Global number field 32.0.40558185231562030583545760465225948212199635509568684161.1

• Label: 32.0.40558185231...4161.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $3^{16}\cdot 7^{16}\cdot 17^{28}$
• Root discriminant: $54.67$
• Ramified primes: $3, 7, 17$
• Class number: $225$ (GRH)
• Class group: $[15, 15]$ (GRH)
• Galois group: $C_2^2\times C_8$ (as 32T37)

Normalized defining polynomial

\( x^{32} - x^{31} - 22 x^{30} + 9 x^{29} + 266 x^{28} - 2 x^{27} - 2597 x^{26} - 54 x^{25} + 20774 x^{24} + 1099 x^{23} - 128438 x^{22} - 27858 x^{21} + 651009 x^{20} + 153366 x^{19} - 2755230 x^{18} - 386184 x^{17} + 9170393 x^{16} + 1793642 x^{15} - 23184576 x^{14} - 6931256 x^{13} + 46258944 x^{12} + 5607104 x^{11} - 67085888 x^{10} + 6766080 x^{9} + 65201664 x^{8} + 3431936 x^{7} - 34820096 x^{6} - 14815232 x^{5} + 13553664 x^{4} + 2523136 x^{3} - 753664 x^{2} + 131072 x + 65536 \)

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(40558185231562030583545760465225948212199635509568684161=3^{16}\cdot 7^{16}\cdot 17^{28}\)
Root discriminant:		$54.67$
Ramified primes:		$3, 7, 17$
This field is Galois and abelian over $\Q$.
Conductor:		\(357=3\cdot 7\cdot 17\)
Dirichlet character group:		$\lbrace$$\chi_{357}(1,·)$, $\chi_{357}(134,·)$, $\chi_{357}(8,·)$, $\chi_{357}(13,·)$, $\chi_{357}(274,·)$, $\chi_{357}(281,·)$, $\chi_{357}(155,·)$, $\chi_{357}(293,·)$, $\chi_{357}(169,·)$, $\chi_{357}(43,·)$, $\chi_{357}(302,·)$, $\chi_{357}(50,·)$, $\chi_{357}(307,·)$, $\chi_{357}(55,·)$, $\chi_{357}(314,·)$, $\chi_{357}(188,·)$, $\chi_{357}(64,·)$, $\chi_{357}(202,·)$, $\chi_{357}(76,·)$, $\chi_{357}(83,·)$, $\chi_{357}(344,·)$, $\chi_{357}(349,·)$, $\chi_{357}(223,·)$, $\chi_{357}(356,·)$, $\chi_{357}(230,·)$, $\chi_{357}(104,·)$, $\chi_{357}(106,·)$, $\chi_{357}(239,·)$, $\chi_{357}(118,·)$, $\chi_{357}(251,·)$, $\chi_{357}(253,·)$, $\chi_{357}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{15} + \frac{1}{4} a^{12} + \frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{19} - \frac{1}{8} a^{18} - \frac{1}{4} a^{17} + \frac{1}{8} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{8} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{20} - \frac{1}{16} a^{19} - \frac{1}{8} a^{18} - \frac{3}{16} a^{17} + \frac{1}{8} a^{16} + \frac{1}{8} a^{15} + \frac{3}{16} a^{14} + \frac{1}{8} a^{13} + \frac{1}{8} a^{12} + \frac{3}{16} a^{11} + \frac{1}{8} a^{10} - \frac{3}{8} a^{9} - \frac{7}{16} a^{8} - \frac{1}{8} a^{7} + \frac{3}{8} a^{6} + \frac{1}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{21} - \frac{1}{32} a^{20} - \frac{1}{16} a^{19} - \frac{3}{32} a^{18} - \frac{3}{16} a^{17} + \frac{1}{16} a^{16} - \frac{5}{32} a^{15} + \frac{1}{16} a^{14} + \frac{1}{16} a^{13} - \frac{5}{32} a^{12} + \frac{1}{16} a^{11} + \frac{5}{16} a^{10} - \frac{15}{32} a^{9} + \frac{7}{16} a^{8} - \frac{5}{16} a^{7} + \frac{1}{4} a^{6} + \frac{1}{32} a^{5} + \frac{5}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{22} - \frac{1}{64} a^{21} - \frac{1}{32} a^{20} - \frac{3}{64} a^{19} - \frac{3}{32} a^{18} - \frac{7}{32} a^{17} + \frac{11}{64} a^{16} + \frac{1}{32} a^{15} - \frac{7}{32} a^{14} + \frac{11}{64} a^{13} + \frac{1}{32} a^{12} - \frac{3}{32} a^{11} + \frac{1}{64} a^{10} - \frac{9}{32} a^{9} - \frac{13}{32} a^{8} + \frac{3}{8} a^{7} + \frac{1}{64} a^{6} + \frac{13}{32} a^{5} + \frac{5}{16} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{128} a^{23} - \frac{1}{128} a^{22} - \frac{1}{64} a^{21} - \frac{3}{128} a^{20} - \frac{3}{64} a^{19} - \frac{7}{64} a^{18} - \frac{21}{128} a^{17} - \frac{15}{64} a^{16} - \frac{7}{64} a^{15} + \frac{11}{128} a^{14} - \frac{15}{64} a^{13} + \frac{29}{64} a^{12} - \frac{63}{128} a^{11} - \frac{25}{64} a^{10} - \frac{13}{64} a^{9} - \frac{5}{16} a^{8} + \frac{33}{128} a^{7} - \frac{19}{64} a^{6} - \frac{11}{32} a^{5} + \frac{3}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{24} - \frac{1}{256} a^{23} - \frac{1}{128} a^{22} - \frac{3}{256} a^{21} - \frac{3}{128} a^{20} - \frac{7}{128} a^{19} - \frac{21}{256} a^{18} + \frac{17}{128} a^{17} + \frac{25}{128} a^{16} + \frac{11}{256} a^{15} - \frac{15}{128} a^{14} - \frac{3}{128} a^{13} + \frac{65}{256} a^{12} + \frac{39}{128} a^{11} - \frac{45}{128} a^{10} - \frac{5}{32} a^{9} - \frac{95}{256} a^{8} - \frac{51}{128} a^{7} + \frac{21}{64} a^{6} - \frac{13}{32} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{512} a^{25} - \frac{1}{512} a^{24} - \frac{1}{256} a^{23} - \frac{3}{512} a^{22} - \frac{3}{256} a^{21} - \frac{7}{256} a^{20} - \frac{21}{512} a^{19} + \frac{17}{256} a^{18} - \frac{39}{256} a^{17} - \frac{117}{512} a^{16} - \frac{15}{256} a^{15} + \frac{61}{256} a^{14} - \frac{63}{512} a^{13} + \frac{39}{256} a^{12} - \frac{109}{256} a^{11} - \frac{21}{64} a^{10} - \frac{95}{512} a^{9} - \frac{115}{256} a^{8} - \frac{11}{128} a^{7} - \frac{13}{64} a^{6} - \frac{1}{2} a^{5} + \frac{7}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{1024} a^{26} - \frac{1}{1024} a^{25} - \frac{1}{512} a^{24} - \frac{3}{1024} a^{23} - \frac{3}{512} a^{22} - \frac{7}{512} a^{21} - \frac{21}{1024} a^{20} + \frac{17}{512} a^{19} - \frac{39}{512} a^{18} - \frac{117}{1024} a^{17} + \frac{113}{512} a^{16} - \frac{67}{512} a^{15} + \frac{193}{1024} a^{14} - \frac{89}{512} a^{13} + \frac{19}{512} a^{12} + \frac{11}{128} a^{11} - \frac{351}{1024} a^{10} + \frac{13}{512} a^{9} - \frac{75}{256} a^{8} - \frac{45}{128} a^{7} - \frac{1}{2} a^{6} + \frac{15}{32} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{137216} a^{27} - \frac{33}{137216} a^{26} - \frac{45}{68608} a^{25} - \frac{99}{137216} a^{24} - \frac{71}{68608} a^{23} + \frac{189}{68608} a^{22} + \frac{99}{137216} a^{21} + \frac{665}{68608} a^{20} - \frac{3907}{68608} a^{19} + \frac{7923}{137216} a^{18} - \frac{2167}{68608} a^{17} + \frac{1057}{68608} a^{16} - \frac{15319}{137216} a^{15} + \frac{13439}{68608} a^{14} - \frac{7249}{68608} a^{13} - \frac{2671}{8576} a^{12} + \frac{17937}{137216} a^{11} - \frac{12507}{68608} a^{10} + \frac{14895}{34304} a^{9} + \frac{653}{8576} a^{8} + \frac{369}{2144} a^{7} + \frac{1093}{4288} a^{6} - \frac{457}{1072} a^{5} + \frac{51}{134} a^{4} - \frac{131}{268} a^{3} - \frac{13}{268} a^{2} - \frac{6}{67} a + \frac{28}{67}$, $\frac{1}{548864} a^{28} + \frac{1}{548864} a^{27} - \frac{35}{137216} a^{26} + \frac{57}{548864} a^{25} + \frac{195}{137216} a^{24} + \frac{455}{274432} a^{23} + \frac{87}{548864} a^{22} - \frac{1021}{68608} a^{21} + \frac{2087}{274432} a^{20} - \frac{6905}{548864} a^{19} - \frac{637}{68608} a^{18} + \frac{54411}{274432} a^{17} + \frac{75853}{548864} a^{16} - \frac{4341}{34304} a^{15} + \frac{5333}{274432} a^{14} - \frac{3669}{137216} a^{13} + \frac{164337}{548864} a^{12} - \frac{34421}{137216} a^{11} - \frac{345}{1072} a^{10} - \frac{15355}{68608} a^{9} + \frac{129}{4288} a^{8} + \frac{329}{8576} a^{7} - \frac{557}{8576} a^{6} + \frac{743}{2144} a^{5} + \frac{255}{1072} a^{4} + \frac{89}{1072} a^{3} + \frac{59}{134} a^{2} + \frac{23}{67} a + \frac{7}{134}$, $\frac{1}{51593216} a^{29} + \frac{21}{51593216} a^{28} - \frac{11}{3224576} a^{27} - \frac{12687}{51593216} a^{26} + \frac{921}{1612288} a^{25} - \frac{46861}{25796608} a^{24} - \frac{157073}{51593216} a^{23} - \frac{14403}{12898304} a^{22} - \frac{302125}{25796608} a^{21} + \frac{305471}{51593216} a^{20} - \frac{627491}{12898304} a^{19} + \frac{2975095}{25796608} a^{18} - \frac{1764059}{51593216} a^{17} + \frac{2186865}{12898304} a^{16} - \frac{73661}{385024} a^{15} - \frac{54075}{12898304} a^{14} - \frac{12558639}{51593216} a^{13} + \frac{532919}{1612288} a^{12} - \frac{2294417}{6449152} a^{11} - \frac{797891}{6449152} a^{10} + \frac{27323}{806144} a^{9} + \frac{229775}{806144} a^{8} - \frac{117017}{806144} a^{7} + \frac{10329}{100768} a^{6} + \frac{43831}{100768} a^{5} - \frac{10171}{100768} a^{4} - \frac{1193}{6298} a^{3} + \frac{625}{6298} a^{2} + \frac{1765}{12596} a - \frac{697}{3149}$, $\frac{1}{78332182895894751785991502594048} a^{30} + \frac{412083897971871436986955}{78332182895894751785991502594048} a^{29} - \frac{25301372054307667927053535}{39166091447947375892995751297024} a^{28} + \frac{69375520222707945212792501}{78332182895894751785991502594048} a^{27} - \frac{7342625629344307996188213069}{39166091447947375892995751297024} a^{26} + \frac{36840673858820351002916445605}{39166091447947375892995751297024} a^{25} + \frac{119374503305148071854455407491}{78332182895894751785991502594048} a^{24} - \frac{127912729924574161827295474901}{39166091447947375892995751297024} a^{23} + \frac{6296059621382011986164446835}{833321094637178210489271304192} a^{22} + \frac{1220635792125630544528968391363}{78332182895894751785991502594048} a^{21} + \frac{13712163062873423428732698453}{833321094637178210489271304192} a^{20} - \frac{1236053473588717909230482142687}{39166091447947375892995751297024} a^{19} + \frac{7451076757215171064984099392217}{78332182895894751785991502594048} a^{18} + \frac{5802319210851991590294795090581}{39166091447947375892995751297024} a^{17} - \frac{4630319816714320344841311324113}{39166091447947375892995751297024} a^{16} - \frac{37243098994933896784860268855}{152992544718544437082014653504} a^{15} - \frac{2509357075619084928996514015119}{78332182895894751785991502594048} a^{14} + \frac{1903724293601399090713708704179}{39166091447947375892995751297024} a^{13} - \frac{7185933383082278375058621658849}{19583045723973687946497875648512} a^{12} + \frac{1312998896410131223671634717785}{9791522861986843973248937824256} a^{11} - \frac{147324200079594810873345331377}{4895761430993421986624468912128} a^{10} - \frac{1042981617311037272361306826689}{2447880715496710993312234456064} a^{9} - \frac{584330420903843006998772397365}{1223940357748355496656117228032} a^{8} - \frac{236818649854104142184482961535}{611970178874177748328058614016} a^{7} - \frac{141087696039465076871164519565}{305985089437088874164029307008} a^{6} + \frac{21200841755553199271656464909}{152992544718544437082014653504} a^{5} + \frac{21018469405556188436149615377}{76496272359272218541007326752} a^{4} + \frac{4382838394766585432545310175}{38248136179636109270503663376} a^{3} - \frac{6163922522506769024948000687}{19124068089818054635251831688} a^{2} - \frac{698065333852785888665780889}{9562034044909027317625915844} a - \frac{341988520852868247900259263}{4781017022454513658812957922}$, $\frac{1}{3858183616950425109152700407363660945123114326566128557362755036053504} a^{31} - \frac{18921629679795381135251704709435198523}{3858183616950425109152700407363660945123114326566128557362755036053504} a^{30} - \frac{1103372309796548568618494208159382909646076186612682574529017}{241136476059401569322043775460228809070194645410383034835172189753344} a^{29} + \frac{3158733049745328449748869255040347894172519420076562132277022025}{3858183616950425109152700407363660945123114326566128557362755036053504} a^{28} - \frac{866533417946398508535247087800597177226998794195054106505358773}{482272952118803138644087550920457618140389290820766069670344379506688} a^{27} - \frac{502651467651897969334397523540392964982532684034582923946443185493}{1929091808475212554576350203681830472561557163283064278681377518026752} a^{26} + \frac{3733006282655966080896617684228763874918975380038884767639133638455}{3858183616950425109152700407363660945123114326566128557362755036053504} a^{25} + \frac{1567032056399536218453872998414811166120773415731496753395117025209}{964545904237606277288175101840915236280778581641532139340688759013376} a^{24} + \frac{1229490731023006705420749419257234731605789469139659436701032650163}{1929091808475212554576350203681830472561557163283064278681377518026752} a^{23} - \frac{1131430404191517812186308325125422791699454096715497070409057295689}{3858183616950425109152700407363660945123114326566128557362755036053504} a^{22} + \frac{12938689672065473467940650590503401351845164161416867055328986446945}{964545904237606277288175101840915236280778581641532139340688759013376} a^{21} - \frac{23551005723859963198419605010877346811958502772149666142671528999401}{1929091808475212554576350203681830472561557163283064278681377518026752} a^{20} - \frac{206546652613298259527124414908749808167434799894036695447499775418723}{3858183616950425109152700407363660945123114326566128557362755036053504} a^{19} - \frac{55317486072047912385398102245867557828437866112751916732882807298595}{964545904237606277288175101840915236280778581641532139340688759013376} a^{18} - \frac{13853237618404926871257722815156320923088833560176120424676837441447}{1929091808475212554576350203681830472561557163283064278681377518026752} a^{17} + \frac{202065290099774895276780192911391164333270491785801553868440518875759}{964545904237606277288175101840915236280778581641532139340688759013376} a^{16} + \frac{945342504431340556103538415006307726488803977440985556332321521064721}{3858183616950425109152700407363660945123114326566128557362755036053504} a^{15} - \frac{27012400630662179678606371490730575637619105048048967776697140913825}{120568238029700784661021887730114404535097322705191517417586094876672} a^{14} + \frac{76416554456832726603152074662734126637429916230991782508935223260665}{482272952118803138644087550920457618140389290820766069670344379506688} a^{13} + \frac{88237664879226239895337502858642571938664717658299515331804927859805}{241136476059401569322043775460228809070194645410383034835172189753344} a^{12} - \frac{25717426356200281122863999805516720226882584588114149244092100424349}{60284119014850392330510943865057202267548661352595758708793047438336} a^{11} - \frac{19216823758348602505396998179613623700226581506875975243653855722219}{60284119014850392330510943865057202267548661352595758708793047438336} a^{10} + \frac{13235314192836960353134555033282481664826798104236029720710643559805}{30142059507425196165255471932528601133774330676297879354396523719168} a^{9} + \frac{2791706214617886450402173665870617471191067443396519128687186403897}{7535514876856299041313867983132150283443582669074469838599130929792} a^{8} - \frac{504609039446916625236472215912907099076174736784883143646881760047}{1883878719214074760328466995783037570860895667268617459649782732448} a^{7} + \frac{559787931659055808257359290874334540270793406466565790228398991079}{1883878719214074760328466995783037570860895667268617459649782732448} a^{6} + \frac{239217161851382877255350901356792563540361300179343884626809892781}{1883878719214074760328466995783037570860895667268617459649782732448} a^{5} + \frac{1409237912624138116601581260452385721803948036385442947770299849}{3514699103011333508075498126460890990412118782217569887406311068} a^{4} - \frac{106558871628829851596542584334036581409292039009397717033189633055}{235484839901759345041058374472879696357611958408577182456222841556} a^{3} + \frac{47214933958163669541165339035141911155543267243119742909360391821}{235484839901759345041058374472879696357611958408577182456222841556} a^{2} - \frac{14648419014184862917278722772852018740809362898376722935504140293}{117742419950879672520529187236439848178805979204288591228111420778} a + \frac{27488532709838105742201918567355433395899774902127095340926956967}{117742419950879672520529187236439848178805979204288591228111420778}$

Class group and class number

$C_{15}\times C_{15}$, which has order $225$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( \frac{659443363150979357230610799923237241405041745669}{141906262761707144846073290918173737419996757317582848} a^{31} - \frac{965635655129314393660165185296123683106685325883}{141906262761707144846073290918173737419996757317582848} a^{30} - \frac{885752487799243471657774030800315308830186504063}{8869141422606696552879580682385858588749797332348928} a^{29} + \frac{12605862944525871026772593992353058969248733734705}{141906262761707144846073290918173737419996757317582848} a^{28} + \frac{43009244641401917466772616168034220775947166799829}{35476565690426786211518322729543434354999189329395712} a^{27} - \frac{40862451226258070639108363127908784070893634429447}{70953131380853572423036645459086868709998378658791424} a^{26} - \frac{1704295570580724286841908810751836497635190819929589}{141906262761707144846073290918173737419996757317582848} a^{25} + \frac{23447670371928124341262827592984928863526075972021}{4434570711303348276439790341192929294374898666174464} a^{24} + \frac{6818843802538628171314093506803348109966268710792517}{70953131380853572423036645459086868709998378658791424} a^{23} - \frac{5549247210769640012486757969517680122039104031624005}{141906262761707144846073290918173737419996757317582848} a^{22} - \frac{1318839403328557605359126995192901525484110955927135}{2217285355651674138219895170596464647187449333087232} a^{21} + \frac{10144159991745299898421312773427603617241734263463705}{70953131380853572423036645459086868709998378658791424} a^{20} + \frac{433899342705868179707297544856905790961425274784286873}{141906262761707144846073290918173737419996757317582848} a^{19} - \frac{11846243222919385212136295469629527997980530819631581}{17738282845213393105759161364771717177499594664697856} a^{18} - \frac{13754163516169501298246259346915321882693910381151995}{1059001960908262274970696200881893562835796696399872} a^{17} + \frac{143098529971015096313479573666511878582733171893111345}{35476565690426786211518322729543434354999189329395712} a^{16} + \frac{6076197189619243242667441280246037660092267423259907725}{141906262761707144846073290918173737419996757317582848} a^{15} - \frac{387198456653180117954049296978801747604824868874210561}{35476565690426786211518322729543434354999189329395712} a^{14} - \frac{3885840649806878296445133501236642558070894423854104933}{35476565690426786211518322729543434354999189329395712} a^{13} + \frac{71765945539057367995310920229974765350811900508828655}{4434570711303348276439790341192929294374898666174464} a^{12} + \frac{497566552617728030916224564171151939091073690653895703}{2217285355651674138219895170596464647187449333087232} a^{11} - \frac{4656875896080541325530287023716371562166547122886953}{66187622556766392185668512555118347677237293524992} a^{10} - \frac{172962839799884191546201126560907182768167639987374175}{554321338912918534554973792649116161796862333271808} a^{9} + \frac{23305471752141151776650046978212805708245236436046827}{138580334728229633638743448162279040449215583317952} a^{8} + \frac{151072631043437680969034651339751363656159156325857789}{554321338912918534554973792649116161796862333271808} a^{7} - \frac{7673427705700680628930625212394253449947305834383949}{69290167364114816819371724081139520224607791658976} a^{6} - \frac{2702791297596020314762864112514773050015697137643733}{17322541841028704204842931020284880056151947914744} a^{5} - \frac{286728896981615931531551065761984262529349959251545}{69290167364114816819371724081139520224607791658976} a^{4} + \frac{735504913033434206288427251283469558043593732574717}{8661270920514352102421465510142440028075973957372} a^{3} - \frac{37321283101723723008233554086306270261497415539642}{2165317730128588025605366377535610007018993489343} a^{2} - \frac{27341113315816878321508862194089187124068991378063}{8661270920514352102421465510142440028075973957372} a + \frac{8864536827058258276908827925356256402479384546417}{4330635460257176051210732755071220014037986978686} \) (order $6$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 8912832207192.955 \) (assuming GRH)

Galois group

$C_2^2\times C_8$ (as 32T37):

An abelian group of order 32

The 32 conjugacy class representatives for $C_2^2\times C_8$

Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{357}) \), \(\Q(\sqrt{-119}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}, \sqrt{-119})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{17}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{-51})\), \(\Q(\sqrt{-7}, \sqrt{17})\), \(\Q(\sqrt{21}, \sqrt{-51})\), 4.4.4913.1, 4.0.44217.1, 4.4.2166633.1, 4.0.240737.1, 8.0.16243247601.1, 8.0.1955143089.1, 8.0.4694298556689.3, 8.8.4694298556689.1, 8.0.4694298556689.2, 8.0.57954303169.1, 8.0.4694298556689.1, 8.8.79803075463713.1, 8.0.985223153873.1, \(\Q(\zeta_{17})^+\), 8.0.33237432513.1, 16.0.22036438939332428546642721.1, 16.0.6368530853467071849979746369.3, 16.0.1104726920056229495169.1, 16.16.6368530853467071849979746369.1, 16.0.6368530853467071849979746369.2, 16.0.6368530853467071849979746369.1, 16.0.970664662927461034900129.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$	R	${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$	R	${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$	R	${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$	${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$	${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$	${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$	${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
3	Data not computed
7	Data not computed
17	Data not computed