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

• Label: 2.71.ae_acd
• 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:	$1 - 4 x - 55 x^{2} - 284 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.0903857707042$, $\pm0.757052437371$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-67})\)
Galois group:	$C_2^2$
Jacobians:	$24$
Isomorphism classes:	64
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4699$	$24787225$	$127537551376$	$645900774690025$	$3255100797623011579$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$68$	$4916$	$356336$	$25417476$	$1804150228$	$128100473678$	$9095126155708$	$645753513992836$	$45848501432044496$	$3255243553662029876$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71^{3}}$.

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-67})\).

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

The base change of $A$ to $\F_{71^{3}}$ is 1.357911.abei^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$

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.e_acd	$2$	(not in LMFDB)
2.71.i_gc	$3$	(not in LMFDB)
2.71.ai_gc	$6$	(not in LMFDB)