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

• Label: 2.71.a_act
• 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:	$1 - 71 x^{2} + 5041 x^{4}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.833333333333$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-71})\)
Galois group:	$C_2^2$
Jacobians:	$13$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4971$	$24710841$	$128100999744$	$646009808058729$	$3255243549205651851$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$4900$	$357912$	$25421764$	$1804229352$	$128101715566$	$9095120158392$	$645753582069124$	$45848500718449032$	$3255243547401422500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 13 curves (of which all are hyperelliptic):

• $y^2=x^6+62 x^5+63 x^4+23 x^3+55 x^2+17 x+51$
• $y^2=9 x^6+66 x^5+52 x^4+6 x^3+70 x^2+59 x+9$
• $y^2=63 x^6+36 x^5+9 x^4+42 x^3+64 x^2+58 x+63$
• $y^2=13 x^6+43 x^5+41 x^4+68 x^3+21 x^2+35 x+13$
• $y^2=20 x^6+17 x^5+3 x^4+50 x^3+5 x^2+32 x+20$
• $y^2=15 x^6+24 x^5+70 x^4+28 x^3+33 x^2+66 x+15$
• $y^2=34 x^6+26 x^5+64 x^4+54 x^3+18 x^2+36 x+34$
• $y^2=12 x^6+26 x^5+49 x^4+17 x^3+28 x^2+46 x+12$
• $y^2=13 x^6+40 x^5+59 x^4+48 x^3+54 x^2+38 x+13$
• $y^2=33 x^6+16 x^5+64 x^4+23 x^3+53 x^2+40 x+33$
• $y^2=18 x^6+41 x^5+22 x^4+19 x^3+16 x^2+67 x+18$
• $y^2=54 x^6+27 x^5+25 x^4+29 x^3+61 x^2+13 x+54$
• $y^2=23 x^6+47 x^5+33 x^4+61 x^3+x^2+20 x+23$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71^{6}}$.

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-71})\).

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

The base change of $A$ to $\F_{71^{6}}$ is 1.128100283921.bosxq^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.act^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{71^{3}}$

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

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_fm	$3$	(not in LMFDB)
2.71.a_ct	$4$	(not in LMFDB)
2.71.a_afm	$12$	(not in LMFDB)