Abelian variety isogeny class 2.67.aj_o over $\F_{67}$

• Label: 2.67.aj_o
• Base field: $\F_{67}$
• 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:	$\F_{67}$
Dimension:	$2$
L-polynomial:	$1 - 9 x + 14 x^{2} - 603 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.0186151165355$, $\pm0.648051550131$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-187})\)
Galois group:	$C_2^2$
Jacobians:	$14$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3892$	$19911472$	$89810499856$	$405943383190464$	$1822815068728494652$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$59$	$4437$	$298604$	$20144953$	$1350108269$	$90457252422$	$6060707649671$	$406067675310961$	$27206533825815188$	$1822837802135063157$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

• $y^2=15 x^6+18 x^5+31 x^4+29 x^3+58 x^2+60 x+25$
• $y^2=13 x^6+47 x^5+11 x^4+58 x^3+31 x^2+4 x+51$
• $y^2=11 x^6+24 x^5+25 x^4+22 x^3+61 x^2+54 x+5$
• $y^2=21 x^6+66 x^5+46 x^4+37 x^3+3 x^2+7 x+51$
• $y^2=18 x^6+29 x^5+38 x^4+33 x^3+7 x^2+13 x+50$
• $y^2=5 x^6+43 x^5+21 x^4+44 x^3+45 x^2+25 x+59$
• $y^2=26 x^6+12 x^5+30 x^4+16 x^3+21 x^2+20 x+58$
• $y^2=61 x^6+26 x^5+41 x^4+49 x^3+2 x^2+8 x+27$
• $y^2=11 x^6+64 x^5+3 x^4+43 x^3+54 x^2+36 x+54$
• $y^2=59 x^6+46 x^5+6 x^4+48 x^3+17 x^2+3 x+52$
• $y^2=13 x^6+35 x^5+53 x^4+31 x^3+4 x^2+16 x+9$
• $y^2=51 x^6+65 x^5+55 x^4+62 x^3+6 x^2+41 x+9$
• $y^2=50 x^6+8 x^5+52 x^4+45 x^3+12 x^2+65 x+66$
• $y^2=65 x^6+7 x^5+14 x^4+34 x^3+34 x^2+25 x+22$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{67^{3}}$.

Endomorphism algebra over $\F_{67}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-187})\).

Endomorphism algebra over $\overline{\F}_{67}$

The base change of $A$ to $\F_{67^{3}}$ is 1.300763.abpo^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-187}) \)$)$

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
2.67.j_o	$2$	(not in LMFDB)
2.67.s_ih	$3$	(not in LMFDB)
2.67.as_ih	$6$	(not in LMFDB)