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

• Label: 2.71.a_ad
• 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:	$1 - 3 x^{2} + 5041 x^{4}$
Frobenius angles:	$\pm0.246637321444$, $\pm0.753362678556$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-139}, \sqrt{145})\)
Galois group:	$C_2^2$
Jacobians:	$88$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5039$	$25391521$	$128100329264$	$646265627280025$	$3255243550629386279$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$5036$	$357912$	$25431828$	$1804229352$	$128100374606$	$9095120158392$	$645753429961828$	$45848500718449032$	$3255243550248891356$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 88 curves (of which all are hyperelliptic):

• $y^2=24 x^6+10 x^5+53 x^4+41 x^3+7 x^2+63 x+3$
• $y^2=26 x^6+70 x^5+16 x^4+3 x^3+49 x^2+15 x+21$
• $y^2=2 x^6+37 x^5+18 x^4+59 x^3+18 x^2+14 x+48$
• $y^2=14 x^6+46 x^5+55 x^4+58 x^3+55 x^2+27 x+52$
• $y^2=21 x^6+38 x^5+23 x^4+20 x^3+7 x^2+9 x+54$
• $y^2=5 x^6+53 x^5+19 x^4+69 x^3+49 x^2+63 x+23$
• $y^2=15 x^6+69 x^5+46 x^4+58 x^3+53 x^2+40 x+68$
• $y^2=34 x^6+57 x^5+38 x^4+51 x^3+16 x^2+67 x+50$
• $y^2=56 x^6+70 x^5+66 x^4+65 x^3+13 x^2+58 x+2$
• $y^2=37 x^6+64 x^5+36 x^4+29 x^3+20 x^2+51 x+14$
• $y^2=40 x^6+45 x^5+48 x^4+18 x^3+46 x^2+51 x+67$
• $y^2=67 x^6+31 x^5+52 x^4+55 x^3+38 x^2+2 x+43$
• $y^2=13 x^6+56 x^5+35 x^4+23 x^3+20 x^2+9 x+56$
• $y^2=20 x^6+37 x^5+32 x^4+19 x^3+69 x^2+63 x+37$
• $y^2=20 x^6+38 x^5+2 x^4+41 x^3+3 x^2+5 x+35$
• $y^2=69 x^6+53 x^5+14 x^4+3 x^3+21 x^2+35 x+32$
• $y^2=69 x^6+70 x^5+23 x^4+6 x^3+22 x^2+44 x+54$
• $y^2=57 x^6+64 x^5+19 x^4+42 x^3+12 x^2+24 x+23$
• $y^2=34 x^6+44 x^5+26 x^4+58 x^3+5 x^2+43 x+25$
• $y^2=25 x^6+24 x^5+40 x^4+51 x^3+35 x^2+17 x+33$
• and

• $y^2=59 x^6+39 x^5+5 x^4+20 x^3+37 x^2+55 x+26$
• $y^2=58 x^6+60 x^5+35 x^4+69 x^3+46 x^2+30 x+40$
• $y^2=64 x^6+58 x^5+67 x^4+10 x^3+24 x^2+29 x+21$
• $y^2=18 x^6+54 x^5+20 x^4+12 x^3+12 x^2+35 x+52$
• $y^2=55 x^6+23 x^5+69 x^4+13 x^3+13 x^2+32 x+9$
• $y^2=26 x^6+59 x^5+61 x^4+65 x^3+51 x^2+65 x+44$
• $y^2=40 x^6+58 x^5+x^4+29 x^3+2 x^2+29 x+24$
• $y^2=13 x^6+64 x^5+43 x^4+69 x^3+62 x^2+24 x+19$
• $y^2=55 x^6+27 x^5+36 x^4+58 x^3+34 x^2+23 x+1$
• $y^2=30 x^6+47 x^5+39 x^4+51 x^3+25 x^2+19 x+7$
• $y^2=61 x^6+69 x^5+51 x^4+43 x^3+56 x^2+10 x+63$
• $y^2=x^6+57 x^5+2 x^4+17 x^3+37 x^2+70 x+15$
• $y^2=25 x^6+21 x^5+67 x^4+66 x^2+7 x+43$
• $y^2=33 x^6+5 x^5+43 x^4+36 x^2+49 x+17$
• $y^2=20 x^6+47 x^5+9 x^4+16 x^3+5 x^2+65 x+62$
• $y^2=69 x^6+45 x^5+63 x^4+41 x^3+35 x^2+29 x+8$
• $y^2=65 x^6+51 x^5+59 x^4+18 x^3+9 x^2+53 x+50$
• $y^2=29 x^6+2 x^5+58 x^4+55 x^3+63 x^2+16 x+66$
• $y^2=39 x^6+55 x^5+45 x^4+60 x^3+34 x^2+42 x+24$
• $y^2=60 x^6+30 x^5+31 x^4+65 x^3+25 x^2+10 x+26$
• $y^2=48 x^6+63 x^5+4 x^4+4 x^3+28 x^2+26 x+24$
• $y^2=52 x^6+15 x^5+28 x^4+28 x^3+54 x^2+40 x+26$
• $y^2=6 x^6+43 x^5+67 x^4+63 x^3+4 x^2+43 x+65$
• $y^2=56 x^6+9 x^5+33 x^4+5 x^3+64 x^2+45 x+3$
• $y^2=66 x^6+29 x^5+10 x^4+49 x^3+28 x^2+31 x+34$
• $y^2=36 x^6+61 x^5+70 x^4+59 x^3+54 x^2+4 x+25$
• $y^2=57 x^6+61 x^5+24 x^4+65 x^3+24 x^2+57 x+15$
• $y^2=44 x^6+x^5+26 x^4+29 x^3+26 x^2+44 x+34$
• $y^2=25 x^6+2 x^5+64 x^4+50 x^3+13 x^2+36 x+17$
• $y^2=33 x^6+14 x^5+22 x^4+66 x^3+20 x^2+39 x+48$
• $y^2=66 x^6+57 x^5+11 x^4+70 x^3+65 x^2+11 x+56$
• $y^2=36 x^6+44 x^5+6 x^4+64 x^3+29 x^2+6 x+37$
• $y^2=40 x^6+53 x^5+17 x^4+40 x^3+22 x^2+37 x+19$
• $y^2=67 x^6+16 x^5+48 x^4+67 x^3+12 x^2+46 x+62$
• $y^2=66 x^6+45 x^5+30 x^4+39 x^3+67 x^2+15 x+16$
• $y^2=36 x^6+50 x^5+57 x^4+63 x^3+53 x^2+22 x+53$
• $y^2=39 x^6+66 x^5+44 x^4+15 x^3+16 x^2+12 x+16$
• $y^2=24 x^6+40 x^5+14 x^4+29 x^3+8 x^2+29 x+14$
• $y^2=39 x^6+66 x^4+25 x^3+35 x+32$
• $y^2=60 x^6+36 x^4+33 x^3+32 x+11$
• $y^2=40 x^6+5 x^5+42 x^4+61 x^3+4 x^2+7 x+12$
• $y^2=67 x^6+35 x^5+10 x^4+x^3+28 x^2+49 x+13$
• $y^2=59 x^6+29 x^5+22 x^4+48 x^3+41 x^2+38 x+58$
• $y^2=58 x^6+61 x^5+12 x^4+52 x^3+3 x^2+53 x+51$
• $y^2=47 x^6+46 x^5+14 x^4+64 x^3+33 x^2+3 x+48$
• $y^2=45 x^6+38 x^5+27 x^4+22 x^3+18 x^2+21 x+52$
• $y^2=52 x^6+24 x^5+21 x^4+45 x^3+49 x^2+53 x+63$
• $y^2=9 x^6+26 x^5+5 x^4+31 x^3+59 x^2+16 x+15$
• $y^2=67 x^6+63 x^5+33 x^4+51 x^3+41 x^2+28 x+34$
• $y^2=43 x^6+15 x^5+18 x^4+2 x^3+3 x^2+54 x+25$
• $y^2=47 x^6+37 x^5+35 x^4+8 x^3+66 x^2+25 x+37$
• $y^2=45 x^6+46 x^5+32 x^4+56 x^3+36 x^2+33 x+46$
• $y^2=52 x^6+19 x^5+5 x^4+35 x^3+66 x^2+2 x+32$
• $y^2=9 x^6+62 x^5+35 x^4+32 x^3+36 x^2+14 x+11$
• $y^2=55 x^6+49 x^5+x^4+69 x^3+69 x^2+20 x+28$
• $y^2=30 x^6+59 x^5+7 x^4+57 x^3+57 x^2+69 x+54$
• $y^2=59 x^6+18 x^5+69 x^4+10 x^3+28 x^2+9 x+24$
• $y^2=58 x^6+55 x^5+57 x^4+70 x^3+54 x^2+63 x+26$
• $y^2=27 x^6+33 x^5+35 x^4+69 x^3+2 x^2+3 x+68$
• $y^2=47 x^6+18 x^5+32 x^4+57 x^3+14 x^2+21 x+50$
• $y^2=27 x^6+9 x^5+45 x^4+37 x^3+25 x^2+14 x+6$
• $y^2=47 x^6+63 x^5+31 x^4+46 x^3+33 x^2+27 x+42$
• $y^2=52 x^6+24 x^5+33 x^4+17 x^3+50 x^2+15 x+60$
• $y^2=52 x^6+30 x^5+34 x^4+9 x^3+x^2+37 x+3$
• $y^2=19 x^6+17 x^4+45 x^3+31 x^2+11 x+40$
• $y^2=62 x^6+48 x^4+31 x^3+4 x^2+6 x+67$
• $y^2=48 x^6+58 x^5+39 x^4+43 x^3+7 x^2+57 x+19$
• $y^2=52 x^6+51 x^5+60 x^4+17 x^3+49 x^2+44 x+62$

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 \(\Q(\sqrt{-139}, \sqrt{145})\).

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

The base change of $A$ to $\F_{71^{2}}$ is 1.5041.ad^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-20155}) \)$)$

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_d	$4$	(not in LMFDB)