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

• Label: 2.61.ap_fg
• 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:	$( 1 - 14 x + 61 x^{2} )( 1 - x + 61 x^{2} )$
	$1 - 15 x + 136 x^{2} - 915 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.146275019398$, $\pm0.479608352732$
Angle rank:	$1$ (numerical)
Jacobians:	$88$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2928$	$14019264$	$51520795200$	$191634871244544$	$713360079065018928$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$47$	$3769$	$226982$	$13840609$	$844616627$	$51521216038$	$3142747908767$	$191707312689889$	$11694146092834142$	$713342913214373329$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 88 curves (of which all are hyperelliptic):

• $y^2=19 x^6+54 x^5+19 x^4+43 x^3+19 x^2+19 x+15$
• $y^2=27 x^6+35 x^5+56 x^4+42 x^3+31 x^2+40 x+21$
• $y^2=5 x^6+12 x^5+8 x^4+11 x^3+12 x^2+2 x+44$
• $y^2=31 x^6+47 x^5+3 x^4+46 x^3+29 x^2+22$
• $y^2=35 x^6+12 x^5+18 x^4+54 x^3+38 x+26$
• $y^2=11 x^6+42 x^5+20 x^4+7 x^3+30 x^2+51 x+29$
• $y^2=60 x^6+37 x^5+45 x^4+x^3+x^2+27 x+36$
• $y^2=37 x^6+2 x^5+24 x^4+40 x^3+43 x^2+14 x+53$
• $y^2=29 x^6+4 x^5+5 x^4+5 x^3+57 x^2+15 x+42$
• $y^2=22 x^6+60 x^5+29 x^4+47 x^2+29 x+55$
• $y^2=6 x^6+42 x^5+49 x^4+18 x^3+6 x^2+23 x+2$
• $y^2=55 x^6+26 x^5+5 x^4+58 x^3+54 x^2+21 x+23$
• $y^2=46 x^6+13 x^5+60 x^4+13 x^3+41 x^2+27 x+36$
• $y^2=51 x^6+58 x^5+19 x^4+24 x^3+14 x^2+48 x+1$
• $y^2=57 x^6+38 x^5+40 x^4+43 x^3+x^2+33 x+28$
• $y^2=22 x^6+46 x^5+53 x^4+56 x^3+23 x^2+30 x+21$
• $y^2=31 x^6+11 x^5+27 x^3+48 x^2+9 x+10$
• $y^2=41 x^6+50 x^5+5 x^4+22 x^3+32 x^2+36 x+30$
• $y^2=10 x^5+60 x^4+29 x^3+45 x^2+56 x+18$
• $y^2=38 x^6+58 x^5+14 x^4+40 x^3+7 x^2+15 x+34$
• and

• $y^2=56 x^6+49 x^5+46 x^4+17 x^3+47 x^2+31 x+22$
• $y^2=x^6+16 x^5+38 x^4+52 x^3+48 x^2+13 x+4$
• $y^2=26 x^6+38 x^5+17 x^4+12 x^3+8 x^2+11 x$
• $y^2=38 x^6+44 x^5+52 x^4+14 x^3+20 x^2+7 x+40$
• $y^2=6 x^6+8 x^5+17 x^4+54 x^3+40 x^2+9 x+14$
• $y^2=37 x^6+21 x^5+44 x^4+12 x^3+16 x^2+10 x$
• $y^2=33 x^6+38 x^5+27 x^4+3 x^3+11 x^2+39 x+50$
• $y^2=31 x^6+59 x^5+22 x^4+52 x^3+10 x^2+32 x+45$
• $y^2=13 x^6+25 x^5+22 x^4+58 x^3+18 x^2+36 x+11$
• $y^2=3 x^6+31 x^5+19 x^4+4 x^3+39 x^2+10 x+57$
• $y^2=11 x^6+42 x^5+5 x^4+37 x^3+13 x^2+29 x+56$
• $y^2=38 x^6+6 x^5+45 x^4+39 x^3+46 x^2+55 x+4$
• $y^2=11 x^6+15 x^5+9 x^4+50 x^3+41 x^2+26 x$
• $y^2=29 x^6+40 x^5+32 x^4+19 x^3+55 x^2+37 x+54$
• $y^2=58 x^6+18 x^5+28 x^4+2 x^3+36 x^2+17 x+37$
• $y^2=60 x^5+42 x^4+39 x^3+18 x^2+20 x+30$
• $y^2=43 x^6+38 x^5+14 x^4+9 x^3+27 x^2+7 x+54$
• $y^2=8 x^6+10 x^5+16 x^4+21 x^3+37 x^2+45 x+7$
• $y^2=38 x^6+35 x^5+44 x^4+28 x^3+42 x^2+24 x+44$
• $y^2=51 x^6+42 x^5+5 x^4+60 x^3+10 x^2+2 x+17$
• $y^2=58 x^6+59 x^5+38 x^4+46 x^3+22 x^2+16 x+5$
• $y^2=37 x^6+23 x^5+42 x^4+27 x^3+46 x^2+19 x+53$
• $y^2=54 x^6+38 x^5+50 x^4+34 x^3+27 x^2+35 x+58$
• $y^2=27 x^6+13 x^5+37 x^4+41 x^3+39 x^2+49$
• $y^2=8 x^6+26 x^5+8 x^4+46 x^3+28 x^2+10 x+23$
• $y^2=7 x^6+60 x^5+11 x^4+52 x^3+41 x^2+58 x+23$
• $y^2=8 x^6+37 x^5+8 x^4+57 x^3+10 x^2+19 x+19$
• $y^2=47 x^6+27 x^5+20 x^4+47 x^3+14 x^2+34 x+55$
• $y^2=4 x^5+6 x^4+15 x^3+51 x^2+21 x+7$
• $y^2=33 x^6+50 x^5+36 x^4+55 x^3+59 x^2+29 x+50$
• $y^2=24 x^6+46 x^5+2 x^4+3 x^3+28 x^2+59 x+54$
• $y^2=30 x^6+18 x^5+14 x^4+45 x^3+44 x^2+34 x+44$
• $y^2=27 x^6+4 x^5+43 x^4+9 x^3+7 x^2+7 x+57$
• $y^2=5 x^6+33 x^5+21 x^4+42 x^3+23 x^2+34 x+38$
• $y^2=47 x^6+22 x^5+50 x^4+43 x^3+16 x^2+10 x+12$
• $y^2=42 x^6+7 x^5+12 x^4+12 x^3+54 x^2+29 x+54$
• $y^2=25 x^6+11 x^5+34 x^4+43 x^3+19 x^2+58 x+23$
• $y^2=32 x^6+8 x^5+15 x^4+26 x^3+7 x^2+20 x$
• $y^2=53 x^6+2 x^5+52 x^4+57 x^3+29 x^2+20 x+29$
• $y^2=19 x^6+28 x^5+54 x^4+12 x^3+3 x^2+37 x+35$
• $y^2=33 x^6+13 x^5+5 x^4+7 x^3+38 x^2+6 x+10$
• $y^2=24 x^6+35 x^5+52 x^4+32 x^3+28 x^2+9 x+2$
• $y^2=59 x^6+14 x^5+6 x^4+48 x^3+34 x^2+5 x+40$
• $y^2=30 x^6+29 x^5+21 x^4+45 x^3+32 x^2+59 x+11$
• $y^2=36 x^6+53 x^5+52 x^4+8 x^3+8 x^2+48 x+29$
• $y^2=12 x^6+10 x^5+33 x^4+31 x^3+34 x^2+48 x+29$
• $y^2=6 x^6+59 x^5+43 x^4+58 x^3+23 x^2+54 x+18$
• $y^2=53 x^6+25 x^5+46 x^4+46 x^3+43 x^2+10 x+26$
• $y^2=36 x^6+43 x^5+51 x^4+22 x^3+21 x^2+15 x+4$
• $y^2=27 x^6+11 x^5+26 x^4+29 x^3+23 x^2+34 x+2$
• $y^2=8 x^6+24 x^5+55 x^4+41 x^3+19 x^2+55 x+35$
• $y^2=34 x^6+2 x^5+39 x^4+28 x^3+60 x^2+24 x+8$
• $y^2=60 x^6+50 x^5+11 x^4+46 x^3+24 x^2+34 x+33$
• $y^2=18 x^6+58 x^5+4 x^4+53 x^3+48 x^2+55$
• $y^2=4 x^6+3 x^5+2 x^4+47 x^3+2 x^2+46 x+24$
• $y^2=9 x^6+22 x^5+42 x^4+29 x^3+20 x^2+22 x$
• $y^2=x^6+x^3+7$
• $y^2=2 x^6+2 x^3+20$
• $y^2=59 x^6+57 x^5+48 x^4+41 x^3+7 x^2+49 x+27$
• $y^2=6 x^5+22 x^4+51 x^3+33 x^2+25 x+18$
• $y^2=47 x^6+41 x^5+3 x^4+8 x^3+24 x^2+45 x+48$
• $y^2=45 x^6+47 x^5+x^4+18 x^3+59 x^2+4 x+33$
• $y^2=14 x^6+12 x^5+18 x^4+10 x^3+59 x^2+10 x+12$
• $y^2=11 x^6+22 x^5+28 x^4+25 x^3+40 x^2+28 x+14$
• $y^2=18 x^6+2 x^5+39 x^4+29 x^3+53 x^2+17 x+50$
• $y^2=46 x^6+13 x^5+57 x^4+52 x^3+24 x^2+29 x+17$
• $y^2=26 x^6+23 x^5+33 x^4+28 x^3+30 x^2+38 x+26$
• $y^2=17 x^6+31 x^5+46 x^4+10 x^3+38 x^2+3 x+44$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61^{6}}$.

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ao $\times$ 1.61.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.61.ao : \(\Q(\sqrt{-3}) \).
• 1.61.ab : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{61^{6}}$ is 1.51520374361.xyoc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{61^{2}}$
  The base change of $A$ to $\F_{61^{2}}$ is 1.3721.acw $\times$ 1.3721.er. The endomorphism algebra for each factor is:

  • 1.3721.acw : \(\Q(\sqrt{-3}) \).
  • 1.3721.er : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{61^{3}}$
  The base change of $A$ to $\F_{61^{3}}$ is 1.226981.aha $\times$ 1.226981.ha. The endomorphism algebra for each factor is:

  • 1.226981.aha : \(\Q(\sqrt{-3}) \).
  • 1.226981.ha : \(\Q(\sqrt{-3}) \).

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.an_ee	$2$	(not in LMFDB)
2.61.n_ee	$2$	(not in LMFDB)
2.61.p_fg	$2$	(not in LMFDB)
2.61.abb_ls	$3$	(not in LMFDB)
2.61.am_ef	$3$	(not in LMFDB)
2.61.a_acw	$3$	(not in LMFDB)
2.61.a_abv	$3$	(not in LMFDB)
2.61.a_er	$3$	(not in LMFDB)
2.61.m_ef	$3$	(not in LMFDB)
2.61.p_fg	$3$	(not in LMFDB)
2.61.bb_ls	$3$	(not in LMFDB)