694$	$45848500718449032$	$3255243558216906254$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 49 curves (of which all are hyperelliptic):

• $y^2=51 x^6+66 x^5+66 x^4+30 x^3+21 x^2+68 x+22$
• $y^2=2 x^6+36 x^5+36 x^4+68 x^3+5 x^2+50 x+12$
• $y^2=36 x^5+8 x^4+7 x^3+55 x^2+16 x+6$
• $y^2=17 x^6+18 x^5+10 x^4+48 x^3+29 x^2+47 x$
• $y^2=12 x^6+5 x^5+62 x^4+12 x^3+24 x^2+4 x+17$
• $y^2=58 x^6+8 x^5+22 x^4+12 x^3+6 x^2+39 x+11$
• $y^2=51 x^6+56 x^5+12 x^4+13 x^3+42 x^2+60 x+6$
• $y^2=29 x^6+29 x^5+48 x^4+42 x^3+67 x^2+53 x+51$
• $y^2=16 x^6+45 x^5+38 x^4+13 x^3+25 x^2+69 x+23$
• $y^2=34 x^6+56 x^5+7 x^4+66 x^3+8 x^2+63 x+64$
• $y^2=47 x^6+31 x^5+42 x^4+44 x^3+32 x^2+35 x+28$
• $y^2=5 x^6+45 x^5+26 x^4+47 x^3+61 x^2+26 x+68$
• $y^2=35 x^6+31 x^5+40 x^4+45 x^3+x^2+40 x+50$
• $y^2=50 x^6+58 x^5+36 x^4+12 x^3+14 x^2+32 x+11$
• $y^2=16 x^6+56 x^5+43 x^4+54 x^3+22 x^2+65 x+68$
• $y^2=58 x^6+39 x^5+49 x^4+60 x^3+21 x^2+26 x+47$
• $y^2=3 x^6+26 x^5+22 x^4+7 x^3+46 x^2+48 x+48$
• $y^2=18 x^6+50 x^5+68 x^4+x^3+25 x^2+31 x+52$
• $y^2=27 x^6+10 x^5+25 x^4+51 x^3+33 x^2+64 x+31$
• $y^2=43 x^6+57 x^5+9 x^4+52 x^3+28 x^2+10 x+7$
• and

• $y^2=7 x^6+18 x^5+9 x^4+38 x^3+17 x^2+20 x+15$
• $y^2=26 x^6+69 x^4+57 x^2+43$
• $y^2=47 x^6+44 x^4+24 x^2+4$
• $y^2=3 x^6+22 x^4+12 x^2+35$
• $y^2=70 x^6+20 x^4+69 x^2+12$
• $y^2=67 x^6+28 x^5+17 x^4+8 x^3+29 x^2+52 x+10$
• $y^2=26 x^6+15 x^5+25 x^4+24 x^3+55 x^2+30 x+15$
• $y^2=48 x^6+39 x^5+13 x^4+40 x^3+70 x^2+x+34$
• $y^2=56 x^6+49 x^5+9 x^4+2 x^3+37 x^2+41 x+48$
• $y^2=33 x^6+50 x^5+36 x^4+8 x^3+46 x^2+48 x+65$
• $y^2=18 x^6+66 x^5+39 x^4+56 x^3+38 x^2+52 x+29$
• $y^2=60 x^6+12 x^5+48 x^4+44 x^3+55 x^2+66 x+70$
• $y^2=65 x^6+13 x^5+52 x^4+24 x^3+30 x^2+36 x+64$
• $y^2=48 x^6+69 x^5+42 x^4+41 x^3+57 x^2+55 x+14$
• $y^2=66 x^6+27 x^5+13 x^4+24 x^3+x^2+x+19$
• $y^2=61 x^6+65 x^5+30 x^4+47 x^3+37 x^2+21 x+23$
• $y^2=62 x^6+45 x^5+51 x^4+57 x^3+5 x^2+25 x+19$
• $y^2=8 x^6+66 x^5+62 x^4+14 x^3+14 x^2+38 x+20$
• $y^2=35 x^6+16 x^5+33 x^4+70 x^3+37 x^2+36 x+27$
• $y^2=48 x^6+23 x^5+31 x^4+x^3+66 x^2+5 x+30$
• $y^2=x^6+25 x^5+43 x^4+52 x^3+2 x^2+63 x+31$
• $y^2=7 x^6+33 x^5+17 x^4+9 x^3+14 x^2+15 x+4$
• $y^2=26 x^6+17 x^5+37 x^4+17 x^3+65 x^2+66 x+27$
• $y^2=10 x^6+32 x^5+51 x^4+36 x^3+40 x^2+20$
• $y^2=62 x^5+52 x^4+17 x^3+25 x^2+15 x+10$
• $y^2=8 x^5+9 x^4+48 x^3+33 x^2+34 x+70$
• $y^2=32 x^6+64 x^5+59 x^4+68 x^3+8 x^2+60 x+51$
• $y^2=36 x^6+23 x^5+18 x^4+47 x^3+50 x^2+24 x+32$
• $y^2=39 x^6+19 x^5+55 x^4+45 x^3+66 x^2+26 x+11$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.aq $\times$ 1.71.q and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.71.aq : \(\Q(\sqrt{-7}) \).
• 1.71.q : \(\Q(\sqrt{-7}) \).

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

The base change of $A$ to $\F_{71^{2}}$ is 1.5041.aek^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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.abg_pi	$2$	(not in LMFDB)
2.71.bg_pi	$2$	(not in LMFDB)
2.71.a_ek	$4$	(not in LMFDB)