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Properties

Label 36.0.519...889.1
Degree $36$
Signature $[0, 18]$
Discriminant $5.192\times 10^{59}$
Root discriminant \(45.58\)
Ramified primes $3,23,199$
Class number not computed
Class group not computed
Galois group $C_2\times A_4\times D_6$ (as 36T334)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 6*x^35 + 54*x^34 - 249*x^33 + 1281*x^32 - 4767*x^31 + 17772*x^30 - 55056*x^29 + 164652*x^28 - 444090*x^27 + 1149975*x^26 - 2811774*x^25 + 6542481*x^24 - 14483877*x^23 + 30345297*x^22 - 60803709*x^21 + 116322072*x^20 - 213103239*x^19 + 373433528*x^18 - 621567384*x^17 + 980981811*x^16 - 1465192428*x^15 + 2079415686*x^14 - 2795268228*x^13 + 3496953564*x^12 - 4091067075*x^11 + 4483655439*x^10 - 4385043570*x^9 + 3744819675*x^8 - 2928547434*x^7 + 2036109996*x^6 - 1081958046*x^5 + 405353103*x^4 - 123233361*x^3 + 32860329*x^2 - 5284761*x + 749197)
 
gp: K = bnfinit(y^36 - 6*y^35 + 54*y^34 - 249*y^33 + 1281*y^32 - 4767*y^31 + 17772*y^30 - 55056*y^29 + 164652*y^28 - 444090*y^27 + 1149975*y^26 - 2811774*y^25 + 6542481*y^24 - 14483877*y^23 + 30345297*y^22 - 60803709*y^21 + 116322072*y^20 - 213103239*y^19 + 373433528*y^18 - 621567384*y^17 + 980981811*y^16 - 1465192428*y^15 + 2079415686*y^14 - 2795268228*y^13 + 3496953564*y^12 - 4091067075*y^11 + 4483655439*y^10 - 4385043570*y^9 + 3744819675*y^8 - 2928547434*y^7 + 2036109996*y^6 - 1081958046*y^5 + 405353103*y^4 - 123233361*y^3 + 32860329*y^2 - 5284761*y + 749197, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 6*x^35 + 54*x^34 - 249*x^33 + 1281*x^32 - 4767*x^31 + 17772*x^30 - 55056*x^29 + 164652*x^28 - 444090*x^27 + 1149975*x^26 - 2811774*x^25 + 6542481*x^24 - 14483877*x^23 + 30345297*x^22 - 60803709*x^21 + 116322072*x^20 - 213103239*x^19 + 373433528*x^18 - 621567384*x^17 + 980981811*x^16 - 1465192428*x^15 + 2079415686*x^14 - 2795268228*x^13 + 3496953564*x^12 - 4091067075*x^11 + 4483655439*x^10 - 4385043570*x^9 + 3744819675*x^8 - 2928547434*x^7 + 2036109996*x^6 - 1081958046*x^5 + 405353103*x^4 - 123233361*x^3 + 32860329*x^2 - 5284761*x + 749197);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 6*x^35 + 54*x^34 - 249*x^33 + 1281*x^32 - 4767*x^31 + 17772*x^30 - 55056*x^29 + 164652*x^28 - 444090*x^27 + 1149975*x^26 - 2811774*x^25 + 6542481*x^24 - 14483877*x^23 + 30345297*x^22 - 60803709*x^21 + 116322072*x^20 - 213103239*x^19 + 373433528*x^18 - 621567384*x^17 + 980981811*x^16 - 1465192428*x^15 + 2079415686*x^14 - 2795268228*x^13 + 3496953564*x^12 - 4091067075*x^11 + 4483655439*x^10 - 4385043570*x^9 + 3744819675*x^8 - 2928547434*x^7 + 2036109996*x^6 - 1081958046*x^5 + 405353103*x^4 - 123233361*x^3 + 32860329*x^2 - 5284761*x + 749197)
 

\( x^{36} - 6 x^{35} + 54 x^{34} - 249 x^{33} + 1281 x^{32} - 4767 x^{31} + 17772 x^{30} - 55056 x^{29} + \cdots + 749197 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(519242561019577792703071300501749617888622898544642417870889\) \(\medspace = 3^{62}\cdot 23^{12}\cdot 199^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(45.58\)
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:  $3^{31/18}23^{1/2}199^{1/2}\approx 448.74464988853384$
Ramified primes:   \(3\), \(23\), \(199\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{131072}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{76}a^{30}-\frac{7}{38}a^{29}+\frac{3}{76}a^{28}+\frac{13}{76}a^{27}-\frac{5}{38}a^{26}+\frac{1}{38}a^{25}+\frac{3}{76}a^{24}-\frac{5}{38}a^{23}-\frac{4}{19}a^{22}-\frac{11}{76}a^{21}+\frac{25}{76}a^{20}+\frac{25}{76}a^{19}-\frac{37}{76}a^{18}+\frac{4}{19}a^{17}-\frac{3}{19}a^{16}-\frac{13}{38}a^{15}-\frac{31}{76}a^{14}+\frac{7}{19}a^{13}-\frac{13}{38}a^{12}-\frac{7}{19}a^{11}+\frac{37}{76}a^{10}+\frac{5}{38}a^{9}+\frac{7}{38}a^{8}+\frac{9}{38}a^{7}-\frac{9}{19}a^{6}-\frac{37}{76}a^{5}-\frac{6}{19}a^{4}+\frac{23}{76}a^{3}-\frac{37}{76}a^{2}-\frac{9}{76}a+\frac{13}{76}$, $\frac{1}{76}a^{31}-\frac{3}{76}a^{29}+\frac{17}{76}a^{28}-\frac{9}{38}a^{27}+\frac{7}{38}a^{26}-\frac{7}{76}a^{25}-\frac{3}{38}a^{24}-\frac{1}{19}a^{23}-\frac{7}{76}a^{22}-\frac{15}{76}a^{21}-\frac{5}{76}a^{20}-\frac{29}{76}a^{19}-\frac{2}{19}a^{18}-\frac{4}{19}a^{17}+\frac{17}{38}a^{16}-\frac{15}{76}a^{15}+\frac{3}{19}a^{14}+\frac{6}{19}a^{13}+\frac{13}{38}a^{12}-\frac{13}{76}a^{11}+\frac{17}{38}a^{10}+\frac{1}{38}a^{9}+\frac{6}{19}a^{8}+\frac{13}{38}a^{7}+\frac{29}{76}a^{6}-\frac{5}{38}a^{5}+\frac{29}{76}a^{4}-\frac{1}{4}a^{3}-\frac{33}{76}a^{2}+\frac{1}{76}a+\frac{15}{38}$, $\frac{1}{76}a^{32}+\frac{13}{76}a^{29}-\frac{9}{76}a^{28}+\frac{15}{76}a^{27}+\frac{1}{76}a^{26}+\frac{5}{76}a^{24}+\frac{1}{76}a^{23}+\frac{13}{76}a^{22}-\frac{15}{38}a^{20}+\frac{29}{76}a^{19}+\frac{25}{76}a^{18}-\frac{8}{19}a^{17}-\frac{13}{76}a^{16}+\frac{5}{38}a^{15}-\frac{31}{76}a^{14}+\frac{17}{38}a^{13}-\frac{15}{76}a^{12}-\frac{3}{19}a^{11}-\frac{1}{76}a^{10}+\frac{4}{19}a^{9}+\frac{15}{38}a^{8}+\frac{7}{76}a^{7}-\frac{1}{19}a^{6}+\frac{8}{19}a^{5}+\frac{23}{76}a^{4}+\frac{9}{19}a^{3}+\frac{1}{19}a^{2}-\frac{35}{76}a+\frac{1}{76}$, $\frac{1}{2888}a^{33}-\frac{1}{2888}a^{32}+\frac{3}{2888}a^{31}-\frac{13}{2888}a^{30}-\frac{85}{2888}a^{29}-\frac{269}{2888}a^{28}+\frac{253}{1444}a^{27}+\frac{453}{2888}a^{26}+\frac{213}{1444}a^{25}+\frac{51}{722}a^{24}+\frac{111}{1444}a^{23}-\frac{37}{1444}a^{22}-\frac{701}{2888}a^{21}+\frac{295}{722}a^{20}+\frac{31}{152}a^{19}+\frac{1337}{2888}a^{18}-\frac{369}{2888}a^{17}-\frac{1}{152}a^{16}+\frac{637}{1444}a^{15}+\frac{337}{2888}a^{14}+\frac{1195}{2888}a^{13}+\frac{567}{2888}a^{12}+\frac{711}{1444}a^{11}+\frac{1209}{2888}a^{10}-\frac{443}{1444}a^{9}-\frac{1113}{2888}a^{8}+\frac{1081}{2888}a^{7}-\frac{309}{2888}a^{6}+\frac{657}{2888}a^{5}+\frac{176}{361}a^{4}-\frac{421}{2888}a^{3}+\frac{187}{722}a^{2}-\frac{791}{2888}a-\frac{743}{2888}$, $\frac{1}{470744}a^{34}+\frac{17}{470744}a^{33}-\frac{2485}{470744}a^{32}+\frac{1181}{470744}a^{31}-\frac{2523}{470744}a^{30}-\frac{113329}{470744}a^{29}-\frac{9217}{235372}a^{28}+\frac{93731}{470744}a^{27}+\frac{22055}{235372}a^{26}+\frac{10423}{117686}a^{25}+\frac{20515}{117686}a^{24}-\frac{14059}{58843}a^{23}+\frac{53}{1304}a^{22}-\frac{2241}{12388}a^{21}+\frac{175197}{470744}a^{20}+\frac{226145}{470744}a^{19}-\frac{14721}{470744}a^{18}+\frac{41979}{470744}a^{17}-\frac{97441}{235372}a^{16}+\frac{82093}{470744}a^{15}-\frac{122965}{470744}a^{14}-\frac{204555}{470744}a^{13}-\frac{5931}{12388}a^{12}+\frac{158209}{470744}a^{11}+\frac{14105}{235372}a^{10}-\frac{48905}{470744}a^{9}+\frac{174923}{470744}a^{8}-\frac{208889}{470744}a^{7}+\frac{23519}{470744}a^{6}+\frac{88089}{235372}a^{5}-\frac{184191}{470744}a^{4}+\frac{84175}{235372}a^{3}-\frac{11601}{24776}a^{2}+\frac{8693}{470744}a+\frac{11709}{58843}$, $\frac{1}{18\!\cdots\!72}a^{35}-\frac{11\!\cdots\!99}{18\!\cdots\!72}a^{34}-\frac{91\!\cdots\!39}{45\!\cdots\!18}a^{33}+\frac{66\!\cdots\!72}{22\!\cdots\!59}a^{32}-\frac{27\!\cdots\!21}{45\!\cdots\!18}a^{31}-\frac{91\!\cdots\!39}{22\!\cdots\!59}a^{30}+\frac{31\!\cdots\!17}{18\!\cdots\!72}a^{29}+\frac{20\!\cdots\!17}{22\!\cdots\!59}a^{28}-\frac{11\!\cdots\!51}{90\!\cdots\!36}a^{27}-\frac{37\!\cdots\!79}{18\!\cdots\!72}a^{26}+\frac{57\!\cdots\!57}{90\!\cdots\!36}a^{25}-\frac{21\!\cdots\!55}{90\!\cdots\!36}a^{24}+\frac{30\!\cdots\!03}{18\!\cdots\!72}a^{23}+\frac{14\!\cdots\!46}{11\!\cdots\!61}a^{22}+\frac{15\!\cdots\!77}{90\!\cdots\!36}a^{21}+\frac{45\!\cdots\!87}{18\!\cdots\!72}a^{20}+\frac{17\!\cdots\!09}{90\!\cdots\!36}a^{19}-\frac{12\!\cdots\!79}{47\!\cdots\!44}a^{18}+\frac{49\!\cdots\!37}{18\!\cdots\!72}a^{17}-\frac{43\!\cdots\!83}{90\!\cdots\!36}a^{16}+\frac{62\!\cdots\!93}{18\!\cdots\!72}a^{15}-\frac{98\!\cdots\!11}{45\!\cdots\!18}a^{14}-\frac{38\!\cdots\!63}{18\!\cdots\!72}a^{13}-\frac{14\!\cdots\!48}{22\!\cdots\!59}a^{12}+\frac{18\!\cdots\!47}{45\!\cdots\!18}a^{11}-\frac{43\!\cdots\!01}{90\!\cdots\!36}a^{10}+\frac{46\!\cdots\!49}{18\!\cdots\!72}a^{9}-\frac{10\!\cdots\!41}{90\!\cdots\!36}a^{8}-\frac{54\!\cdots\!73}{45\!\cdots\!18}a^{7}-\frac{76\!\cdots\!95}{18\!\cdots\!72}a^{6}+\frac{94\!\cdots\!47}{45\!\cdots\!18}a^{5}+\frac{42\!\cdots\!43}{90\!\cdots\!36}a^{4}+\frac{14\!\cdots\!61}{90\!\cdots\!36}a^{3}+\frac{70\!\cdots\!31}{18\!\cdots\!72}a^{2}+\frac{38\!\cdots\!35}{18\!\cdots\!72}a+\frac{87\!\cdots\!59}{18\!\cdots\!72}$ Copy content Toggle raw display

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

not computed

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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{1886304961813595305873254712731358986969189297766771702058227114784577081481994104647003790904002824479919264161911831595684508307797}{376783333113864169329141570357576299677928964040735007593556254641806612521321449612119032610215787303178504765074408058882327929539746389} a^{35} - \frac{77967746865208256486748312046111185597398294416030011440739789407226827538943232526629903560881005924056291745734550459101826251198485}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{34} + \frac{749701104507639421727270051103361913357431590614098444144586179745982785699398699415172932981414678089955724947885197517330827370854329}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{33} - \frac{3132231900390581868213415442355690861461271697158845648143346447495955324881153883645508134243704130702645078488598849972529665537961767}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{32} + \frac{16712708118907385102813401369581645552381363534995035521504129393908108090980127809590281701886126172791008361822786648546140153982057913}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{31} - \frac{57990757985036010328322494819710506712067032037318917485162695458884287223667686663218658341878076927563054841797952559653393716920152645}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{30} + \frac{219726637566669501675623623538468174729048555504895174506298800902503343011870283908338705720873687124736187795911674163086360198931157709}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{29} - \frac{80931341950926420396633833931511833439160582232230500704682241887790685303256971475533297439503687201928982932476812596008224083811569293}{376783333113864169329141570357576299677928964040735007593556254641806612521321449612119032610215787303178504765074408058882327929539746389} a^{28} + \frac{1943841477575190217457337591378097541915840818583141774724992945851101195982428545319805576333176487959923615175016444442947376891889106405}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{27} - \frac{2539768765142268421694200445610888259154770495541717919354080243955841219486613189130966065780978306166274518945422208374968749608770739145}{1507133332455456677316566281430305198711715856162940030374225018567226450085285798448476130440863149212714019060297632235529311718158985556} a^{26} + \frac{1639126942630455045118328968727222628548464976826846051865201093871183157703218822137494661846388831339997549619427716113907308142649768620}{376783333113864169329141570357576299677928964040735007593556254641806612521321449612119032610215787303178504765074408058882327929539746389} a^{25} - \frac{7872984136010526856607226396830479162167541959485749342830010193505464813121662183243026594313410316385361274860980656004629088844307015891}{753566666227728338658283140715152599355857928081470015187112509283613225042642899224238065220431574606357009530148816117764655859079492778} a^{24} + \frac{36226500703352193415429637843724833823756857366734171697184763440496667654842171727797259301335883893663368486207901572935651353986150860791}{1507133332455456677316566281430305198711715856162940030374225018567226450085285798448476130440863149212714019060297632235529311718158985556} a^{23} - \frac{8323288000459375312195243515486395340286976067605081068658973876496038946912237031232002208654687312622933380210722433916413319118073982649}{158645613942679650243849082255821599864391142753993687407813159849181731587924820889313276888511910443443580953715540235318874917700945848} a^{22} + \frac{163024659516578825983397416930116913403937362305078541429268928047249415172308321638894573547245157922555655012303047533761396623580627450729}{1507133332455456677316566281430305198711715856162940030374225018567226450085285798448476130440863149212714019060297632235529311718158985556} a^{21} - \frac{645756453769965060622797832947641494402089952983641142192701286609235394640049521644677356040419361285083311018970315986514329228988176550479}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{20} + \frac{1217232537504995639857973833157564245184919224193610690509467849428798426785669297889650802807944496665035465031322788248425651851629023654061}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{19} - \frac{115882645862942076786403557388502631919073292244663415964720644600714174977196159441707957013702302366583760380678973107408167243970228586859}{158645613942679650243849082255821599864391142753993687407813159849181731587924820889313276888511910443443580953715540235318874917700945848} a^{18} + \frac{3801799022735422442032021267291467114649078518609609103766677873505087002657846635264694607629472755851905340547927365498826179190315079862039}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{17} - \frac{3107498734509703109842233271853693226068771755105531825207417931073771870370917436170787464983250787327516449823957712951768077287529582503043}{1507133332455456677316566281430305198711715856162940030374225018567226450085285798448476130440863149212714019060297632235529311718158985556} a^{16} + \frac{9632563497456901840216433028869154347978940197786829989129837108623747562055571140475527617426614864449230252905483703984008925765539511639091}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{15} - \frac{14101256397318967206177922285976485866936961953996327104155587069956310413542505634744376769100191324691363891506243828403343221062996264171617}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{14} + \frac{19662996403169869923610315684072932269864631264818064218133491041147922118705566037727003459739505487508136385266033348554750132201938609734545}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{13} - \frac{12929037278233319368448965962332882190684499748707406145313726077960158816527492123754991326581926990767230218721642587574742649494307569748041}{1507133332455456677316566281430305198711715856162940030374225018567226450085285798448476130440863149212714019060297632235529311718158985556} a^{12} + \frac{31321321218140442637448584342580735615849176098660611994650157295739461970048348103294450262013475537589094467455669732039631170796252362575063}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{11} - \frac{8947209729322393512063436044800650570666805181085821499619357213640309461123113045192413263250483069121260031139776756367636467488552788416831}{753566666227728338658283140715152599355857928081470015187112509283613225042642899224238065220431574606357009530148816117764655859079492778} a^{10} + \frac{38064322492319160965885511290701791117885460562244120167213488128627415408398986372458206516189336641276263325193731030235289641345297998938549}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{9} - \frac{34738726393052382027658018682692443216804043889175353896099652834778480832300900363471984728823503709437102645089503500045295598576954496754985}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{8} + \frac{27915742865891351522272262805780726659436028644530663540195159827241950950718771367116278216776922378085035489195965506094146432442412814904113}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{7} - \frac{21358926566361520659297329475445832695408461808084262085831208743236387297531115051283817507285381745402548308028357738074635488726768435136893}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{6} + \frac{6660991472743662117642765740722773358828116171221555412902422302378639571573087398484466048584803723915467751577564224236974005743426374309791}{1507133332455456677316566281430305198711715856162940030374225018567226450085285798448476130440863149212714019060297632235529311718158985556} a^{5} - \frac{5530607692557302604353048247679492785307136109060600376053149110294769510233059282631828247116829706818127363709285266389386468707628073712439}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{4} + \frac{884980077174407702063994737732961255800275127648813579868772298009284473850987853251276815393862574863375747090343117363057235613417129760217}{1507133332455456677316566281430305198711715856162940030374225018567226450085285798448476130440863149212714019060297632235529311718158985556} a^{3} - \frac{553021330233518391724667549542050531253245925641449409475778362016039198872274539843236470122962186163939223238956557414072206361861620966855}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a^{2} + \frac{82297219611672300176461916754710986493083245053914912555941099626949435712070787191825312223104306671992904986815067306627946490607487229649}{3014266664910913354633132562860610397423431712325880060748450037134452900170571596896952260881726298425428038120595264471058623436317971112} a - \frac{8246420678232517446700285024334748274699107553780119576825677662496759729371843037389505120375670728354031438028911816488263104017973139891}{1507133332455456677316566281430305198711715856162940030374225018567226450085285798448476130440863149212714019060297632235529311718158985556} \)  (order $18$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - 6*x^35 + 54*x^34 - 249*x^33 + 1281*x^32 - 4767*x^31 + 17772*x^30 - 55056*x^29 + 164652*x^28 - 444090*x^27 + 1149975*x^26 - 2811774*x^25 + 6542481*x^24 - 14483877*x^23 + 30345297*x^22 - 60803709*x^21 + 116322072*x^20 - 213103239*x^19 + 373433528*x^18 - 621567384*x^17 + 980981811*x^16 - 1465192428*x^15 + 2079415686*x^14 - 2795268228*x^13 + 3496953564*x^12 - 4091067075*x^11 + 4483655439*x^10 - 4385043570*x^9 + 3744819675*x^8 - 2928547434*x^7 + 2036109996*x^6 - 1081958046*x^5 + 405353103*x^4 - 123233361*x^3 + 32860329*x^2 - 5284761*x + 749197)
 
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^36 - 6*x^35 + 54*x^34 - 249*x^33 + 1281*x^32 - 4767*x^31 + 17772*x^30 - 55056*x^29 + 164652*x^28 - 444090*x^27 + 1149975*x^26 - 2811774*x^25 + 6542481*x^24 - 14483877*x^23 + 30345297*x^22 - 60803709*x^21 + 116322072*x^20 - 213103239*x^19 + 373433528*x^18 - 621567384*x^17 + 980981811*x^16 - 1465192428*x^15 + 2079415686*x^14 - 2795268228*x^13 + 3496953564*x^12 - 4091067075*x^11 + 4483655439*x^10 - 4385043570*x^9 + 3744819675*x^8 - 2928547434*x^7 + 2036109996*x^6 - 1081958046*x^5 + 405353103*x^4 - 123233361*x^3 + 32860329*x^2 - 5284761*x + 749197, 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^36 - 6*x^35 + 54*x^34 - 249*x^33 + 1281*x^32 - 4767*x^31 + 17772*x^30 - 55056*x^29 + 164652*x^28 - 444090*x^27 + 1149975*x^26 - 2811774*x^25 + 6542481*x^24 - 14483877*x^23 + 30345297*x^22 - 60803709*x^21 + 116322072*x^20 - 213103239*x^19 + 373433528*x^18 - 621567384*x^17 + 980981811*x^16 - 1465192428*x^15 + 2079415686*x^14 - 2795268228*x^13 + 3496953564*x^12 - 4091067075*x^11 + 4483655439*x^10 - 4385043570*x^9 + 3744819675*x^8 - 2928547434*x^7 + 2036109996*x^6 - 1081958046*x^5 + 405353103*x^4 - 123233361*x^3 + 32860329*x^2 - 5284761*x + 749197);
 
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^36 - 6*x^35 + 54*x^34 - 249*x^33 + 1281*x^32 - 4767*x^31 + 17772*x^30 - 55056*x^29 + 164652*x^28 - 444090*x^27 + 1149975*x^26 - 2811774*x^25 + 6542481*x^24 - 14483877*x^23 + 30345297*x^22 - 60803709*x^21 + 116322072*x^20 - 213103239*x^19 + 373433528*x^18 - 621567384*x^17 + 980981811*x^16 - 1465192428*x^15 + 2079415686*x^14 - 2795268228*x^13 + 3496953564*x^12 - 4091067075*x^11 + 4483655439*x^10 - 4385043570*x^9 + 3744819675*x^8 - 2928547434*x^7 + 2036109996*x^6 - 1081958046*x^5 + 405353103*x^4 - 123233361*x^3 + 32860329*x^2 - 5284761*x + 749197);
 
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

$C_2\times A_4\times D_6$ (as 36T334):

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 288
The 48 conjugacy class representatives for $C_2\times A_4\times D_6$
Character table for $C_2\times A_4\times D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 3.3.621.1, 6.6.3916917.1, 6.0.1305639.1, 6.0.1156923.1, \(\Q(\zeta_{9})\), 9.9.174583151469.1, 12.0.15342238784889.1, 18.0.91437830330543390573883.1, 18.0.240194958088349971071050598639.1, 18.18.720584874265049913213151795917.1

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: deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, some 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 ${\href{/padicField/2.6.0.1}{6} }^{6}$ R ${\href{/padicField/5.6.0.1}{6} }^{6}$ ${\href{/padicField/7.6.0.1}{6} }^{4}{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{12}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{14}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ R ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.3.0.1}{3} }^{12}$ ${\href{/padicField/37.2.0.1}{2} }^{14}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{4}{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $36$$18$$2$$62$
\(23\) Copy content Toggle raw display 23.6.0.1$x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
23.6.0.1$x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
23.12.6.1$x^{12} + 140 x^{10} + 18 x^{9} + 8092 x^{8} + 20 x^{7} + 244001 x^{6} - 56140 x^{5} + 4059563 x^{4} - 1721304 x^{3} + 36312468 x^{2} - 14694000 x + 141161628$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
23.12.6.1$x^{12} + 140 x^{10} + 18 x^{9} + 8092 x^{8} + 20 x^{7} + 244001 x^{6} - 56140 x^{5} + 4059563 x^{4} - 1721304 x^{3} + 36312468 x^{2} - 14694000 x + 141161628$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(199\) Copy content Toggle raw display $\Q_{199}$$x + 196$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 196$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 196$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 196$$1$$1$$0$Trivial$[\ ]$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.1.1$x^{2} + 597$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.1.1$x^{2} + 597$$2$$1$$1$$C_2$$[\ ]_{2}$
199.4.2.1$x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.4.2.1$x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$