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

• Label: 2.71.a_a
• Base field: $\F_{71}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$1 + 5041 x^{4}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.750000000000$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(i, \sqrt{142})\)
Galois group:	$C_2^2$
Jacobians:	$73$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5042$	$25421764$	$128100283922$	$646266084871696$	$3255243551009881202$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$5042$	$357912$	$25431846$	$1804229352$	$128100283922$	$9095120158392$	$645753429599038$	$45848500718449032$	$3255243551009881202$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 73 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71^{4}}$.

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{142})\).

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

The base change of $A$ to $\F_{71^{4}}$ is 1.25411681.oxu^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $71$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{71^{2}}$

  The base change of $A$ to $\F_{71^{2}}$ is 1.5041.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.a_afm	$8$	(not in LMFDB)
2.71.a_fm	$8$	(not in LMFDB)
2.71.a_act	$24$	(not in LMFDB)