Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 23*x^22 - 58*x^21 + 100*x^20 - 160*x^19 + 242*x^18 - 156*x^17 - 228*x^16 + 452*x^15 + 164*x^14 - 1190*x^13 + 1288*x^12 - 326*x^11 - 800*x^10 + 832*x^9 + 514*x^8 - 1302*x^7 + 500*x^6 + 664*x^5 - 155*x^4 - 1418*x^3 + 1951*x^2 - 1112*x + 241)
gp: K = bnfinit(y^24 - 6*y^23 + 23*y^22 - 58*y^21 + 100*y^20 - 160*y^19 + 242*y^18 - 156*y^17 - 228*y^16 + 452*y^15 + 164*y^14 - 1190*y^13 + 1288*y^12 - 326*y^11 - 800*y^10 + 832*y^9 + 514*y^8 - 1302*y^7 + 500*y^6 + 664*y^5 - 155*y^4 - 1418*y^3 + 1951*y^2 - 1112*y + 241, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 6*x^23 + 23*x^22 - 58*x^21 + 100*x^20 - 160*x^19 + 242*x^18 - 156*x^17 - 228*x^16 + 452*x^15 + 164*x^14 - 1190*x^13 + 1288*x^12 - 326*x^11 - 800*x^10 + 832*x^9 + 514*x^8 - 1302*x^7 + 500*x^6 + 664*x^5 - 155*x^4 - 1418*x^3 + 1951*x^2 - 1112*x + 241);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 6*x^23 + 23*x^22 - 58*x^21 + 100*x^20 - 160*x^19 + 242*x^18 - 156*x^17 - 228*x^16 + 452*x^15 + 164*x^14 - 1190*x^13 + 1288*x^12 - 326*x^11 - 800*x^10 + 832*x^9 + 514*x^8 - 1302*x^7 + 500*x^6 + 664*x^5 - 155*x^4 - 1418*x^3 + 1951*x^2 - 1112*x + 241)
\( x^{24} - 6 x^{23} + 23 x^{22} - 58 x^{21} + 100 x^{20} - 160 x^{19} + 242 x^{18} - 156 x^{17} + \cdots + 241 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(18075490334784000000000000000000\)
\(\medspace = 2^{24}\cdot 3^{24}\cdot 5^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.06\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2\cdot 3^{3/2}5^{3/4}\approx 34.748765558648074$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{19}a^{20}-\frac{4}{19}a^{19}-\frac{7}{19}a^{18}-\frac{7}{19}a^{17}+\frac{7}{19}a^{16}+\frac{6}{19}a^{15}+\frac{9}{19}a^{14}+\frac{1}{19}a^{13}-\frac{6}{19}a^{12}+\frac{9}{19}a^{11}+\frac{7}{19}a^{10}-\frac{3}{19}a^{9}+\frac{2}{19}a^{8}+\frac{2}{19}a^{7}-\frac{6}{19}a^{6}-\frac{5}{19}a^{5}-\frac{3}{19}a^{4}+\frac{9}{19}a^{2}+\frac{1}{19}a-\frac{3}{19}$, $\frac{1}{19}a^{21}-\frac{4}{19}a^{19}+\frac{3}{19}a^{18}-\frac{2}{19}a^{17}-\frac{4}{19}a^{16}-\frac{5}{19}a^{15}-\frac{1}{19}a^{14}-\frac{2}{19}a^{13}+\frac{4}{19}a^{12}+\frac{5}{19}a^{11}+\frac{6}{19}a^{10}+\frac{9}{19}a^{9}-\frac{9}{19}a^{8}+\frac{2}{19}a^{7}+\frac{9}{19}a^{6}-\frac{4}{19}a^{5}+\frac{7}{19}a^{4}+\frac{9}{19}a^{3}-\frac{1}{19}a^{2}+\frac{1}{19}a+\frac{7}{19}$, $\frac{1}{520391}a^{22}+\frac{4421}{520391}a^{21}-\frac{652}{520391}a^{20}-\frac{229390}{520391}a^{19}+\frac{251383}{520391}a^{18}-\frac{42842}{520391}a^{17}+\frac{253826}{520391}a^{16}+\frac{244452}{520391}a^{15}-\frac{93874}{520391}a^{14}+\frac{241295}{520391}a^{13}-\frac{55259}{520391}a^{12}-\frac{117101}{520391}a^{11}-\frac{253938}{520391}a^{10}+\frac{4691}{27389}a^{9}-\frac{143170}{520391}a^{8}+\frac{79622}{520391}a^{7}+\frac{228524}{520391}a^{6}-\frac{3747}{8531}a^{5}+\frac{228049}{520391}a^{4}+\frac{5531}{520391}a^{3}-\frac{106620}{520391}a^{2}-\frac{195112}{520391}a-\frac{231627}{520391}$, $\frac{1}{41\!\cdots\!89}a^{23}-\frac{10\!\cdots\!84}{41\!\cdots\!89}a^{22}-\frac{10\!\cdots\!26}{41\!\cdots\!89}a^{21}+\frac{85\!\cdots\!55}{41\!\cdots\!89}a^{20}+\frac{28\!\cdots\!69}{22\!\cdots\!31}a^{19}-\frac{18\!\cdots\!30}{41\!\cdots\!89}a^{18}+\frac{10\!\cdots\!71}{41\!\cdots\!89}a^{17}+\frac{25\!\cdots\!62}{41\!\cdots\!89}a^{16}+\frac{23\!\cdots\!47}{41\!\cdots\!89}a^{15}+\frac{98\!\cdots\!83}{41\!\cdots\!89}a^{14}-\frac{16\!\cdots\!97}{41\!\cdots\!89}a^{13}-\frac{32\!\cdots\!60}{41\!\cdots\!89}a^{12}+\frac{14\!\cdots\!54}{41\!\cdots\!89}a^{11}-\frac{56\!\cdots\!30}{14\!\cdots\!41}a^{10}-\frac{85\!\cdots\!66}{41\!\cdots\!89}a^{9}-\frac{15\!\cdots\!36}{41\!\cdots\!89}a^{8}+\frac{30\!\cdots\!82}{41\!\cdots\!89}a^{7}-\frac{14\!\cdots\!48}{41\!\cdots\!89}a^{6}-\frac{21\!\cdots\!81}{41\!\cdots\!89}a^{5}-\frac{17\!\cdots\!35}{41\!\cdots\!89}a^{4}+\frac{10\!\cdots\!70}{41\!\cdots\!89}a^{3}+\frac{10\!\cdots\!22}{41\!\cdots\!89}a^{2}-\frac{11\!\cdots\!45}{41\!\cdots\!89}a+\frac{22\!\cdots\!29}{91\!\cdots\!91}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{73616946842978946}{385899254913200159} a^{23} - \frac{7462656356153571980}{7332085843350803021} a^{22} + \frac{27199219556983244939}{7332085843350803021} a^{21} - \frac{63003113424945170569}{7332085843350803021} a^{20} + \frac{97853678341237425469}{7332085843350803021} a^{19} - \frac{158509176010303132348}{7332085843350803021} a^{18} + \frac{232899148742777208807}{7332085843350803021} a^{17} - \frac{62853260300608451637}{7332085843350803021} a^{16} - \frac{361334797096169948987}{7332085843350803021} a^{15} + \frac{390773395557468131143}{7332085843350803021} a^{14} + \frac{491465161484057999336}{7332085843350803021} a^{13} - \frac{1336747531522870070326}{7332085843350803021} a^{12} + \frac{907671804585560377208}{7332085843350803021} a^{11} + \frac{129018987937036181}{6326217293659019} a^{10} - \frac{1014351005174727814448}{7332085843350803021} a^{9} + \frac{485370744649705117064}{7332085843350803021} a^{8} + \frac{1045162766334852803796}{7332085843350803021} a^{7} - \frac{1122011553221285078722}{7332085843350803021} a^{6} - \frac{55062761327452545061}{7332085843350803021} a^{5} + \frac{886012307751874158164}{7332085843350803021} a^{4} + \frac{19877686242970701284}{385899254913200159} a^{3} - \frac{1734124202687164364017}{7332085843350803021} a^{2} + \frac{82737670931200852412}{385899254913200159} a - \frac{508948454126450352214}{7332085843350803021} \)
(order $4$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{82\!\cdots\!79}{41\!\cdots\!89}a^{23}-\frac{43\!\cdots\!71}{41\!\cdots\!89}a^{22}+\frac{15\!\cdots\!39}{41\!\cdots\!89}a^{21}-\frac{36\!\cdots\!93}{41\!\cdots\!89}a^{20}+\frac{56\!\cdots\!73}{41\!\cdots\!89}a^{19}-\frac{92\!\cdots\!44}{41\!\cdots\!89}a^{18}+\frac{13\!\cdots\!27}{41\!\cdots\!89}a^{17}-\frac{35\!\cdots\!86}{41\!\cdots\!89}a^{16}-\frac{21\!\cdots\!50}{41\!\cdots\!89}a^{15}+\frac{22\!\cdots\!57}{41\!\cdots\!89}a^{14}+\frac{29\!\cdots\!80}{41\!\cdots\!89}a^{13}-\frac{77\!\cdots\!37}{41\!\cdots\!89}a^{12}+\frac{52\!\cdots\!85}{41\!\cdots\!89}a^{11}+\frac{32\!\cdots\!72}{14\!\cdots\!41}a^{10}-\frac{59\!\cdots\!65}{41\!\cdots\!89}a^{9}+\frac{27\!\cdots\!31}{41\!\cdots\!89}a^{8}+\frac{61\!\cdots\!58}{41\!\cdots\!89}a^{7}-\frac{34\!\cdots\!01}{22\!\cdots\!31}a^{6}-\frac{39\!\cdots\!47}{41\!\cdots\!89}a^{5}+\frac{51\!\cdots\!04}{41\!\cdots\!89}a^{4}+\frac{23\!\cdots\!82}{41\!\cdots\!89}a^{3}-\frac{10\!\cdots\!86}{41\!\cdots\!89}a^{2}+\frac{90\!\cdots\!40}{41\!\cdots\!89}a-\frac{11\!\cdots\!31}{17\!\cdots\!29}$, $\frac{10\!\cdots\!89}{41\!\cdots\!89}a^{23}-\frac{53\!\cdots\!56}{41\!\cdots\!89}a^{22}+\frac{19\!\cdots\!68}{41\!\cdots\!89}a^{21}-\frac{45\!\cdots\!59}{41\!\cdots\!89}a^{20}+\frac{71\!\cdots\!52}{41\!\cdots\!89}a^{19}-\frac{11\!\cdots\!57}{41\!\cdots\!89}a^{18}+\frac{89\!\cdots\!46}{22\!\cdots\!31}a^{17}-\frac{48\!\cdots\!38}{41\!\cdots\!89}a^{16}-\frac{25\!\cdots\!01}{41\!\cdots\!89}a^{15}+\frac{28\!\cdots\!31}{41\!\cdots\!89}a^{14}+\frac{34\!\cdots\!48}{41\!\cdots\!89}a^{13}-\frac{97\!\cdots\!93}{41\!\cdots\!89}a^{12}+\frac{67\!\cdots\!91}{41\!\cdots\!89}a^{11}+\frac{34\!\cdots\!10}{14\!\cdots\!41}a^{10}-\frac{73\!\cdots\!55}{41\!\cdots\!89}a^{9}+\frac{36\!\cdots\!35}{41\!\cdots\!89}a^{8}+\frac{74\!\cdots\!65}{41\!\cdots\!89}a^{7}-\frac{43\!\cdots\!25}{22\!\cdots\!31}a^{6}-\frac{28\!\cdots\!08}{41\!\cdots\!89}a^{5}+\frac{65\!\cdots\!61}{41\!\cdots\!89}a^{4}+\frac{25\!\cdots\!67}{41\!\cdots\!89}a^{3}-\frac{12\!\cdots\!12}{41\!\cdots\!89}a^{2}+\frac{11\!\cdots\!14}{41\!\cdots\!89}a-\frac{15\!\cdots\!62}{17\!\cdots\!29}$, $\frac{28\!\cdots\!32}{41\!\cdots\!89}a^{23}-\frac{15\!\cdots\!83}{41\!\cdots\!89}a^{22}+\frac{29\!\cdots\!61}{22\!\cdots\!31}a^{21}-\frac{13\!\cdots\!59}{41\!\cdots\!89}a^{20}+\frac{20\!\cdots\!78}{41\!\cdots\!89}a^{19}-\frac{33\!\cdots\!26}{41\!\cdots\!89}a^{18}+\frac{48\!\cdots\!97}{41\!\cdots\!89}a^{17}-\frac{14\!\cdots\!44}{41\!\cdots\!89}a^{16}-\frac{73\!\cdots\!92}{41\!\cdots\!89}a^{15}+\frac{83\!\cdots\!91}{41\!\cdots\!89}a^{14}+\frac{97\!\cdots\!88}{41\!\cdots\!89}a^{13}-\frac{27\!\cdots\!48}{41\!\cdots\!89}a^{12}+\frac{19\!\cdots\!24}{41\!\cdots\!89}a^{11}+\frac{85\!\cdots\!43}{14\!\cdots\!41}a^{10}-\frac{21\!\cdots\!65}{41\!\cdots\!89}a^{9}+\frac{10\!\cdots\!83}{41\!\cdots\!89}a^{8}+\frac{21\!\cdots\!34}{41\!\cdots\!89}a^{7}-\frac{24\!\cdots\!60}{41\!\cdots\!89}a^{6}-\frac{39\!\cdots\!11}{41\!\cdots\!89}a^{5}+\frac{18\!\cdots\!28}{41\!\cdots\!89}a^{4}+\frac{69\!\cdots\!80}{41\!\cdots\!89}a^{3}-\frac{35\!\cdots\!15}{41\!\cdots\!89}a^{2}+\frac{33\!\cdots\!08}{41\!\cdots\!89}a-\frac{48\!\cdots\!57}{17\!\cdots\!29}$, $\frac{11\!\cdots\!95}{41\!\cdots\!89}a^{23}-\frac{25\!\cdots\!68}{22\!\cdots\!31}a^{22}+\frac{36\!\cdots\!15}{93\!\cdots\!61}a^{21}-\frac{32\!\cdots\!89}{41\!\cdots\!89}a^{20}+\frac{41\!\cdots\!87}{41\!\cdots\!89}a^{19}-\frac{80\!\cdots\!02}{41\!\cdots\!89}a^{18}+\frac{10\!\cdots\!25}{41\!\cdots\!89}a^{17}+\frac{58\!\cdots\!82}{41\!\cdots\!89}a^{16}-\frac{21\!\cdots\!50}{41\!\cdots\!89}a^{15}+\frac{36\!\cdots\!69}{41\!\cdots\!89}a^{14}+\frac{48\!\cdots\!91}{41\!\cdots\!89}a^{13}-\frac{55\!\cdots\!65}{41\!\cdots\!89}a^{12}-\frac{60\!\cdots\!74}{41\!\cdots\!89}a^{11}+\frac{75\!\cdots\!43}{14\!\cdots\!41}a^{10}-\frac{46\!\cdots\!54}{41\!\cdots\!89}a^{9}-\frac{28\!\cdots\!05}{41\!\cdots\!89}a^{8}+\frac{83\!\cdots\!72}{41\!\cdots\!89}a^{7}+\frac{27\!\cdots\!01}{41\!\cdots\!89}a^{6}-\frac{80\!\cdots\!52}{41\!\cdots\!89}a^{5}+\frac{31\!\cdots\!90}{41\!\cdots\!89}a^{4}+\frac{45\!\cdots\!40}{41\!\cdots\!89}a^{3}-\frac{79\!\cdots\!11}{41\!\cdots\!89}a^{2}+\frac{37\!\cdots\!86}{41\!\cdots\!89}a-\frac{88\!\cdots\!91}{17\!\cdots\!29}$, $\frac{52\!\cdots\!87}{41\!\cdots\!89}a^{23}-\frac{46\!\cdots\!60}{68\!\cdots\!49}a^{22}+\frac{10\!\cdots\!15}{41\!\cdots\!89}a^{21}-\frac{23\!\cdots\!01}{41\!\cdots\!89}a^{20}+\frac{37\!\cdots\!72}{41\!\cdots\!89}a^{19}-\frac{60\!\cdots\!20}{41\!\cdots\!89}a^{18}+\frac{88\!\cdots\!17}{41\!\cdots\!89}a^{17}-\frac{25\!\cdots\!54}{41\!\cdots\!89}a^{16}-\frac{13\!\cdots\!59}{41\!\cdots\!89}a^{15}+\frac{15\!\cdots\!20}{41\!\cdots\!89}a^{14}+\frac{18\!\cdots\!58}{41\!\cdots\!89}a^{13}-\frac{51\!\cdots\!88}{41\!\cdots\!89}a^{12}+\frac{35\!\cdots\!51}{41\!\cdots\!89}a^{11}+\frac{19\!\cdots\!32}{14\!\cdots\!41}a^{10}-\frac{38\!\cdots\!98}{41\!\cdots\!89}a^{9}+\frac{18\!\cdots\!11}{41\!\cdots\!89}a^{8}+\frac{39\!\cdots\!39}{41\!\cdots\!89}a^{7}-\frac{43\!\cdots\!31}{41\!\cdots\!89}a^{6}-\frac{18\!\cdots\!97}{41\!\cdots\!89}a^{5}+\frac{18\!\cdots\!76}{22\!\cdots\!31}a^{4}+\frac{14\!\cdots\!39}{41\!\cdots\!89}a^{3}-\frac{66\!\cdots\!92}{41\!\cdots\!89}a^{2}+\frac{52\!\cdots\!84}{36\!\cdots\!71}a-\frac{80\!\cdots\!07}{17\!\cdots\!29}$, $\frac{34\!\cdots\!74}{80\!\cdots\!79}a^{23}-\frac{91\!\cdots\!73}{42\!\cdots\!41}a^{22}+\frac{61\!\cdots\!32}{80\!\cdots\!79}a^{21}-\frac{13\!\cdots\!83}{80\!\cdots\!79}a^{20}+\frac{18\!\cdots\!30}{80\!\cdots\!79}a^{19}-\frac{29\!\cdots\!07}{80\!\cdots\!79}a^{18}+\frac{43\!\cdots\!26}{80\!\cdots\!79}a^{17}+\frac{50\!\cdots\!65}{80\!\cdots\!79}a^{16}-\frac{98\!\cdots\!64}{80\!\cdots\!79}a^{15}+\frac{60\!\cdots\!39}{80\!\cdots\!79}a^{14}+\frac{17\!\cdots\!16}{80\!\cdots\!79}a^{13}-\frac{15\!\cdots\!26}{42\!\cdots\!41}a^{12}+\frac{83\!\cdots\!98}{80\!\cdots\!79}a^{11}+\frac{14\!\cdots\!55}{80\!\cdots\!79}a^{10}-\frac{20\!\cdots\!60}{80\!\cdots\!79}a^{9}+\frac{76\!\cdots\!21}{80\!\cdots\!79}a^{8}+\frac{29\!\cdots\!46}{80\!\cdots\!79}a^{7}-\frac{10\!\cdots\!26}{42\!\cdots\!41}a^{6}-\frac{19\!\cdots\!00}{80\!\cdots\!79}a^{5}+\frac{16\!\cdots\!45}{80\!\cdots\!79}a^{4}+\frac{24\!\cdots\!36}{80\!\cdots\!79}a^{3}-\frac{38\!\cdots\!33}{80\!\cdots\!79}a^{2}+\frac{19\!\cdots\!14}{80\!\cdots\!79}a+\frac{12\!\cdots\!65}{33\!\cdots\!19}$, $\frac{13\!\cdots\!18}{23\!\cdots\!81}a^{23}-\frac{43\!\cdots\!71}{14\!\cdots\!41}a^{22}+\frac{15\!\cdots\!93}{14\!\cdots\!41}a^{21}-\frac{36\!\cdots\!95}{14\!\cdots\!41}a^{20}+\frac{57\!\cdots\!01}{14\!\cdots\!41}a^{19}-\frac{93\!\cdots\!58}{14\!\cdots\!41}a^{18}+\frac{71\!\cdots\!26}{76\!\cdots\!39}a^{17}-\frac{37\!\cdots\!15}{14\!\cdots\!41}a^{16}-\frac{21\!\cdots\!64}{14\!\cdots\!41}a^{15}+\frac{23\!\cdots\!45}{14\!\cdots\!41}a^{14}+\frac{28\!\cdots\!30}{14\!\cdots\!41}a^{13}-\frac{78\!\cdots\!28}{14\!\cdots\!41}a^{12}+\frac{53\!\cdots\!51}{14\!\cdots\!41}a^{11}+\frac{88\!\cdots\!75}{14\!\cdots\!41}a^{10}-\frac{59\!\cdots\!08}{14\!\cdots\!41}a^{9}+\frac{28\!\cdots\!46}{14\!\cdots\!41}a^{8}+\frac{61\!\cdots\!63}{14\!\cdots\!41}a^{7}-\frac{66\!\cdots\!71}{14\!\cdots\!41}a^{6}-\frac{50\!\cdots\!43}{23\!\cdots\!81}a^{5}+\frac{52\!\cdots\!25}{14\!\cdots\!41}a^{4}+\frac{21\!\cdots\!79}{14\!\cdots\!41}a^{3}-\frac{10\!\cdots\!50}{14\!\cdots\!41}a^{2}+\frac{92\!\cdots\!82}{14\!\cdots\!41}a-\frac{12\!\cdots\!82}{60\!\cdots\!01}$, $\frac{21\!\cdots\!11}{41\!\cdots\!89}a^{23}-\frac{11\!\cdots\!92}{41\!\cdots\!89}a^{22}+\frac{41\!\cdots\!07}{41\!\cdots\!89}a^{21}-\frac{50\!\cdots\!11}{22\!\cdots\!31}a^{20}+\frac{15\!\cdots\!10}{41\!\cdots\!89}a^{19}-\frac{24\!\cdots\!70}{41\!\cdots\!89}a^{18}+\frac{35\!\cdots\!86}{41\!\cdots\!89}a^{17}-\frac{10\!\cdots\!46}{41\!\cdots\!89}a^{16}-\frac{54\!\cdots\!15}{41\!\cdots\!89}a^{15}+\frac{60\!\cdots\!03}{41\!\cdots\!89}a^{14}+\frac{73\!\cdots\!24}{41\!\cdots\!89}a^{13}-\frac{20\!\cdots\!35}{41\!\cdots\!89}a^{12}+\frac{14\!\cdots\!87}{41\!\cdots\!89}a^{11}+\frac{73\!\cdots\!76}{14\!\cdots\!41}a^{10}-\frac{15\!\cdots\!34}{41\!\cdots\!89}a^{9}+\frac{76\!\cdots\!73}{41\!\cdots\!89}a^{8}+\frac{83\!\cdots\!06}{22\!\cdots\!31}a^{7}-\frac{17\!\cdots\!76}{41\!\cdots\!89}a^{6}-\frac{53\!\cdots\!40}{41\!\cdots\!89}a^{5}+\frac{13\!\cdots\!74}{41\!\cdots\!89}a^{4}+\frac{54\!\cdots\!55}{41\!\cdots\!89}a^{3}-\frac{26\!\cdots\!07}{41\!\cdots\!89}a^{2}+\frac{24\!\cdots\!75}{41\!\cdots\!89}a-\frac{33\!\cdots\!75}{17\!\cdots\!29}$, $\frac{22\!\cdots\!17}{41\!\cdots\!89}a^{23}-\frac{12\!\cdots\!09}{41\!\cdots\!89}a^{22}+\frac{44\!\cdots\!52}{41\!\cdots\!89}a^{21}-\frac{10\!\cdots\!65}{41\!\cdots\!89}a^{20}+\frac{16\!\cdots\!58}{41\!\cdots\!89}a^{19}-\frac{26\!\cdots\!83}{41\!\cdots\!89}a^{18}+\frac{38\!\cdots\!39}{41\!\cdots\!89}a^{17}-\frac{10\!\cdots\!82}{41\!\cdots\!89}a^{16}-\frac{59\!\cdots\!55}{41\!\cdots\!89}a^{15}+\frac{65\!\cdots\!05}{41\!\cdots\!89}a^{14}+\frac{13\!\cdots\!35}{68\!\cdots\!49}a^{13}-\frac{22\!\cdots\!28}{41\!\cdots\!89}a^{12}+\frac{15\!\cdots\!51}{41\!\cdots\!89}a^{11}+\frac{84\!\cdots\!11}{14\!\cdots\!41}a^{10}-\frac{16\!\cdots\!98}{41\!\cdots\!89}a^{9}+\frac{81\!\cdots\!65}{41\!\cdots\!89}a^{8}+\frac{16\!\cdots\!94}{41\!\cdots\!89}a^{7}-\frac{18\!\cdots\!26}{41\!\cdots\!89}a^{6}-\frac{92\!\cdots\!38}{41\!\cdots\!89}a^{5}+\frac{14\!\cdots\!98}{41\!\cdots\!89}a^{4}+\frac{60\!\cdots\!92}{41\!\cdots\!89}a^{3}-\frac{28\!\cdots\!20}{41\!\cdots\!89}a^{2}+\frac{26\!\cdots\!25}{41\!\cdots\!89}a-\frac{35\!\cdots\!24}{17\!\cdots\!29}$, $\frac{75\!\cdots\!31}{41\!\cdots\!89}a^{23}-\frac{40\!\cdots\!81}{41\!\cdots\!89}a^{22}+\frac{14\!\cdots\!56}{41\!\cdots\!89}a^{21}-\frac{33\!\cdots\!53}{41\!\cdots\!89}a^{20}+\frac{52\!\cdots\!38}{41\!\cdots\!89}a^{19}-\frac{85\!\cdots\!13}{41\!\cdots\!89}a^{18}+\frac{12\!\cdots\!38}{41\!\cdots\!89}a^{17}-\frac{32\!\cdots\!31}{41\!\cdots\!89}a^{16}-\frac{19\!\cdots\!09}{41\!\cdots\!89}a^{15}+\frac{20\!\cdots\!17}{41\!\cdots\!89}a^{14}+\frac{26\!\cdots\!27}{41\!\cdots\!89}a^{13}-\frac{71\!\cdots\!14}{41\!\cdots\!89}a^{12}+\frac{48\!\cdots\!70}{41\!\cdots\!89}a^{11}+\frac{26\!\cdots\!66}{14\!\cdots\!41}a^{10}-\frac{54\!\cdots\!74}{41\!\cdots\!89}a^{9}+\frac{25\!\cdots\!11}{41\!\cdots\!89}a^{8}+\frac{55\!\cdots\!54}{41\!\cdots\!89}a^{7}-\frac{59\!\cdots\!26}{41\!\cdots\!89}a^{6}-\frac{17\!\cdots\!75}{41\!\cdots\!89}a^{5}+\frac{48\!\cdots\!40}{41\!\cdots\!89}a^{4}+\frac{20\!\cdots\!61}{41\!\cdots\!89}a^{3}-\frac{92\!\cdots\!54}{41\!\cdots\!89}a^{2}+\frac{84\!\cdots\!62}{41\!\cdots\!89}a-\frac{11\!\cdots\!36}{17\!\cdots\!29}$, $\frac{25\!\cdots\!90}{41\!\cdots\!89}a^{23}-\frac{11\!\cdots\!75}{41\!\cdots\!89}a^{22}+\frac{39\!\cdots\!46}{41\!\cdots\!89}a^{21}-\frac{79\!\cdots\!47}{41\!\cdots\!89}a^{20}+\frac{97\!\cdots\!23}{41\!\cdots\!89}a^{19}-\frac{16\!\cdots\!58}{41\!\cdots\!89}a^{18}+\frac{23\!\cdots\!20}{41\!\cdots\!89}a^{17}+\frac{17\!\cdots\!26}{41\!\cdots\!89}a^{16}-\frac{71\!\cdots\!94}{41\!\cdots\!89}a^{15}+\frac{18\!\cdots\!68}{41\!\cdots\!89}a^{14}+\frac{13\!\cdots\!15}{41\!\cdots\!89}a^{13}-\frac{16\!\cdots\!15}{41\!\cdots\!89}a^{12}-\frac{12\!\cdots\!60}{41\!\cdots\!89}a^{11}+\frac{39\!\cdots\!22}{14\!\cdots\!41}a^{10}-\frac{13\!\cdots\!74}{41\!\cdots\!89}a^{9}-\frac{20\!\cdots\!70}{41\!\cdots\!89}a^{8}+\frac{23\!\cdots\!17}{41\!\cdots\!89}a^{7}-\frac{67\!\cdots\!33}{41\!\cdots\!89}a^{6}-\frac{11\!\cdots\!99}{41\!\cdots\!89}a^{5}+\frac{11\!\cdots\!48}{41\!\cdots\!89}a^{4}+\frac{19\!\cdots\!74}{41\!\cdots\!89}a^{3}-\frac{20\!\cdots\!35}{41\!\cdots\!89}a^{2}+\frac{28\!\cdots\!72}{41\!\cdots\!89}a+\frac{25\!\cdots\!18}{17\!\cdots\!29}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 535957.6069656424 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 535957.6069656424 \cdot 2}{4\cdot\sqrt{18075490334784000000000000000000}}\cr\approx \mathstrut & 0.238623834579320 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 23*x^22 - 58*x^21 + 100*x^20 - 160*x^19 + 242*x^18 - 156*x^17 - 228*x^16 + 452*x^15 + 164*x^14 - 1190*x^13 + 1288*x^12 - 326*x^11 - 800*x^10 + 832*x^9 + 514*x^8 - 1302*x^7 + 500*x^6 + 664*x^5 - 155*x^4 - 1418*x^3 + 1951*x^2 - 1112*x + 241)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^24 - 6*x^23 + 23*x^22 - 58*x^21 + 100*x^20 - 160*x^19 + 242*x^18 - 156*x^17 - 228*x^16 + 452*x^15 + 164*x^14 - 1190*x^13 + 1288*x^12 - 326*x^11 - 800*x^10 + 832*x^9 + 514*x^8 - 1302*x^7 + 500*x^6 + 664*x^5 - 155*x^4 - 1418*x^3 + 1951*x^2 - 1112*x + 241, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^24 - 6*x^23 + 23*x^22 - 58*x^21 + 100*x^20 - 160*x^19 + 242*x^18 - 156*x^17 - 228*x^16 + 452*x^15 + 164*x^14 - 1190*x^13 + 1288*x^12 - 326*x^11 - 800*x^10 + 832*x^9 + 514*x^8 - 1302*x^7 + 500*x^6 + 664*x^5 - 155*x^4 - 1418*x^3 + 1951*x^2 - 1112*x + 241);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 6*x^23 + 23*x^22 - 58*x^21 + 100*x^20 - 160*x^19 + 242*x^18 - 156*x^17 - 228*x^16 + 452*x^15 + 164*x^14 - 1190*x^13 + 1288*x^12 - 326*x^11 - 800*x^10 + 832*x^9 + 514*x^8 - 1302*x^7 + 500*x^6 + 664*x^5 - 155*x^4 - 1418*x^3 + 1951*x^2 - 1112*x + 241);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$S_3\times C_{12}$ (as 24T65):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 72 |
| The 36 conjugacy class representatives for $S_3\times C_{12}$ |
| Character table for $S_3\times C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{15})^+\), 4.0.18000.1, \(\Q(i, \sqrt{5})\), 6.0.648000.1, 8.0.324000000.1, 12.0.419904000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.12.0.1}{12} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $24$ | $2$ | $12$ | $24$ | |||
|
\(3\)
| 3.12.18.61 | $x^{12} + 12 x^{11} + 42 x^{10} + 42 x^{9} + 54 x^{8} + 18 x^{7} + 21 x^{6} + 72 x^{5} + 108 x^{4} + 36 x^{3} + 180$ | $6$ | $2$ | $18$ | $C_{12}$ | $[2]_{2}^{2}$ |
| 3.12.6.1 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ | |
|
\(5\)
| 5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| 5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.45.6t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.180.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.52488000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.180.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.52488000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.36.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.36.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.60.4t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 5 $ | 4.0.18000.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.60.4t1.a.b | $1$ | $ 2^{2} \cdot 3 \cdot 5 $ | 4.0.18000.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| 1.180.12t1.a.a | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.3099363912000000000.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.b.a | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.b.b | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.180.12t1.a.b | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.3099363912000000000.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.180.12t1.a.c | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.3099363912000000000.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.b.c | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.180.12t1.a.d | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.3099363912000000000.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.b.d | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 2.1620.3t2.b.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5 $ | 3.1.1620.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.1620.6t3.g.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5 $ | 6.2.13122000.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.180.12t18.a.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.419904000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.180.12t18.a.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.419904000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.180.6t5.b.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.648000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.180.6t5.b.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.648000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| 2.8100.12t11.a.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5^{2}$ | 12.4.193710244500000000.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.8100.12t11.a.b | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5^{2}$ | 12.4.193710244500000000.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.2700.24t65.a.a | $2$ | $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ | 24.0.18075490334784000000000000000000.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2700.24t65.a.b | $2$ | $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ | 24.0.18075490334784000000000000000000.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2700.24t65.a.c | $2$ | $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ | 24.0.18075490334784000000000000000000.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2700.24t65.a.d | $2$ | $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ | 24.0.18075490334784000000000000000000.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.