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

• Label: 2.71.a_cu
• 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:	$1 + 72 x^{2} + 5041 x^{4}$
Frobenius angles:	$\pm0.334630599408$, $\pm0.665369400592$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{70}, \sqrt{-214})\)
Galois group:	$C_2^2$
Jacobians:	$60$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5114$	$26152996$	$128099568314$	$646002538896400$	$3255243552685288154$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$5186$	$357912$	$25421478$	$1804229352$	$128098852706$	$9095120158392$	$645753584911678$	$45848500718449032$	$3255243554360695106$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 60 curves (of which all are hyperelliptic):

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

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

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 \(\Q(\sqrt{70}, \sqrt{-214})\).

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

The base change of $A$ to $\F_{71^{2}}$ is 1.5041.cu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3745}) \)$)$

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_acu	$4$	(not in LMFDB)