Irrationalities and their p-modular reductions

Common Irrationalities in Characteristic 7

C1  =  X + 4
C2  =  X2 + 6X + 3
C3  =  X3 + 6X2 + 4
C4  =  X4 + 5X2 + 4X + 3
C5  =  X5 + X + 4
C6  =  X6 + X4 + 5X3 + 4X2 + 6X + 3
C8  =  X8 + 4X3 + 6X2 + 2X + 3
C9  =  X9 + 6X4 + X3 + 6X + 4
C12  =  X12 + 2X8 + 5X7 + 3X6 + 2X5 + 4X4 + 5X2 + 3


ξ f Cn
b5 2X + 2 C2
b5∗ 5X + 4 C2
b11 6X C2
b11∗∗ X + 6 C2
b13 4X + 1 C2
b13∗ 3X + 5 C2
b15 3X + 5 C2
b15∗∗ 4X + 1 C2
b17 X + 6 C2
b17∗ 6X C2
b19 5 C1
b19∗∗ 1 C1
b23 5X + 4 C2
b23∗∗ 2X + 2 C2
b27 0 C1
b27∗∗ 6 C1
b29 0 C1
b29∗ 6 C1
b31 2 C1
b31∗∗ 4 C1
b33 5X + 4 C2
b33∗ 2X + 2 C2
b37 5 C1
b37∗ 1 C1
b39 6X C2
b39∗∗ X + 6 C2
b45 6X C2
b45∗ X + 6 C2
b47 5 C1
b47∗∗ 1 C1
b59 2 C1
b59∗∗ 4 C1
b87 2 C1
b87∗∗ 4 C1
b95 X + 6 C2
b95∗∗ 6X C2
b117 5X + 4 C2
b117∗ 2X + 2 C2
c13 2X2 + 5X + 2 C3
c13∗2 2X2 + 6 C3
c13∗4 3X2 + 2X + 5 C3
c19 0 C1
c19∗2 4 C1
c19∗4 2 C1
c31 4X2 + 3 C3
c31∗3 4X2 + 3X + 2 C3
c31∗5 6X2 + 4X + 1 C3
c37 4X + 3 C3
c37∗2 2X2 + 4X C3
c37∗3 5X2 + 6X + 3 C3
c43 3X2 + 5X + 4 C3
c43∗3 X + 4 C3
c43∗7 4X2 + X + 5 C3
d17 4X3 + 2X2 + 2X + 1 C4
d17∗2 6X3 + X + 2 C4
d17∗3 3X3 + 3X C4
d17∗6 X3 + 5X2 + X + 3 C4
e31 5X4 + 4X3 + 3X2 + 1 C5
e31∗2 6X4 + 6X3 + 4X + 6 C5
e31∗3 4X4 + 3X3 + 3X2 + 6X + 3 C5
e31∗4 3X4 + 4X3 + 2X2 + 6X + 5 C5
e31∗8 3X4 + 4X3 + 6X2 + 5X + 5 C5
f31 6X2 + X + 3 C3
f31∗3 2X2 + 4X + 4 C3
f31∗5 6X2 + 1 C3
f31∗7 2X2 + 6X + 5 C3
f31∗11 4X C3
f31∗15 5X2 + 6X C3
f37 X2 + 4X C3
f37∗2 2X2 + 6X C3
f37∗3 6X2 + 5X + 5 C3
f37∗5 6X2 + X + 5 C3
f37∗6 6X2 + 3 C3
f37∗9 5X C3
i 6X + 4 C2
i2 4X + 5 C2
i3 5 C1
i5 4 C1
i6 1 C1
i8 X + 3 C2
i10 5 C1
i11 5X + 1 C2
i15 6X + 4 C2
i19 4 C1
i26 3 C1
k13 5X3 + X2 + 2X C4
k13∗2 X3 + 2X2 + 1 C4
k13∗4 2X3 + 6X2 + 5X C4
k13∗7 6X3 + 5X2 + 6 C4
k'52 X3 + X2 + X + 2 C4
k'52∗3 6X3 + 6X2 + 6X + 5 C4
k'52∗5 6X3 + X2 + 4X + 3 C4
k'52∗15 X3 + 6X2 + 3X + 4 C4
m5 X3 + 4X2 + 5X + 6 C4
m5∗2 3X3 + X2 + 5X + 1 C4
m5∗3 4X3 + 6X2 + 2X + 6 C4
m5∗4 6X3 + 3X2 + 2X + 1 C4
m16 6 C1
m16∗3 3 C1
m16∗9 1 C1
m16∗11 4 C1
m20 4X C2
m20∗3 3X + 4 C2
m20∗11 3X C2
m20∗13 4X + 3 C2
r2 3 C1
r3 5X + 1 C2
r5 4X + 5 C2
r6 X + 3 C2
r10 5X + 1 C2
r11 2 C1
r13 X + 3 C2
r15 1 C1
r17 2X + 6 C2
r30 3 C1
s51 3X3 + 6X2 + 3X + 5 C4
s51∗5 4X3 + 4X2 + X + 3 C4
s51∗11 X2 + 3X + 1 C4
s51∗19 3X2 + 6 C4
u''52 X3 + X2 + X + 2 C4
u''52∗3 6X3 + 6X2 + 6X + 5 C4
u''52∗5 6X3 + X2 + 4X + 3 C4
u''52∗15 X3 + 6X2 + 3X + 4 C4
v33 3X C2
v33∗2 2X + 5 C2
v33∗5 5X C2
v33∗7 4X + 3 C2
w'40 6X + 3 C2
w'40∗3 6X + 5 C2
w'40∗7 X + 2 C2
w'40∗17 X + 4 C2
w'80 3X3 + 6X2 + 3 C4
w'80∗3 4X3 + 4X2 + 4X + 1 C4
w'80∗7 X3 + 3X2 + 6X C4
w'80∗11 6X3 + 4X2 + X C4
w'80∗13 4X3 + X2 + 4 C4
w'80∗17 3X3 + 4X2 + 2X + 5 C4
w'80∗31 3X3 + 3X2 + 3X + 6 C4
w'80∗33 4X3 + 3X2 + 5X + 2 C4
x13 5X3 + X2 + 5 C4
x13∗2 5X3 + 4X2 + X + 2 C4
x13∗4 5X + 5 C4
x13∗7 4X3 + 2X2 + X + 1 C4
x19 4 C1
x19∗2 3 C1
x19∗4 0 C1
x19∗5 1 C1
x19∗8 3 C1
x19∗10 2 C1
x43 X C2
x43∗2 3X + 2 C2
x43∗3 5X + 5 C2
x43∗4 6X + 2 C2
x43∗5 2X + 6 C2
x43∗7 6X + 1 C2
x43∗9 3X + 3 C2
x43∗10 4X + 1 C2
x43∗13 5X + 1 C2
x43∗14 4X + 5 C2
x43∗19 X + 1 C2
x43∗20 4X + 6 C2
x43∗21 2X + 3 C2
x43∗26 3X + 5 C2
x57 4 C1
x57∗2 2 C1
x57∗4 6 C1
x57∗5 5 C1
x57∗8 0 C1
x57∗10 2 C1
x57∗11 1 C1
x57∗16 6 C1
x57∗22 1 C1
x57∗23 5 C1
x57∗29 4 C1
x57∗31 0 C1
x73 3X7 + 3X6 + 4X5 + 4X4 + 3X3 + X2 + 4X C8
x73∗2 X7 + 3X6 + 2X4 + 3X3 + 5X2 + 2X + 6 C8
x73∗3 3X7 + 2X6 + 5X3 + 4X2 + 2X + 3 C8
x73∗4 6X7 + 2X6 + 4X5 + 3X4 + 2X3 + 5X2 + 2X C8
x73∗5 5X5 + 6X4 + 6X3 + 3X2 + X + 4 C8
x73∗6 3X7 + 6X6 + 2X5 + X4 + 2X3 + 6X2 + 2X C8
x73∗7 4X7 + X6 + 5X5 + 6X4 + 6X3 + X2 + 6X + 4 C8
x73∗9 3X6 + X4 + 5X3 + 4X + 4 C8
x73∗11 X7 + 2X6 + X5 + X4 + 5X3 + 5X2 + 3 C8
x73∗12 X7 + 3X6 + 3X3 + 3X2 + 5X + 5 C8
x73∗13 2X7 + X6 + 2X5 + 3X4 + 4X3 + 4X2 + 4X + 2 C8
x73∗14 X7 + 4X6 + 6X5 + 6X4 + 2X2 + 5X + 1 C8
x73∗17 2X6 + 5X5 + X4 + 5X3 + 2X2 + 4X + 5 C8
x73∗18 6X7 + 3X6 + 4X4 + 6X3 + 2X + 6 C8
x73∗21 X7 + 5X6 + 6X5 + 5X4 + X3 + 6X C8
x73∗25 5X7 + 2X6 + 4X5 + 4X4 + 4X3 + 4X2 + X + 1 C8
x73∗26 5X7 + 3X6 + 2X4 + 4X3 + 4X2 + 2X + 5 C8
x73∗27 4X7 + 6X6 + 4X3 + 5X2 + 4X C8
x73∗33 2X6 + 2X5 + X4 + 5X3 + X2 + 4X + 2 C8
x73∗34 X7 + 4X6 + 6X5 + 4X4 + 3X3 + 2X2 + 2X + 1 C8
x73∗35 6X7 + 5X6 + 4X4 + 4X3 + 6X2 + 3X C8
x73∗36 2X7 + 4X6 + 2X5 + 4X4 + 6X3 + X2 + 4X + 4 C8
x73∗42 2X7 + 5X6 + X5 + 4X4 + 6X3 + 5X2 + 3X C8
x73∗43 6X7 + 6X6 + X5 + 4X4 + 6X3 + X2 + 5X + 6 C8
x171 1 C1
x171∗2 6 C1
x171∗4 0 C1
x171∗5 1 C1
x171∗8 1 C1
x171∗10 5 C1
x171∗11 3 C1
x171∗13 6 C1
x171∗16 4 C1
x171∗17 4 C1
x171∗20 5 C1
x171∗22 6 C1
x171∗23 5 C1
x171∗29 3 C1
x171∗31 6 C1
x171∗34 3 C1
x171∗37 5 C1
x171∗40 5 C1
x171∗41 3 C1
x171∗43 2 C1
x171∗44 6 C1
x171∗47 3 C1
x171∗58 2 C1
x171∗59 5 C1
x171∗61 0 C1
x171∗62 2 C1
x171∗65 2 C1
x171∗68 6 C1
x171∗73 1 C1
x171∗86 6 C1
x171∗94 3 C1
x171∗97 3 C1
x171∗115 4 C1
x171∗118 0 C1
x171∗122 4 C1
x171∗143 5 C1
y9 5X2 + 3X + 2 C3
y9∗2 3X + 6 C3
y9∗4 2X2 + X + 6 C3
y13 X5 + 4X4 + 6X3 + 6X2 + 6X + 6 C6
y13∗2 3X5 + 3X4 + 2X3 + 2X2 + 5X + 1 C6
y13∗3 2X5 + 4X3 + 2X2 + 4X + 4 C6
y13∗4 2X5 + X4 + 3X3 + 2 C6
y13∗5 4X5 + 2X4 + 2X3 + X2 + 4X + 5 C6
y13∗6 2X5 + 4X4 + 4X3 + 3X2 + 2X + 2 C6
y15 6X3 + 3X2 + X + 3 C4
y15∗2 5X3 + 2X2 + 2X + 1 C4
y15∗4 X2 + 5X + 1 C4
y15∗7 3X3 + X2 + 6X + 3 C4
y16 3X + 2 C2
y16∗3 6X + 4 C2
y16∗5 X + 3 C2
y16∗7 4X + 5 C2
y20 5X2 + 2X + 2 C4
y20∗3 5X3 + 3X + 1 C4
y20∗7 2X3 + 4X + 6 C4
y20∗9 2X2 + 5X + 5 C4
y24 4X + 3 C2
y24∗5 4X C2
y24∗7 3X C2
y24∗11 3X + 4 C2
y25 3X + 3 C2
y25∗2 6X + 1 C2
y25∗3 X + 2 C2
y25∗4 6X + 3 C2
y25∗6 5X + 6 C2
y25∗7 4X + 6 C2
y25∗8 2X + 4 C2
y25∗9 6X + 2 C2
y25∗11 X C2
y25∗12 X + 1 C2
y32 4X3 + 2X2 + 6X + 3 C4
y32∗3 3X3 + 2X2 + 4X C4
y32∗5 2X3 + 4X2 + 2 C4
y32∗7 5X3 + 4X2 + 6X + 4 C4
y32∗9 2X3 + 3X2 + X + 3 C4
y32∗11 5X3 + 3X2 + 5 C4
y32∗13 4X3 + 5X2 + 3X C4
y32∗15 3X3 + 5X2 + X + 4 C4
y48 3X + 5 C2
y48∗5 5X C2
y48∗7 4X + 1 C2
y48∗11 5X + 2 C2
y48∗13 2X + 5 C2
y48∗17 3X + 6 C2
y48∗19 2X C2
y48∗23 4X + 2 C2
y52 4X11 + 4X10 + X9 + 4X7 + 4X6 + 4X4 + X3 + 4X2 + 2X + 2 C12
y52∗3 X11 + 2X10 + 3X9 + 5X8 + 3X7 + 6X6 + 3X5 + 5X4 + 6X3 + 2X2 + 4X + 6 C12
y52∗5 6X11 + 4X10 + X9 + 3X8 + 6X7 + 4X6 + X5 + X3 + 5X + 5 C12
y52∗7 2X11 + 2X10 + 2X9 + 2X8 + 6X7 + 6X6 + 5X5 + 4X4 + 4X3 + X2 + X + 2 C12
y52∗9 3X11 + 3X10 + 3X9 + 5X8 + 2X6 + 5X5 + 5X4 + X3 + 3X2 + 2X + 3 C12
y52∗11 6X11 + X10 + 2X9 + 6X8 + 3X6 + 4X5 + X4 + X3 + 5X2 + 5X + 1 C12
y52∗15 X11 + 6X10 + 5X9 + X8 + 4X6 + 3X5 + 6X4 + 6X3 + 2X2 + 2X + 6 C12
y52∗17 4X11 + 4X10 + 4X9 + 2X8 + 5X6 + 2X5 + 2X4 + 6X3 + 4X2 + 5X + 4 C12
y52∗19 5X11 + 5X10 + 5X9 + 5X8 + X7 + X6 + 2X5 + 3X4 + 3X3 + 6X2 + 6X + 5 C12
y52∗21 X11 + 3X10 + 6X9 + 4X8 + X7 + 3X6 + 6X5 + 6X3 + 2X + 2 C12
y52∗23 6X11 + 5X10 + 4X9 + 2X8 + 4X7 + X6 + 4X5 + 2X4 + X3 + 5X2 + 3X + 1 C12
y52∗25 3X11 + 3X10 + 6X9 + 3X7 + 3X6 + 3X4 + 6X3 + 3X2 + 5X + 5 C12
y60 X3 + 6X2 + 6 C4
y60∗7 5X3 + 6X + 3 C4
y60∗11 3X3 + 6X2 + 4X + 5 C4
y60∗13 2X3 + 5X2 + 3X + 6 C4
y60∗17 5X3 + 2X2 + 4X + 1 C4
y60∗19 4X3 + X2 + 3X + 2 C4
y60∗23 2X3 + X + 4 C4
y60∗29 6X3 + X2 + 1 C4
y96 6X3 + 5X + 4 C4
y96∗5 6X3 + 3X2 + 2X + 1 C4
y96∗7 6X2 + X + 1 C4
y96∗11 X3 + 6X2 + 3X + 4 C4
y96∗13 3X3 + X2 + 5X + 1 C4
y96∗17 4X3 + 3X2 + 5X + 2 C4
y96∗19 X3 + X2 + X + 2 C4
y96∗23 4X3 + 4X2 + 4X + 1 C4
y96∗25 3X3 + 3X2 + 3X + 6 C4
y96∗29 6X3 + 6X2 + 6X + 5 C4
y96∗31 3X3 + 4X2 + 2X + 5 C4
y96∗35 4X3 + 6X2 + 2X + 6 C4
y96∗37 6X3 + X2 + 4X + 3 C4
y96∗41 X2 + 6X + 6 C4
y96∗43 X3 + 4X2 + 5X + 6 C4
y96∗47 X3 + 2X + 3 C4
y100 2X3 + X2 + 3X + 5 C4
y100∗3 2X3 + 5X2 + 6X + 1 C4
y100∗7 6X3 + 2X2 + 3X + 2 C4
y100∗9 2X3 + 6X2 + 5X C4
y100∗11 6X3 + 4X2 + X C4
y100∗13 6X3 + 5X2 + 6 C4
y100∗17 3X2 + 4X + 4 C4
y100∗19 4X3 + X + 5 C4
y100∗21 2X3 + 2X2 + 2X + 4 C4
y100∗23 3X3 + 6X2 + 3 C4
y100∗27 4X3 + X2 + 4 C4
y100∗29 5X3 + 5X2 + 5X + 3 C4
y100∗31 3X3 + 6X + 2 C4
y100∗33 4X2 + 3X + 3 C4
y100∗37 X3 + 2X2 + 1 C4
y100∗39 X3 + 3X2 + 6X C4
y100∗41 5X3 + X2 + 2X C4
y100∗43 X3 + 5X2 + 4X + 5 C4
y100∗47 5X3 + 2X2 + X + 6 C4
y100∗49 5X3 + 6X2 + 4X + 2 C4
y'40 4X C2
y'40∗3 4X + 3 C2
y'40∗7 3X + 4 C2
y'40∗11 5X + 5 C2
y'40∗13 2X + 3 C2
y'40∗17 5X + 4 C2
y'40∗21 3X C2
y'40∗31 2X + 2 C2
y'48 1 C1
y'48∗5 3 C1
y'48∗11 4 C1
y'48∗13 2 C1
y'48∗17 2 C1
y'48∗19 5 C1
y'48∗25 6 C1
y'48∗41 5 C1
y'80 2X3 + 2X2 + 2X + 4 C4
y'80∗3 2X3 + 4X2 + 2 C4
y'80∗7 5X3 + 2X2 + X + 6 C4
y'80∗11 4X3 + 6X2 + 2X + 6 C4
y'80∗13 6X3 + 3X2 + 2X + 1 C4
y'80∗17 2X2 + 5X + 5 C4
y'80∗21 3X3 + 2X2 + 4X C4
y'80∗23 2X3 + 5X2 + 6X + 1 C4
y'80∗31 5X3 + 3X + 1 C4
y'80∗33 5X2 + 2X + 2 C4
y'80∗41 5X3 + 5X2 + 5X + 3 C4
y'80∗43 5X3 + 3X2 + 5 C4
y'80∗51 3X3 + X2 + 5X + 1 C4
y'80∗53 X3 + 4X2 + 5X + 6 C4
y'80∗61 4X3 + 5X2 + 3X C4
y'80∗71 2X3 + 4X + 6 C4
y''48 6X C2
y''48∗5 4X + 4 C2
y''48∗7 X + 6 C2
y''48∗11 4X + 6 C2
y''48∗17 X C2
y''48∗23 6X + 1 C2
y''48∗29 3X + 3 C2
y''48∗35 3X + 1 C2
y'''24 2 C1
y'''24∗5 6 C1
y'''24∗13 5 C1
y'''24∗17 1 C1
z3 2 C1
z3∗∗ 4 C1
z8 2X + 4 C2
z8∗3 2X + 1 C2
z8∗5 5X + 3 C2
z8∗7 5X + 6 C2
z9 2X2 + 2X + 1 C3
z9∗2 6X2 + 2X + 2 C3
z9∗4 4X2 + 4X + 2 C3
z9∗5 5X2 + 4X + 4 C3
z9∗7 X2 + X + 4 C3
z9∗8 3X2 + X + 1 C3
z27 6X8 + 3X7 + 6X5 + X4 + 5X3 + 5X2 + 3X + 3 C9
z27∗2 4X8 + 5X7 + 4X6 + 5X5 + 5X4 + 3X3 + 2X2 + 6X + 1 C9
z27∗4 3X8 + 5X7 + 4X6 + 2X5 + 4X3 + 5X2 + 2X + 2 C9
z27∗5 5X8 + 6X7 + 6X6 + 5X5 + 3X4 + 6X3 + 2X2 + 4X + 3 C9
z27∗7 2X8 + 6X7 + 4X6 + 6X5 + 6X4 + 5X3 + 5X2 + 4X + 3 C9
z27∗8 2X8 + 6X7 + 3X6 + 6X5 + 6X4 + 3X3 C9
z27∗10 5X8 + 6X7 + 5X5 + 2X4 + 3X3 + 3X2 + 6X + 6 C9
z27∗11 X8 + 3X7 + X6 + 3X5 + 3X4 + 6X3 + 4X2 + 5X + 2 C9
z27∗13 6X8 + 3X7 + X6 + 4X5 + X3 + 3X2 + 4X + 4 C9
z27∗14 3X8 + 5X7 + 5X6 + 3X5 + 6X4 + 5X3 + 4X2 + X + 6 C9
z27∗16 4X8 + 5X7 + X6 + 5X5 + 5X4 + 3X3 + 3X2 + X + 6 C9
z27∗17 4X8 + 5X7 + 6X6 + 5X5 + 5X4 + 6X3 C9
z27∗19 3X8 + 5X7 + 3X5 + 4X4 + 6X3 + 6X2 + 5X + 5 C9
z27∗20 2X8 + 6X7 + 2X6 + 6X5 + 6X4 + 5X3 + X2 + 3X + 4 C9
z27∗22 5X8 + 6X7 + 2X6 + X5 + 2X3 + 6X2 + X + 1 C9
z27∗23 6X8 + 3X7 + 3X6 + 6X5 + 5X4 + 3X3 + X2 + 2X + 5 C9
z27∗25 X8 + 3X7 + 2X6 + 3X5 + 3X4 + 6X3 + 6X2 + 2X + 5 C9
z27∗26 X8 + 3X7 + 5X6 + 3X5 + 3X4 + 5X3 C9




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