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

• Label: 2.71.ag_abj
• 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:	$1 - 6 x - 35 x^{2} - 426 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.0507952241045$, $\pm0.717461890771$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-62})\)
Galois group:	$C_2^2$
Jacobians:	$24$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4575$	$24883425$	$127341922500$	$645782832246825$	$3255095020997589375$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$66$	$4936$	$355788$	$25412836$	$1804147026$	$128099459878$	$9095123531406$	$645753481754116$	$45848500603517268$	$3255243554178992776$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

• $y^2=56 x^6+55 x^5+70 x^4+48 x^3+4 x^2+66 x+34$
• $y^2=19 x^6+9 x^5+69 x^4+70 x^3+25 x^2+34 x+19$
• $y^2=53 x^6+51 x^5+42 x^4+4 x^3+3 x^2+11 x+66$
• $y^2=22 x^6+26 x^5+41 x^4+30 x^3+28 x^2+35 x+22$
• $y^2=50 x^6+38 x^5+67 x^4+18 x^3+59 x^2+49 x+50$
• $y^2=27 x^6+62 x^5+38 x^4+34 x^3+4 x^2+15 x+34$
• $y^2=2 x^6+35 x^5+44 x^4+4 x^3+41 x^2+48 x+2$
• $y^2=54 x^6+24 x^5+43 x^4+9 x^3+23 x^2+16 x+54$
• $y^2=14 x^6+30 x^5+41 x^4+46 x^3+43 x^2+47 x+30$
• $y^2=27 x^6+26 x^5+20 x^4+46 x^3+52 x^2+30 x+52$
• $y^2=12 x^6+64 x^5+x^4+49 x^3+37 x^2+11 x+64$
• $y^2=56 x^6+9 x^5+11 x^4+40 x^3+25 x^2+43 x+56$
• $y^2=14 x^6+8 x^5+x^4+18 x^3+8 x^2+61 x+27$
• $y^2=40 x^6+67 x^5+51 x^4+25 x^3+32 x^2+31 x+40$
• $y^2=49 x^6+41 x^5+54 x^4+38 x^3+16 x^2+40 x+49$
• $y^2=48 x^6+26 x^5+32 x^4+52 x^3+29 x^2+x+16$
• $y^2=34 x^6+66 x^5+10 x^4+58 x^3+39 x^2+33 x+46$
• $y^2=10 x^6+9 x^5+58 x^4+47 x^3+36 x^2+20 x+41$
• $y^2=45 x^6+6 x^5+7 x^4+50 x^3+6 x^2+66 x+6$
• $y^2=27 x^6+35 x^5+58 x^4+38 x^3+64 x^2+20 x+28$
• $y^2=39 x^6+36 x^5+49 x^4+44 x^3+14 x^2+24 x+24$
• $y^2=58 x^6+42 x^5+56 x^4+57 x^3+61 x^2+52 x+42$
• $y^2=44 x^6+17 x^5+60 x^4+25 x^3+63 x^2+57 x+6$
• $y^2=65 x^6+27 x^5+11 x^4+51 x^3+56 x^2+62 x+41$

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 \(\Q(\sqrt{-3}, \sqrt{-62})\).

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

The base change of $A$ to $\F_{71^{3}}$ is 1.357911.abow^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-62}) \)$)$

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.g_abj	$2$	(not in LMFDB)
2.71.m_gw	$3$	(not in LMFDB)
2.71.am_gw	$6$	(not in LMFDB)