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

• Label: 2.71.a_afa
• 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:	$1 - 130 x^{2} + 5041 x^{4}$
Frobenius angles:	$\pm0.0659004698611$, $\pm0.934099530139$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{17})\)
Galois group:	$C_2^2$
Jacobians:	$42$
Isomorphism classes:	190

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4912$	$24127744$	$128100052912$	$645407114858496$	$3255243552738373552$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$4782$	$357912$	$25398046$	$1804229352$	$128099821902$	$9095120158392$	$645753539922238$	$45848500718449032$	$3255243554466865902$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

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

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

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 \(\Q(\sqrt{-3}, \sqrt{17})\).

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

The base change of $A$ to $\F_{71^{2}}$ is 1.5041.afa^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$

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.ag_df	$3$	(not in LMFDB)
2.71.g_df	$3$	(not in LMFDB)
2.71.a_fa	$4$	(not in LMFDB)