Abelian variety isogeny class 2.83.be_pb over $\F_{83}$

• Label: 2.83.be_pb
• Base field: $\F_{83}$
• 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:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$( 1 + 15 x + 83 x^{2} )^{2}$
	$1 + 30 x + 391 x^{2} + 2490 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.807831363318$, $\pm0.807831363318$
Angle rank:	$1$ (numerical)
Jacobians:	$18$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9801$	$46662561$	$326529959184$	$2253269789767161$	$15515059798626412761$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$114$	$6772$	$571068$	$47478916$	$3938791494$	$326942401318$	$27136041249858$	$2252292209915908$	$186940256409288324$	$15516041171924154772$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

• $y^2=73 x^6+60 x^5+62 x^4+8 x^3+62 x^2+60 x+73$
• $y^2=x^6+76 x^5+19 x^4+19 x^2+76 x+1$
• $y^2=9 x^6+26 x^5+48 x^4+60 x^3+42 x+48$
• $y^2=30 x^6+32 x^5+34 x^4+23 x^3+47 x^2+60 x+21$
• $y^2=2 x^6+4 x^5+13 x^4+49 x^3+78 x^2+26 x+11$
• $y^2=30 x^6+26 x^5+4 x^4+15 x^3+4 x^2+26 x+30$
• $y^2=63 x^6+52 x^5+59 x^4+25 x^3+59 x^2+52 x+63$
• $y^2=15 x^6+81 x^5+72 x^4+45 x^3+72 x^2+81 x+15$
• $y^2=31 x^6+34 x^5+16 x^4+39 x^3+16 x^2+34 x+31$
• $y^2=73 x^6+29 x^5+67 x^4+35 x^3+21 x^2+16 x+44$
• $y^2=25 x^6+48 x^5+57 x^4+63 x^3+46 x^2+81 x+75$
• $y^2=26 x^6+43 x^5+14 x^4+46 x^3+14 x^2+43 x+26$
• $y^2=69 x^6+72 x^5+39 x^4+40 x^3+45 x^2+5 x+59$
• $y^2=72 x^6+75 x^5+64 x^4+52 x^3+58 x^2+58 x+75$
• $y^2=70 x^6+24 x^5+43 x^4+61 x^3+26 x^2+6 x+75$
• $y^2=48 x^6+79 x^5+58 x^4+60 x^3+81 x^2+27 x+60$
• $y^2=32 x^6+57 x^5+61 x^4+78 x^3+61 x^2+57 x+32$
• $y^2=38 x^6+82 x^5+63 x^4+8 x^3+63 x^2+82 x+38$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{83}$.

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.p^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-107}) \)$)$

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
2.83.abe_pb	$2$	(not in LMFDB)
2.83.a_ach	$2$	(not in LMFDB)
2.83.ap_fm	$3$	(not in LMFDB)