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

• Label: 2.61.c_acf
• 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:	$1 + 2 x - 57 x^{2} + 122 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.207534254477$, $\pm0.874200921144$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:	$C_2^2$
Jacobians:	$28$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3789$	$13416849$	$51683475600$	$191797089915609$	$713372305669967709$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$3604$	$227698$	$13852324$	$844631104$	$51521025958$	$3142740061504$	$191707327321924$	$11694145697044378$	$713342911184869204$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

• $y^2=28 x^6+56 x^5+27 x^4+47 x^3+27 x^2+48 x+53$
• $y^2=x^6+x^3+22$
• $y^2=45 x^6+52 x^5+22 x^4+35 x^3+28 x^2+31 x+29$
• $y^2=23 x^6+15 x^5+4 x^4+40 x^3+56 x^2+32 x+58$
• $y^2=35 x^6+13 x^5+43 x^4+19 x^2+60 x+58$
• $y^2=45 x^6+27 x^5+31 x^4+39 x^3+42 x^2+5 x+16$
• $y^2=51 x^6+16 x^5+2 x^3+57 x^2+32 x+35$
• $y^2=39 x^6+47 x^5+34 x^4+23 x^3+55 x^2+55 x+4$
• $y^2=39 x^6+29 x^5+28 x^4+45 x^3+42 x^2+31 x+1$
• $y^2=23 x^6+34 x^5+57 x^4+33 x^3+43 x^2+24 x+58$
• $y^2=22 x^6+32 x^5+22 x^4+16 x^3+33 x^2+47 x+33$
• $y^2=22 x^6+9 x^5+3 x^4+42 x^3+40 x^2+13 x+30$
• $y^2=22 x^6+9 x^5+31 x^4+30 x^3+34 x^2+17 x+46$
• $y^2=9 x^6+18 x^5+25 x^4+12 x^3+37 x^2+50 x+10$
• $y^2=7 x^6+26 x^5+10 x^4+15 x^3+18 x^2+9 x+45$
• $y^2=x^6+2 x^3+49$
• $y^2=46 x^6+34 x^5+42 x^4+57 x^3+56 x^2+49 x+55$
• $y^2=18 x^6+30 x^5+4 x^4+50 x^3+25 x^2+35 x+49$
• $y^2=45 x^6+50 x^5+57 x^4+29 x^3+21 x^2+14 x+60$
• $y^2=14 x^6+59 x^5+4 x^4+49 x^3+4 x^2+38 x+31$
• $y^2=17 x^6+29 x^5+27 x^4+28 x^3+30 x^2+37 x+27$
• $y^2=22 x^6+19 x^5+47 x^4+39 x^3+59 x^2+17 x+30$
• $y^2=31 x^6+60 x^5+x^4+8 x^3+14 x^2+57 x+50$
• $y^2=x^6+x^3+45$
• $y^2=x^6+x^3+13$
• $y^2=20 x^6+21 x^5+48 x^4+34 x^3+6 x^2+41 x+24$
• $y^2=10 x^6+36 x^5+27 x^4+12 x^3+26 x^2+17 x+8$
• $y^2=36 x^6+29 x^5+27 x^4+59 x^3+16 x^2+6 x+51$

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 \(\Q(\sqrt{-3}, \sqrt{5})\).

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

The base change of $A$ to $\F_{61^{3}}$ is 1.226981.nu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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.ac_acf	$2$	(not in LMFDB)
2.61.ae_ew	$3$	(not in LMFDB)
2.61.ac_acf	$6$	(not in LMFDB)