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

• Label: 2.71.p_ew
• Base field: $\F_{71}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$( 1 - x + 71 x^{2} )( 1 + 16 x + 71 x^{2} )$
	$1 + 15 x + 126 x^{2} + 1065 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.481100681038$, $\pm0.898333180169$
Angle rank:	$2$ (numerical)
Jacobians:	$54$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6248$	$25541824$	$128423266400$	$645430551910144$	$3255194705163903128$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$87$	$5069$	$358812$	$25398969$	$1804202277$	$128101197278$	$9095118871947$	$645753528380689$	$45848500087278132$	$3255243557604280229$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 54 curves (of which all are hyperelliptic):

• $y^2=19 x^6+63 x^5+13 x^4+14 x^3+68 x^2+27 x+16$
• $y^2=37 x^6+60 x^5+31 x^4+55 x^3+53 x^2+62 x+16$
• $y^2=18 x^6+39 x^5+59 x^4+70 x^3+14 x^2+23 x+70$
• $y^2=57 x^6+41 x^5+27 x^4+33 x^3+25 x+65$
• $y^2=70 x^6+30 x^5+7 x^4+45 x^3+27 x^2+64 x+61$
• $y^2=28 x^5+70 x^4+50 x^3+46 x^2+15 x+10$
• $y^2=47 x^6+47 x^5+62 x^4+27 x^3+29 x^2+27 x+1$
• $y^2=38 x^6+55 x^5+22 x^4+33 x^3+47 x^2+61 x+56$
• $y^2=10 x^6+44 x^5+26 x^4+51 x^3+15 x^2+21 x+61$
• $y^2=58 x^6+39 x^5+26 x^4+61 x^3+7 x^2+63 x+31$
• $y^2=34 x^6+54 x^5+12 x^4+58 x^3+63 x^2+19 x+5$
• $y^2=16 x^6+45 x^5+43 x^4+2 x^3+5 x^2+50 x+24$
• $y^2=46 x^6+20 x^5+70 x^4+37 x^3+30 x^2+43 x+12$
• $y^2=4 x^6+49 x^5+40 x^4+x^3+14 x^2+63 x+21$
• $y^2=58 x^6+12 x^5+3 x^4+4 x^3+66 x^2+65 x+64$
• $y^2=33 x^6+26 x^5+51 x^4+41 x^3+30 x^2+6 x+7$
• $y^2=48 x^6+64 x^5+55 x^4+15 x^3+4 x^2+45 x+18$
• $y^2=39 x^6+22 x^5+36 x^4+53 x^3+19 x^2+16 x+37$
• $y^2=9 x^6+55 x^5+59 x^4+70 x^3+61 x^2+34 x+4$
• $y^2=39 x^6+45 x^5+54 x^4+39 x^3+36 x^2+2 x+47$
• and

• $y^2=57 x^6+35 x^5+13 x^4+20 x^3+25 x^2+21 x+5$
• $y^2=20 x^6+62 x^4+35 x^3+26 x^2+7 x+2$
• $y^2=40 x^5+48 x^4+24 x^3+70 x^2+36 x+20$
• $y^2=49 x^6+13 x^5+8 x^4+4 x^3+60 x^2+67 x+39$
• $y^2=60 x^6+27 x^5+62 x^4+36 x^3+36 x^2+31 x+59$
• $y^2=55 x^6+53 x^5+7 x^4+45 x^3+65 x^2+4 x+51$
• $y^2=24 x^6+44 x^5+6 x^4+54 x^3+44 x^2+5 x+40$
• $y^2=24 x^6+43 x^5+63 x^4+67 x^3+17 x^2+7 x+55$
• $y^2=40 x^6+21 x^5+40 x^4+6 x^3+43 x^2+20 x+8$
• $y^2=20 x^6+36 x^5+68 x^4+44 x^3+67 x^2+52 x+56$
• $y^2=64 x^6+31 x^5+7 x^4+30 x^3+23 x^2+57 x+4$
• $y^2=5 x^6+63 x^5+7 x^4+10 x^3+18 x+49$
• $y^2=45 x^6+59 x^5+7 x^4+60 x^3+58 x^2+61 x+6$
• $y^2=13 x^6+35 x^5+68 x^4+9 x^3+5 x^2+39 x$
• $y^2=30 x^6+15 x^5+62 x^4+41 x^3+46 x^2+37 x$
• $y^2=69 x^6+13 x^5+60 x^4+70 x^3+36 x^2+16 x+19$
• $y^2=45 x^6+48 x^5+28 x^4+38 x^3+13 x^2+26 x+53$
• $y^2=45 x^6+16 x^5+69 x^4+35 x^3+28 x^2+49 x+45$
• $y^2=58 x^6+24 x^5+57 x^4+22 x^3+46 x^2+45 x+52$
• $y^2=14 x^6+66 x^5+48 x^4+25 x^3+6 x^2+28 x+15$
• $y^2=13 x^6+16 x^5+50 x^4+67 x^3+40 x^2+69 x+38$
• $y^2=6 x^6+19 x^5+50 x^4+16 x^3+68 x^2+47 x+45$
• $y^2=24 x^6+63 x^5+7 x^4+13 x^3+47 x^2+59 x+53$
• $y^2=6 x^6+14 x^5+8 x^4+4 x^3+15 x^2+58 x+43$
• $y^2=44 x^6+6 x^5+69 x^4+13 x^3+56 x^2+16 x+42$
• $y^2=27 x^6+33 x^5+44 x^4+39 x^3+x^2+7 x+64$
• $y^2=47 x^6+10 x^5+5 x^4+34 x^3+43 x^2+10 x+49$
• $y^2=12 x^6+31 x^5+8 x^4+6 x^3+7 x^2+50 x+6$
• $y^2=26 x^6+46 x^5+12 x^4+11 x^3+29 x^2+63 x+3$
• $y^2=56 x^6+51 x^5+44 x^4+20 x^3+8 x^2+35 x+37$
• $y^2=23 x^6+37 x^5+22 x^4+66 x^3+41 x^2+58 x+16$
• $y^2=9 x^6+51 x^5+41 x^4+18 x^3+52 x^2+3 x+61$
• $y^2=38 x^6+4 x^5+63 x^4+6 x^3+50 x^2+62 x+26$
• $y^2=50 x^6+37 x^5+64 x^4+69 x^3+37 x^2+6 x+52$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.ab $\times$ 1.71.q and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.71.ab : \(\Q(\sqrt{-283}) \).
• 1.71.q : \(\Q(\sqrt{-7}) \).

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.ar_gc	$2$	(not in LMFDB)
2.71.ap_ew	$2$	(not in LMFDB)
2.71.r_gc	$2$	(not in LMFDB)