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

• Label: 2.61.h_am
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 + 7 x - 12 x^{2} + 427 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.314576171490$, $\pm0.981242838156$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{65})\)
Galois group:	$C_2^2$
Jacobians:	$20$
Isomorphism classes:	72
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4144$	$13575744$	$51947526400$	$191678047352064$	$713378746305641584$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$69$	$3649$	$228858$	$13843729$	$844638729$	$51519522598$	$3142743229149$	$191707289770369$	$11694146465972418$	$713342911773740329$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=15 x^6+34 x^5+50 x^4+27 x^3+36 x^2+45 x+45$
• $y^2=28 x^6+53 x^5+19 x^4+37 x^3+40 x^2+34 x+12$
• $y^2=52 x^6+50 x^5+29 x^4+12 x^3+24 x^2+14 x+45$
• $y^2=60 x^6+5 x^5+46 x^4+28 x^3+53 x^2+49 x+49$
• $y^2=59 x^6+2 x^5+17 x^4+43 x^3+12 x^2+40 x+16$
• $y^2=2 x^6+4 x^3+28$
• $y^2=5 x^6+56 x^5+54 x^4+43 x^3+21 x^2+45 x+7$
• $y^2=28 x^6+59 x^5+14 x^4+53 x^3+60 x^2+55 x+45$
• $y^2=44 x^6+24 x^5+43 x^4+45 x^3+28 x^2+56 x+29$
• $y^2=32 x^6+51 x^5+51 x^4+13 x^3+28 x^2+44 x+27$
• $y^2=34 x^6+59 x^4+38 x^3+3 x^2+15 x+15$
• $y^2=60 x^6+37 x^5+6 x^4+29 x^3+15 x^2+24 x+9$
• $y^2=2 x^6+2 x^3+10$
• $y^2=31 x^6+44 x^5+53 x^4+43 x^3+38 x^2+5 x+27$
• $y^2=50 x^6+44 x^5+34 x^4+43 x^3+4 x^2+4 x+7$
• $y^2=15 x^6+35 x^5+2 x^4+31 x^3+11 x^2+34 x+27$
• $y^2=22 x^6+28 x^5+30 x^4+53 x^3+4 x^2+24 x+5$
• $y^2=58 x^6+17 x^5+7 x^4+27 x^3+5 x^2+28 x+13$
• $y^2=2 x^6+29 x^5+29 x^4+33 x^3+25 x^2+27 x+14$
• $y^2=7 x^6+8 x^5+19 x^4+44 x^3+42 x^2+24 x+15$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61^{3}}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{65})\).

Endomorphism algebra over $\overline{\F}_{61}$

The base change of $A$ to $\F_{61^{3}}$ is 1.226981.bkc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-195}) \)$)$

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
2.61.ah_am	$2$	(not in LMFDB)
2.61.ao_gp	$3$	(not in LMFDB)
2.61.a_cv	$6$	(not in LMFDB)