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

• Label: 2.61.aw_ji
• 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:	$( 1 - 12 x + 61 x^{2} )( 1 - 10 x + 61 x^{2} )$
	$1 - 22 x + 242 x^{2} - 1342 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.221142061624$, $\pm0.278857938376$
Angle rank:	$1$ (numerical)
Jacobians:	$25$
Isomorphism classes:	88

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2600$	$13852800$	$51815839400$	$191900067840000$	$713405328370865000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$3722$	$228280$	$13859758$	$844670200$	$51520374362$	$3142738798120$	$191707271553118$	$11694145883260360$	$713342911662882602$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 25 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61^{4}}$.

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.am $\times$ 1.61.ak 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.am : \(\Q(\sqrt{-1}) \).
• 1.61.ak : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{61^{4}}$ is 1.13845841.khq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

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

  • 1.3721.aw : \(\Q(\sqrt{-1}) \).
  • 1.3721.w : \(\Q(\sqrt{-1}) \).

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_c	$2$	(not in LMFDB)
2.61.c_c	$2$	(not in LMFDB)
2.61.w_ji	$2$	(not in LMFDB)