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

• Label: 2.71.a_adg
• 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:	$1 - 84 x^{2} + 5041 x^{4}$
Frobenius angles:	$\pm0.149258255063$, $\pm0.850741744937$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-58}, \sqrt{226})\)
Galois group:	$C_2^2$
Jacobians:	$14$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4958$	$24581764$	$128100961550$	$645907382725264$	$3255243551093960078$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$4874$	$357912$	$25417734$	$1804229352$	$128101639178$	$9095120158392$	$645753614579134$	$45848500718449032$	$3255243551178038954$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-58}, \sqrt{226})\).

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

The base change of $A$ to $\F_{71^{2}}$ is 1.5041.adg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3277}) \)$)$

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_dg	$4$	(not in LMFDB)