Fields with Small GRD

A new home for this data

The tables below have been superseded by the searchable database of number fields where one can include a bound for the Galois Root Discriminant of the field. In particular, there may be additional fields in that database.

Guide to these tables

Below are are tables of fields with small Galois Root Discriminant (GRD). They are sorted by Galois group, with an emphasis on non-abelian groups. Files can be downloaded as gp files of polynomials, or printable pdf, which has other data included.

The number of distinct Galois fields is given. In some cases, polynomials for non-isomorphic fields with isomorphic Galois closures are listed. Groups only appear for their lowest degree permutation representation. In some cases, tables will contain polynomials of higher degree which have the same Galois closure.

Status flags are either C meaning that the table is proven complete by an exhaustive computation; L for likely to be complete, but not proven; <B for some bound B to indicate that they are proven complete up that the given bound. A blank status flag indicates that we have no information about the completeness of the corresponding table.

Credits

Most of the tables assembled here come from joint work of Jones and Roberts, growing out of Galois Number Fields with Small Root Discriminant, to appear in J. Number Theory.

Rachel Wallington has certified the completeness of the entries for 6T10 and 6T13, and contributed new complete entries for 7T2, 7T3, 7T4, solvable octics, and 12T5.

The SL2(16) field comes from a paper by Johan Bosman.

Degree 2
Download # Status Group
pdf gp 1220 C T1 S2 = C2
Degree 3
Download # Status Group
pdf gp 47 C T1 A3 = C3
pdf gp 610 C T2 S3 = GL2(F2)
Degree 4
Download # Status Group
pdf gp 228 C T1 C4
pdf gp 2421 C T2 V4 = C22
pdf gp 1425 C T3 D4
pdf gp 59 C T4 A4 = PSL2(F3)
pdf gp 527 C T5 S4 = PGL2(F3)
Degree 5
Download # Status Group
pdf gp 7 C T1 C5
pdf gp 146 C T2 D5 = C5:C2
pdf gp 102 C T3 F5 = M20 = C5:C4
pdf gp 78 C T4 A5 = L(6) =PSL2(F5)
pdf gp 192 C T5 S5 = L(6):2 = PGL2(F5)
Degree 6
Download # Status Group
pdf gp 399 C T1 C6 = C3C2
pdf gp 1795 C T3 D6 = S3C2
pdf gp 254 C T5 S3C3 = 3 wr 2
pdf gp 243 C T6 A4C2 = 2 wr 3
pdf gp 445 C T9 S32 = C32:V4
pdf gp 17 C T10 C32:C4
pdf gp 1098 C T11 S4C2 = 2 wr S3
pdf gp 137 C T13 C32:D4 = S3 wr 2
pdf gp 5 C T15 A6
pdf gp 13 C T16 S6
Degree 7
Download # Status Group
pdf gp 4 C T1 C7
pdf gp 80 C T2 D7 = C7:C2
pdf gp 2 C T3 C7:C3
pdf gp 94 C T4 F7 = M42 = C7:C6
pdf gp 17 < 39.52 T5 L(7) = L(8) = GL3(F2) = PSL2(F7) = G168
pdf gp 1 < 39.52 T6 A7
pdf gp 1   T7 S7
Degree 8
Download # Status Group
pdf gp 23 C T1 C8
pdf gp 581 C T2 C4C2
pdf gp 908 C T3 C23
pdf gp 7 C T5 Q8
pdf gp 354 C T6 D8
pdf gp 55 C T7 1/2[23]4 = 8:{1,5}
pdf gp 121 C T8 QD16 = Q8:C2
pdf gp 1477 C T9 D4C2
pdf gp 310 C T10 C22:C4
pdf gp 307 C T11 Q8:C2
pdf gp 4 C T12 SL2(F3)
pdf gp 818 C T15 (D4C2):C2
pdf gp 38 C T16 8T7:C2
pdf gp 126 C T17 C42:C2
pdf gp 318 C T18 C23:C22
pdf gp 110 C T19 C23:C4 = 8T20 = 8T21
pdf gp 147 C T22 C23:C22
pdf gp 194 C T23 GL2(F3)
pdf gp 1 C T25 C23:C7
pdf gp 210 C T26 Q8:D4
pdf gp 80 C T27 C24:C4=8T28
pdf gp 229 C T29 C23:D4=8T31
pdf gp 35 C T30 C42:C4
pdf gp 8 C T32 C23:A4
pdf gp 12 C T33 C23:A4
pdf gp 55 C T34 C24:S3
pdf gp 183 C T35 D4 wr 2
pdf gp 4 C T36 C23:(C7:C3)
pdf gp 23 C T38 C24:A4
pdf gp 14 C T39 C23:S4
pdf gp 49 C T40 Q8:S4
pdf gp 111 C T41 C23:S4
pdf gp 14 C T42 A4 wr 2
pdf gp 20   T43 L(8):2 = PGL2(F7) = G168:2
pdf gp 84 C T44 24S4
pdf gp 39 C T45 C24:S32
— — 0 C T46 A42:C4
pdf gp 15 C T47 S4 wr 2
pdf gp 3 T48 23.GL3(F2)
Degree 9
Download # Status Group
pdf gp 3 C T1 C9
pdf gp 9 C T2 C32
pdf gp 105 C T3 D9
pdf gp 48 C T5 C32:C2
pdf gp 2 C T6 C9:C3
— — 0 C T7 C32:C3
pdf gp 69 C T10 C9:C6
pdf gp 64 C T11 C32:C6=9T13
pdf gp 37 C T12 C32:S3
pdf gp 2 C T14 C32:Q8
pdf gp 3 C T15 C32:C8
— — 0 C T17 3 wr 3
pdf gp 145 C T18 C32:D6
pdf gp 33 C T19 C32:QD16
pdf gp 10 C T20 3 wr S3
pdf gp 42 C T21 C33:S3
pdf gp 17 C T22 (C32:C3):S3
— — 0 C T23 C32: SL2(F3)
pdf gp 37 C T24 (C32:C3):D6
pdf gp 4 C T25 C33:A4
pdf gp 51 C T26 C32:GL2(F3)
pdf gp 14   T27 L(9) = SL2(F8) = G504
pdf gp 7 C T28 S3 wr 3
pdf gp 2 C T29 C33:S4
pdf gp 35 C T30 C33:S4
pdf gp 15 C T31 S3 wr S3
pdf gp 15   T32 L(9):3 = G504:3
Degree 10
Download # Status Group
pdf gp 69 C T1 C10 = C5C2
pdf gp 384 C T3 D10 = D5C2
pdf gp 6 C T8 C24:C5
pdf gp 15 C T14 2 wr 5
pdf gp 24 C T15 C24:D5 = 10T16
pdf gp 40 C T23 2 wr D5
pdf gp 3   T30 L(10):2 = PGL2(F9)
pdf gp 5 C T34 24.A5
pdf gp 14   T35 L(10):22 = PGammaL2(F9) = PGL2(F9).2
pdf gp 8 C T36 25.A5
pdf gp 17 C T37 24.S5 = T38
pdf gp 15 C T39 25.S5
pdf gp 1   T40 A52.2
Degree 11
Download # Status Group
pdf gp 1 C T1 C11
pdf gp 32 C T2 D11
— — 0 C T3 C11:C5
pdf gp 1   T5 L(11) = L(12) = PSL2(F11) = G660
Degree 12
Download # Status Group
pdf gp 66 C T1 C12=C4C3
pdf gp 391 C T2 C6C2
pdf gp 15 C T5 C3:C4
pdf gp 854 C T10 S3C22
pdf gp 333 C T12 D12
pdf gp 396 C T13 (C6 C2): C2=T15
pdf gp 652 C T28 D4 S3
Degree 13
Download # Status Group
pdf gp 1 C T1 C13
pdf gp 18 C T1 D13
Degree 14
Download # Status Group
pdf gp 8 C T1 C14=C7C2
Degree 15
Download # Status Group
pdf gp 4 C T1 C15=C5C3
Degree 16
Download # Status Group
pdf gp 9 C T1 C16
pdf gp 195 C T2 C4C22
pdf gp 73 C T3 C24
pdf gp 16 C T4 C42
pdf gp 30 C T5 C8C2
pdf gp 7 C T7 Q8C2
pdf gp 11 C T8 C4:C4
— — 0 C T14 Q16
Degree 17
Download # Status Group
— — 0 C T1 C17
pdf gp 10 C T2 D17
pdf gp 1   T6 SL2(F16)
Degree 19
Download # Status Group
— — 0 C T1 C19
pdf gp 9 C T2 D19
Degree 23
Download # Status Group
pdf gp 1 C T1 C23
pdf gp 8 C T2 D23
Last Update: March 15, 2015
Credits © 2009 John Jones

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Last Modified: March 15, 2015