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

• Label: 2.61.ad_aca
• 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:	$1 - 3 x - 52 x^{2} - 183 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.105151362908$, $\pm0.771818029575$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-235})\)
Galois group:	$C_2^2$
Jacobians:	$26$
Isomorphism classes:	88

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3484$	$13434304$	$51284131600$	$191781084328704$	$713302521976375084$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$59$	$3609$	$225938$	$13851169$	$844548479$	$51520737318$	$3142746297659$	$191707313682529$	$11694146519265338$	$713342912260729329$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{-235})\).

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

The base change of $A$ to $\F_{61^{3}}$ is 1.226981.auc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-235}) \)$)$

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.d_aca	$2$	(not in LMFDB)
2.61.g_fb	$3$	(not in LMFDB)
2.61.ag_fb	$6$	(not in LMFDB)