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

• Label: 2.61.ah_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:	$\pm0.0187571618436$, $\pm0.685423828510$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{65})\)
Galois group:	$C_2^2$
Jacobians:	$20$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3276$	$13575744$	$51095889936$	$191678047352064$	$713307078931031676$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$55$	$3649$	$225106$	$13843729$	$844553875$	$51519522598$	$3142742442895$	$191707289770369$	$11694145719695866$	$713342911773740329$

Jacobians and polarizations

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

• $y^2=38 x^6+17 x^5+25 x^4+44 x^3+18 x^2+53 x+53$
• $y^2=56 x^6+45 x^5+38 x^4+13 x^3+19 x^2+7 x+24$
• $y^2=43 x^6+39 x^5+58 x^4+24 x^3+48 x^2+28 x+29$
• $y^2=59 x^6+10 x^5+31 x^4+56 x^3+45 x^2+37 x+37$
• $y^2=60 x^6+x^5+39 x^4+52 x^3+6 x^2+20 x+8$
• $y^2=x^6+2 x^3+14$
• $y^2=10 x^6+51 x^5+47 x^4+25 x^3+42 x^2+29 x+14$
• $y^2=14 x^6+60 x^5+7 x^4+57 x^3+30 x^2+58 x+53$
• $y^2=27 x^6+48 x^5+25 x^4+29 x^3+56 x^2+51 x+58$
• $y^2=16 x^6+56 x^5+56 x^4+37 x^3+14 x^2+22 x+44$
• $y^2=7 x^6+57 x^4+15 x^3+6 x^2+30 x+30$
• $y^2=30 x^6+49 x^5+3 x^4+45 x^3+38 x^2+12 x+35$
• $y^2=x^6+x^3+5$
• $y^2=x^6+27 x^5+45 x^4+25 x^3+15 x^2+10 x+54$
• $y^2=39 x^6+27 x^5+7 x^4+25 x^3+8 x^2+8 x+14$
• $y^2=38 x^6+48 x^5+x^4+46 x^3+36 x^2+17 x+44$
• $y^2=44 x^6+56 x^5+60 x^4+45 x^3+8 x^2+48 x+10$
• $y^2=29 x^6+39 x^5+34 x^4+44 x^3+33 x^2+14 x+37$
• $y^2=4 x^6+58 x^5+58 x^4+5 x^3+50 x^2+54 x+28$
• $y^2=14 x^6+16 x^5+38 x^4+27 x^3+23 x^2+48 x+30$

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.abkc^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.h_am	$2$	(not in LMFDB)
2.61.o_gp	$3$	(not in LMFDB)
2.61.ao_gp	$6$	(not in LMFDB)