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

• Label: 2.61.ac_et
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$( 1 - x + 61 x^{2} )^{2}$
	$1 - 2 x + 123 x^{2} - 122 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.479608352732$, $\pm0.479608352732$
Angle rank:	$1$ (numerical)
Jacobians:	$43$
Cyclic group of points:	no
Non-cyclic primes:	$61$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3721$	$14768649$	$51603482896$	$191508040081449$	$713311999773156001$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$3964$	$227346$	$13831444$	$844559700$	$51521216038$	$3142745910420$	$191707264729444$	$11694145857028026$	$713342914371414604$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 43 curves (of which all are hyperelliptic):

• $y^2=2 x^6+2 x^3+54$
• $y^2=27 x^6+39 x^5+22 x^4+51 x^3+52 x^2+49 x+58$
• $y^2=7 x^6+15 x^5+40 x^4+31 x^3+30 x+36$
• $y^2=44 x^6+9 x^5+29 x^4+9 x^3+23 x^2+34 x+37$
• $y^2=x^6+59 x^5+21 x^4+24 x^2+42 x+58$
• $y^2=42 x^6+21 x^5+39 x^4+21 x^3+9 x^2+17 x+36$
• $y^2=2 x^6+45 x^5+57 x^4+60 x^3+3 x^2+52 x+43$
• $y^2=23 x^6+9 x^5+59 x^4+45 x^3+14 x^2+56 x+41$
• $y^2=40 x^6+60 x^5+14 x^4+x^3+58 x^2+4 x+43$
• $y^2=43 x^6+27 x^5+22 x^4+36 x^3+55 x^2+29 x+46$
• $y^2=36 x^6+25 x^5+19 x^4+28 x^3+41 x^2+13 x+49$
• $y^2=54 x^6+19 x^5+5 x^4+9 x^3+56 x^2+19 x+7$
• $y^2=29 x^6+3 x^5+7 x^4+19 x^3+38 x^2+5 x+30$
• $y^2=22 x^6+37 x^5+25 x^4+10 x^3+42 x^2+8 x+56$
• $y^2=31 x^6+40 x^5+39 x^4+44 x^3+16 x^2+40 x+12$
• $y^2=26 x^6+18 x^5+6 x^4+19 x^3+23 x^2+44 x+27$
• $y^2=12 x^6+47 x^5+13 x^4+58 x^3+45 x^2+12 x+14$
• $y^2=47 x^6+54 x^5+38 x^4+58 x^3+4 x^2+54 x+6$
• $y^2=33 x^6+23 x^5+12 x^4+5 x^3+12 x^2+23 x+33$
• $y^2=21 x^6+59 x^5+58 x^4+46 x^2+38 x+58$
• and

• $y^2=58 x^6+24 x^5+26 x^4+15 x^3+17 x^2+33 x+41$
• $y^2=46 x^6+46 x^5+9 x^4+16 x^3+58 x^2+6 x+34$
• $y^2=7 x^6+36 x^5+19 x^4+51 x^3+40 x^2+38 x+51$
• $y^2=2 x^6+2 x^3+21$
• $y^2=41 x^6+20 x^5+23 x^4+40 x^3+54 x^2+12 x+1$
• $y^2=39 x^6+30 x^5+52 x^4+39 x^3+46 x^2+8 x+31$
• $y^2=7 x^6+15 x^5+6 x^4+9 x^3+58 x^2+5 x+35$
• $y^2=6 x^6+18 x^5+43 x^4+56 x^3+28 x^2+30 x+59$
• $y^2=26 x^6+47 x^5+39 x^4+37 x^3+4 x^2+56 x+30$
• $y^2=32 x^6+6 x^5+24 x^4+53 x^3+32 x^2+18 x+31$
• $y^2=51 x^6+36 x^5+33 x^4+54 x^3+38 x^2+19 x+35$
• $y^2=13 x^6+39 x^5+24 x^4+34 x^3+29 x^2+49 x+46$
• $y^2=47 x^6+43 x^5+23 x^4+27 x^3+28 x^2+59 x+4$
• $y^2=24 x^6+28 x^5+54 x^4+22 x^3+4 x^2+20$
• $y^2=2 x^6+33 x^3+2$
• $y^2=6 x^6+15 x^5+54 x^4+46 x^3+26 x^2+9 x+18$
• $y^2=2 x^6+24 x^3+40$
• $y^2=8 x^6+44 x^5+x^4+53 x^3+20 x^2+32 x+11$
• $y^2=23 x^6+23 x^5+42 x^4+22 x^3+41 x^2+54 x+10$
• $y^2=49 x^6+6 x^5+47 x^4+4 x^3+13 x^2+17 x+49$
• $y^2=60 x^6+59 x^5+22 x^4+10 x^3+48 x^2+54 x+9$
• $y^2=18 x^6+27 x^5+51 x^4+54 x^3+60 x^2+51$
• $y^2=12 x^6+42 x^5+20 x^4+29 x^3+48 x^2+15 x+19$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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.a_er	$2$	(not in LMFDB)
2.61.c_et	$2$	(not in LMFDB)
2.61.aba_lf	$3$	(not in LMFDB)
2.61.ao_ff	$3$	(not in LMFDB)
2.61.b_aci	$3$	(not in LMFDB)
2.61.n_ee	$3$	(not in LMFDB)
2.61.bc_mg	$3$	(not in LMFDB)