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

• Label: 2.71.g_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:	$\pm0.282538109229$, $\pm0.949204775895$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-62})\)
Galois group:	$C_2^2$
Jacobians:	$24$
Isomorphism classes:	136
Cyclic group of points:	no
Non-cyclic primes:	$7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5439$	$24883425$	$128862332676$	$645782832246825$	$3255392090968854879$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$78$	$4936$	$360036$	$25412836$	$1804311678$	$128099459878$	$9095116785378$	$645753481754116$	$45848500833380796$	$3255243554178992776$

Jacobians and polarizations

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

• $y^2=8 x^6+18 x^5+10 x^4+17 x^3+31 x^2+50 x+15$
• $y^2=23 x^6+52 x^5+20 x^4+10 x^3+34 x^2+15 x+23$
• $y^2=16 x^6+2 x^5+10 x^4+28 x^3+21 x^2+6 x+36$
• $y^2=12 x^6+40 x^5+3 x^4+68 x^3+54 x^2+32 x+12$
• $y^2=68 x^6+46 x^5+40 x^4+33 x^3+49 x^2+7 x+68$
• $y^2=47 x^6+8 x^5+53 x^4+25 x^3+28 x^2+34 x+25$
• $y^2=14 x^6+32 x^5+24 x^4+28 x^3+3 x^2+52 x+14$
• $y^2=28 x^6+44 x^5+67 x^4+52 x^3+54 x^2+53 x+28$
• $y^2=27 x^6+68 x^5+3 x^4+38 x^3+17 x^2+45 x+68$
• $y^2=14 x^6+24 x^5+13 x^4+37 x^3+48 x^2+55 x+48$
• $y^2=13 x^6+22 x^5+7 x^4+59 x^3+46 x^2+6 x+22$
• $y^2=8 x^6+52 x^5+32 x^4+26 x^3+34 x^2+67 x+8$
• $y^2=27 x^6+56 x^5+7 x^4+55 x^3+56 x^2+x+47$
• $y^2=67 x^6+43 x^5+2 x^4+33 x^3+11 x^2+4 x+67$
• $y^2=7 x^6+16 x^5+28 x^4+46 x^3+53 x^2+26 x+7$
• $y^2=52 x^6+40 x^5+11 x^4+9 x^3+61 x^2+7 x+41$
• $y^2=15 x^6+50 x^5+42 x^4+59 x^3+36 x^2+25 x+37$
• $y^2=70 x^6+63 x^5+51 x^4+45 x^3+39 x^2+69 x+3$
• $y^2=31 x^6+42 x^5+49 x^4+66 x^3+42 x^2+36 x+42$
• $y^2=47 x^6+32 x^5+51 x^4+53 x^3+22 x^2+69 x+54$
• $y^2=36 x^6+66 x^5+7 x^4+57 x^3+2 x^2+44 x+44$
• $y^2=59 x^6+6 x^5+8 x^4+69 x^3+29 x^2+48 x+6$
• $y^2=24 x^6+48 x^5+65 x^4+33 x^3+15 x^2+44 x+42$
• $y^2=29 x^6+47 x^5+6 x^4+2 x^3+37 x^2+8 x+3$

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.bow^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.ag_abj	$2$	(not in LMFDB)
2.71.am_gw	$3$	(not in LMFDB)
2.71.a_ec	$6$	(not in LMFDB)