Abelian variety isogeny class 2.61.an_ge over $\F_{61}$

• Label: 2.61.an_ge
• Base field: $\F_{61}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 - 13 x + 160 x^{2} - 793 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.315352364581$, $\pm0.408291131055$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-59 +8 \sqrt{17}})\)
Galois group:	$D_{4}$
Jacobians:	$50$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3076$	$14420288$	$51899207104$	$191735667798272$	$713282514536735956$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$49$	$3873$	$228646$	$13847889$	$844524789$	$51519874854$	$3142743216025$	$191707333531905$	$11694146168973790$	$713342911527169993$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 50 curves (of which all are hyperelliptic):

• $y^2=11 x^6+52 x^5+57 x^4+46 x^3+2 x^2+30 x+15$
• $y^2=36 x^6+59 x^5+42 x^4+12 x^3+39 x^2+29 x+43$
• $y^2=6 x^6+45 x^5+21 x^4+28 x^3+9 x^2+56 x+51$
• $y^2=11 x^6+28 x^5+44 x^4+51 x^3+48 x^2+28 x+2$
• $y^2=45 x^6+5 x^5+21 x^4+43 x^3+10 x^2+31 x+53$
• $y^2=14 x^6+33 x^5+10 x^4+3 x^3+49 x^2+55 x+42$
• $y^2=43 x^5+30 x^4+38 x^3+55 x^2+12 x+59$
• $y^2=33 x^6+23 x^5+52 x^4+25 x^3+12 x^2+47 x+12$
• $y^2=50 x^6+49 x^5+8 x^4+47 x^3+55 x^2+55 x+30$
• $y^2=54 x^6+24 x^5+13 x^4+52 x^3+45 x^2+39 x+1$
• $y^2=29 x^6+33 x^5+14 x^4+52 x^3+42 x^2+21 x+20$
• $y^2=25 x^6+54 x^5+4 x^4+11 x^3+57 x^2+25 x+33$
• $y^2=13 x^6+52 x^5+46 x^4+34 x^3+27 x^2+34 x+13$
• $y^2=16 x^6+49 x^5+58 x^4+12 x^3+30 x^2+31 x+22$
• $y^2=20 x^6+49 x^5+55 x^4+51 x^3+4 x^2+18 x$
• $y^2=6 x^6+48 x^5+10 x^4+13 x^3+27 x^2+29 x+44$
• $y^2=15 x^6+39 x^5+45 x^4+59 x^3+50 x^2+12 x+17$
• $y^2=40 x^6+38 x^5+3 x^4+34 x^3+47 x^2+7 x+15$
• $y^2=31 x^6+60 x^5+37 x^4+50 x^3+54 x^2+57 x+10$
• $y^2=35 x^6+18 x^5+25 x^4+40 x^3+58 x^2+53 x+21$
• and

• $y^2=31 x^6+23 x^5+42 x^4+6 x^3+25 x^2+51 x+48$
• $y^2=21 x^6+6 x^5+10 x^4+17 x^3+16 x^2+2 x+7$
• $y^2=21 x^6+7 x^5+13 x^3+36 x^2+42 x+29$
• $y^2=44 x^6+23 x^5+8 x^4+17 x^3+22 x^2+x+28$
• $y^2=4 x^6+39 x^5+12 x^4+11 x^3+33 x^2+10 x+3$
• $y^2=38 x^6+11 x^5+31 x^4+39 x^3+20 x^2+53 x+58$
• $y^2=40 x^6+25 x^4+39 x^3+21 x^2+12 x+2$
• $y^2=30 x^6+50 x^5+55 x^4+46 x^3+42 x^2+25 x+3$
• $y^2=x^6+52 x^5+14 x^4+24 x^3+42 x^2+42 x+15$
• $y^2=9 x^6+38 x^5+59 x^4+49 x^3+59 x^2+9 x+50$
• $y^2=59 x^6+41 x^5+45 x^4+7 x^3+47 x^2+18 x+18$
• $y^2=28 x^6+31 x^5+58 x^4+43 x^3+47 x^2+31 x+40$
• $y^2=46 x^6+18 x^5+9 x^4+28 x^3+52 x^2+24 x+21$
• $y^2=35 x^6+19 x^5+10 x^4+47 x^3+6 x^2+29 x$
• $y^2=14 x^6+45 x^5+50 x^4+x^3+9 x^2+34 x+30$
• $y^2=55 x^6+52 x^5+37 x^4+3 x^3+33 x^2+36 x+11$
• $y^2=59 x^6+59 x^5+50 x^4+58 x^3+9 x^2+55 x+23$
• $y^2=24 x^6+36 x^5+9 x^4+37 x^3+39 x^2+9 x+58$
• $y^2=40 x^6+25 x^5+55 x^4+32 x^3+26 x^2+5 x+32$
• $y^2=28 x^6+50 x^5+19 x^4+19 x^3+35 x^2+24 x+23$
• $y^2=58 x^6+42 x^5+58 x^4+24 x^3+44 x^2+44 x+48$
• $y^2=49 x^6+31 x^5+51 x^4+30 x^3+17 x^2+33 x+51$
• $y^2=60 x^6+50 x^5+53 x^3+40 x^2+8 x+32$
• $y^2=23 x^6+35 x^5+35 x^4+32 x^3+13$
• $y^2=55 x^6+24 x^5+47 x^4+16 x^3+59 x^2+34 x+11$
• $y^2=11 x^6+29 x^5+50 x^4+17 x^3+29 x^2+45 x+31$
• $y^2=23 x^6+49 x^5+28 x^4+58 x^3+60 x^2+48 x$
• $y^2=29 x^6+48 x^5+24 x^4+46 x^3+x+35$
• $y^2=9 x^6+21 x^5+43 x^4+26 x^3+59 x^2+36 x+29$
• $y^2=28 x^6+58 x^5+31 x^4+45 x^3+8 x^2+33 x+27$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-59 +8 \sqrt{17}})\).

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
2.61.n_ge	$2$	(not in LMFDB)