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Properties

Label 32.0.59816430901...0000.9
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $79.30$
Ramified primes $2, 3, 5, 13$
Class number $5120$ (GRH)
Class group $[2, 2, 2, 8, 80]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![531854181, -302934816, 372745611, -1413093708, 449157015, -102727494, -1842800025, 2487107298, 2900961157, -2987674244, -1693247590, 1385729356, 756846581, -223742052, -309446035, -58409400, 112998102, 41887806, -31683220, -11544686, 6700777, 2047834, -1101976, -277890, 143331, 30390, -14536, -2358, 1076, 106, -49, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 49*x^30 + 106*x^29 + 1076*x^28 - 2358*x^27 - 14536*x^26 + 30390*x^25 + 143331*x^24 - 277890*x^23 - 1101976*x^22 + 2047834*x^21 + 6700777*x^20 - 11544686*x^19 - 31683220*x^18 + 41887806*x^17 + 112998102*x^16 - 58409400*x^15 - 309446035*x^14 - 223742052*x^13 + 756846581*x^12 + 1385729356*x^11 - 1693247590*x^10 - 2987674244*x^9 + 2900961157*x^8 + 2487107298*x^7 - 1842800025*x^6 - 102727494*x^5 + 449157015*x^4 - 1413093708*x^3 + 372745611*x^2 - 302934816*x + 531854181)
 
gp: K = bnfinit(x^32 - 2*x^31 - 49*x^30 + 106*x^29 + 1076*x^28 - 2358*x^27 - 14536*x^26 + 30390*x^25 + 143331*x^24 - 277890*x^23 - 1101976*x^22 + 2047834*x^21 + 6700777*x^20 - 11544686*x^19 - 31683220*x^18 + 41887806*x^17 + 112998102*x^16 - 58409400*x^15 - 309446035*x^14 - 223742052*x^13 + 756846581*x^12 + 1385729356*x^11 - 1693247590*x^10 - 2987674244*x^9 + 2900961157*x^8 + 2487107298*x^7 - 1842800025*x^6 - 102727494*x^5 + 449157015*x^4 - 1413093708*x^3 + 372745611*x^2 - 302934816*x + 531854181, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} - 49 x^{30} + 106 x^{29} + 1076 x^{28} - 2358 x^{27} - 14536 x^{26} + 30390 x^{25} + 143331 x^{24} - 277890 x^{23} - 1101976 x^{22} + 2047834 x^{21} + 6700777 x^{20} - 11544686 x^{19} - 31683220 x^{18} + 41887806 x^{17} + 112998102 x^{16} - 58409400 x^{15} - 309446035 x^{14} - 223742052 x^{13} + 756846581 x^{12} + 1385729356 x^{11} - 1693247590 x^{10} - 2987674244 x^{9} + 2900961157 x^{8} + 2487107298 x^{7} - 1842800025 x^{6} - 102727494 x^{5} + 449157015 x^{4} - 1413093708 x^{3} + 372745611 x^{2} - 302934816 x + 531854181 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5981643090147991811559885370844487936000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.30$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(131,·)$, $\chi_{780}(649,·)$, $\chi_{780}(779,·)$, $\chi_{780}(493,·)$, $\chi_{780}(623,·)$, $\chi_{780}(157,·)$, $\chi_{780}(671,·)$, $\chi_{780}(421,·)$, $\chi_{780}(551,·)$, $\chi_{780}(47,·)$, $\chi_{780}(181,·)$, $\chi_{780}(311,·)$, $\chi_{780}(313,·)$, $\chi_{780}(287,·)$, $\chi_{780}(577,·)$, $\chi_{780}(707,·)$, $\chi_{780}(73,·)$, $\chi_{780}(203,·)$, $\chi_{780}(337,·)$, $\chi_{780}(83,·)$, $\chi_{780}(469,·)$, $\chi_{780}(599,·)$, $\chi_{780}(733,·)$, $\chi_{780}(443,·)$, $\chi_{780}(229,·)$, $\chi_{780}(359,·)$, $\chi_{780}(109,·)$, $\chi_{780}(697,·)$, $\chi_{780}(239,·)$, $\chi_{780}(467,·)$, $\chi_{780}(541,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{16} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{17} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{12} a^{20} - \frac{1}{6} a^{18} + \frac{1}{12} a^{16} - \frac{1}{2} a^{15} + \frac{1}{3} a^{14} - \frac{1}{2} a^{13} - \frac{1}{6} a^{12} - \frac{1}{2} a^{11} - \frac{1}{12} a^{10} - \frac{1}{2} a^{9} + \frac{5}{12} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{12} a^{4} - \frac{1}{2} a^{3} + \frac{5}{12} a^{2} - \frac{1}{4}$, $\frac{1}{12} a^{21} - \frac{1}{6} a^{19} + \frac{1}{12} a^{17} - \frac{1}{2} a^{16} + \frac{1}{3} a^{15} - \frac{1}{2} a^{14} - \frac{1}{6} a^{13} - \frac{1}{2} a^{12} - \frac{1}{12} a^{11} - \frac{1}{2} a^{10} + \frac{5}{12} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{5} - \frac{1}{2} a^{4} + \frac{5}{12} a^{3} - \frac{1}{4} a$, $\frac{1}{12} a^{22} + \frac{1}{12} a^{18} - \frac{1}{2} a^{17} + \frac{1}{6} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{12} a^{12} - \frac{1}{2} a^{11} - \frac{1}{12} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{12} a^{4} - \frac{1}{12} a^{2} - \frac{1}{2}$, $\frac{1}{36} a^{23} - \frac{1}{36} a^{22} - \frac{1}{12} a^{19} + \frac{1}{36} a^{18} - \frac{1}{3} a^{17} - \frac{1}{9} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{36} a^{13} + \frac{1}{12} a^{12} + \frac{1}{4} a^{11} - \frac{1}{36} a^{10} - \frac{1}{6} a^{9} + \frac{4}{9} a^{8} + \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{5}{12} a^{5} - \frac{7}{36} a^{4} + \frac{7}{36} a^{3} - \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{36} a^{24} - \frac{1}{36} a^{22} - \frac{1}{18} a^{19} - \frac{5}{36} a^{18} - \frac{4}{9} a^{17} + \frac{11}{36} a^{16} - \frac{1}{6} a^{15} + \frac{1}{36} a^{14} - \frac{7}{18} a^{13} - \frac{1}{2} a^{12} - \frac{5}{18} a^{11} + \frac{7}{18} a^{10} - \frac{2}{9} a^{9} + \frac{5}{18} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{7}{18} a^{5} - \frac{5}{12} a^{4} - \frac{7}{18} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{4}$, $\frac{1}{36} a^{25} - \frac{1}{36} a^{22} + \frac{1}{36} a^{20} + \frac{1}{9} a^{19} + \frac{1}{12} a^{18} - \frac{13}{36} a^{17} + \frac{5}{36} a^{16} + \frac{7}{36} a^{15} - \frac{7}{18} a^{14} + \frac{13}{36} a^{13} + \frac{11}{36} a^{12} - \frac{7}{36} a^{11} - \frac{1}{18} a^{9} - \frac{5}{36} a^{8} + \frac{1}{12} a^{7} + \frac{11}{36} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{5}{36} a^{3} - \frac{1}{3} a^{2} + \frac{5}{12} a + \frac{1}{4}$, $\frac{1}{1991916} a^{26} + \frac{3178}{497979} a^{25} - \frac{125}{663972} a^{24} - \frac{26305}{1991916} a^{23} + \frac{20177}{497979} a^{22} + \frac{16033}{1991916} a^{21} + \frac{14191}{995958} a^{20} + \frac{73897}{1991916} a^{19} - \frac{16633}{1991916} a^{18} + \frac{907063}{1991916} a^{17} - \frac{36727}{331986} a^{16} + \frac{289705}{995958} a^{15} + \frac{166910}{497979} a^{14} + \frac{78803}{221324} a^{13} - \frac{28813}{995958} a^{12} + \frac{84272}{497979} a^{11} - \frac{132635}{1991916} a^{10} + \frac{690269}{1991916} a^{9} + \frac{28123}{1991916} a^{8} + \frac{326315}{1991916} a^{7} + \frac{650255}{1991916} a^{6} - \frac{16292}{165993} a^{5} + \frac{111109}{1991916} a^{4} - \frac{295507}{995958} a^{3} + \frac{51825}{110662} a^{2} + \frac{142643}{663972} a + \frac{93057}{221324}$, $\frac{1}{1991916} a^{27} + \frac{737}{55331} a^{25} - \frac{17771}{1991916} a^{24} - \frac{2009}{663972} a^{23} + \frac{29335}{995958} a^{22} + \frac{28145}{995958} a^{21} - \frac{23543}{663972} a^{20} + \frac{72769}{497979} a^{19} + \frac{65509}{497979} a^{18} + \frac{230513}{1991916} a^{17} + \frac{213818}{497979} a^{16} + \frac{593785}{1991916} a^{15} + \frac{891637}{1991916} a^{14} - \frac{169717}{995958} a^{13} - \frac{30319}{663972} a^{12} - \frac{583975}{1991916} a^{11} + \frac{433075}{995958} a^{10} + \frac{382547}{1991916} a^{9} - \frac{620297}{1991916} a^{8} + \frac{3897}{221324} a^{7} + \frac{311213}{663972} a^{6} + \frac{27715}{995958} a^{5} + \frac{83751}{221324} a^{4} + \frac{15920}{497979} a^{3} + \frac{13445}{331986} a^{2} - \frac{47273}{165993} a + \frac{9049}{55331}$, $\frac{1}{5975748} a^{28} + \frac{1}{5975748} a^{27} - \frac{1}{5975748} a^{26} - \frac{18145}{2987874} a^{25} + \frac{77159}{5975748} a^{24} + \frac{881}{1991916} a^{23} - \frac{56035}{5975748} a^{22} + \frac{15677}{497979} a^{21} - \frac{4305}{221324} a^{20} - \frac{3073}{331986} a^{19} + \frac{219227}{5975748} a^{18} + \frac{920959}{5975748} a^{17} + \frac{1819219}{5975748} a^{16} + \frac{2548207}{5975748} a^{15} - \frac{2597479}{5975748} a^{14} - \frac{33825}{110662} a^{13} - \frac{28017}{221324} a^{12} - \frac{242681}{663972} a^{11} - \frac{535033}{5975748} a^{10} - \frac{637357}{1991916} a^{9} + \frac{749027}{5975748} a^{8} - \frac{1940177}{5975748} a^{7} - \frac{2178955}{5975748} a^{6} - \frac{690236}{1493937} a^{5} + \frac{1959973}{5975748} a^{4} + \frac{63571}{995958} a^{3} + \frac{304837}{1991916} a^{2} + \frac{20939}{55331} a - \frac{65303}{663972}$, $\frac{1}{782822988} a^{29} + \frac{1}{260940996} a^{28} - \frac{11}{782822988} a^{27} - \frac{53}{391411494} a^{26} + \frac{2584894}{195705747} a^{25} - \frac{1432463}{195705747} a^{24} + \frac{3604553}{782822988} a^{23} + \frac{11508349}{782822988} a^{22} + \frac{8667341}{260940996} a^{21} + \frac{422828}{65235249} a^{20} - \frac{99770641}{782822988} a^{19} + \frac{16113767}{195705747} a^{18} + \frac{17201516}{65235249} a^{17} - \frac{8904023}{130470498} a^{16} - \frac{49845461}{195705747} a^{15} - \frac{178964507}{782822988} a^{14} + \frac{51635843}{130470498} a^{13} - \frac{73922231}{260940996} a^{12} - \frac{73041106}{195705747} a^{11} + \frac{198422623}{782822988} a^{10} - \frac{9878215}{782822988} a^{9} + \frac{252460883}{782822988} a^{8} + \frac{128951603}{391411494} a^{7} - \frac{12503267}{391411494} a^{6} - \frac{19970417}{86980332} a^{5} - \frac{338315845}{782822988} a^{4} - \frac{57623443}{130470498} a^{3} + \frac{15311377}{130470498} a^{2} + \frac{13122817}{43490166} a + \frac{30170351}{86980332}$, $\frac{1}{286513213608} a^{30} + \frac{91}{286513213608} a^{29} - \frac{16777}{286513213608} a^{28} + \frac{13349}{143256606804} a^{27} + \frac{58931}{286513213608} a^{26} + \frac{791381351}{95504404536} a^{25} - \frac{1463871173}{143256606804} a^{24} + \frac{764191867}{95504404536} a^{23} - \frac{312400655}{47752202268} a^{22} + \frac{2466955}{95504404536} a^{21} - \frac{805502545}{286513213608} a^{20} - \frac{14155813225}{143256606804} a^{19} - \frac{11142517675}{143256606804} a^{18} - \frac{4318056295}{143256606804} a^{17} + \frac{5658423911}{71628303402} a^{16} + \frac{14709855911}{47752202268} a^{15} - \frac{18027176885}{47752202268} a^{14} + \frac{10214820875}{23876101134} a^{13} - \frac{120163912585}{286513213608} a^{12} + \frac{4756301975}{95504404536} a^{11} + \frac{104230605983}{286513213608} a^{10} + \frac{4758143677}{71628303402} a^{9} - \frac{72042419383}{286513213608} a^{8} - \frac{43937172767}{286513213608} a^{7} - \frac{110691053201}{286513213608} a^{6} + \frac{6422461781}{23876101134} a^{5} - \frac{845803211}{11938050567} a^{4} - \frac{873347}{60753438} a^{3} + \frac{8394092693}{31834801512} a^{2} - \frac{3801654893}{10611600504} a + \frac{27252187}{173960664}$, $\frac{1}{424368243990777096958076354577145110409339838322880973004319970834987056219851801680689279374646999297870320395857702608504} a^{31} + \frac{587718768973221665305203252189836313763808000437668887016937116118571257840643339743097700539552694088955259349}{424368243990777096958076354577145110409339838322880973004319970834987056219851801680689279374646999297870320395857702608504} a^{30} - \frac{110756591089461897870232937868541418685688432539913584647939794509409646458468040572005045291633294920282474893849}{424368243990777096958076354577145110409339838322880973004319970834987056219851801680689279374646999297870320395857702608504} a^{29} - \frac{5139450878714349835870961700805043082978808205019498062823189690086922165400264086064153773767665970239515508214057}{106092060997694274239519088644286277602334959580720243251079992708746764054962950420172319843661749824467580098964425652126} a^{28} - \frac{90817549288554347378657467183078707068631258272553749795658269757545947721022022673096055032610720009816724186871515}{424368243990777096958076354577145110409339838322880973004319970834987056219851801680689279374646999297870320395857702608504} a^{27} + \frac{1996752238757809819394788749280716758559585201471521599090398176262302273088725390791077879204804214852498464137191}{15717342370028781368817642762116485570716290308254850852011850771666187267401918580766269606468407381402604459105840837352} a^{26} - \frac{1041092275229171873617041668218478981166553054900533493822773830841708472269181605404895140449737579985262784874899816101}{212184121995388548479038177288572555204669919161440486502159985417493528109925900840344639687323499648935160197928851304252} a^{25} + \frac{251440495104325929581241791600138089443430081696293169027214724168663646948269337840990630315061982215667967634975858617}{141456081330259032319358784859048370136446612774293657668106656944995685406617267226896426458215666432623440131952567536168} a^{24} - \frac{128587150927234768968659286750388836779361093836747135969732903151082411189973661685481899993966488859760689336703221763}{35364020332564758079839696214762092534111653193573414417026664236248921351654316806724106614553916608155860032988141884042} a^{23} - \frac{4679793917238662982978741923104551730352482860215553679786968521317601919143096312974560990543536957209099252083796692353}{141456081330259032319358784859048370136446612774293657668106656944995685406617267226896426458215666432623440131952567536168} a^{22} - \frac{13399906322117217182652275137312193528684803197084640881782525888725966499757839526371945417608145460423468771325715988355}{424368243990777096958076354577145110409339838322880973004319970834987056219851801680689279374646999297870320395857702608504} a^{21} + \frac{2510930960534054584845374935626360497468855473028757565324345998655966648655035487691337497612711769148413387868649014447}{212184121995388548479038177288572555204669919161440486502159985417493528109925900840344639687323499648935160197928851304252} a^{20} + \frac{17729469867178061153847887578626568547922712975333096004281191027935153750144005341995963079229256552867944141927858370069}{212184121995388548479038177288572555204669919161440486502159985417493528109925900840344639687323499648935160197928851304252} a^{19} + \frac{17130905694270356010834627319151270136206314946526205474834169129485204763060600286182739166048001946139142621285473637347}{106092060997694274239519088644286277602334959580720243251079992708746764054962950420172319843661749824467580098964425652126} a^{18} - \frac{16196452073757898207088466497874138920220668260455229088325476376460774445284561602361994603587547419752290533987253256745}{106092060997694274239519088644286277602334959580720243251079992708746764054962950420172319843661749824467580098964425652126} a^{17} - \frac{1647477790955648183289022192770416752106961812146553401100603978036375494541283484089486777958585165077830045336416052900}{17682010166282379039919848107381046267055826596786707208513332118124460675827158403362053307276958304077930016494070942021} a^{16} - \frac{14257788082359862131196755236196249864309046042142204638569783349265187035735331490464119454931485078630105807245641084679}{35364020332564758079839696214762092534111653193573414417026664236248921351654316806724106614553916608155860032988141884042} a^{15} - \frac{8800753696480040926715256405917118075198453073339527571716338764028065035370221423295008423681162690178871802854307497359}{35364020332564758079839696214762092534111653193573414417026664236248921351654316806724106614553916608155860032988141884042} a^{14} + \frac{166468461581101156253698652909634128421647703926699457673069482651054421686701171017555409981704390598787933059078617030739}{424368243990777096958076354577145110409339838322880973004319970834987056219851801680689279374646999297870320395857702608504} a^{13} + \frac{33628440274795773694359882549483855298631403461094165560673560436683415167878728719481296244003992445634992176439072046601}{141456081330259032319358784859048370136446612774293657668106656944995685406617267226896426458215666432623440131952567536168} a^{12} - \frac{58687250985110358298796048398843546970994555822506440461712884677914375311079688952773928861913088486322120083562210443677}{424368243990777096958076354577145110409339838322880973004319970834987056219851801680689279374646999297870320395857702608504} a^{11} - \frac{11436078213562034812702427504755085080726404912961134319969160293735538718669422847920878590509649737712746234654458846309}{212184121995388548479038177288572555204669919161440486502159985417493528109925900840344639687323499648935160197928851304252} a^{10} + \frac{53950450722386655033351473535382171904361967017644346050950263723499885123235248580517574951803094411838461231917464484305}{424368243990777096958076354577145110409339838322880973004319970834987056219851801680689279374646999297870320395857702608504} a^{9} - \frac{42384978337645821271162269963320682127079397125111317483140201130922512016406828550348284478821188435767029704903236373297}{424368243990777096958076354577145110409339838322880973004319970834987056219851801680689279374646999297870320395857702608504} a^{8} + \frac{125094597338337563504040811813701247097722257124115805258588894826196440904243908146598103353544173199536970252169558074053}{424368243990777096958076354577145110409339838322880973004319970834987056219851801680689279374646999297870320395857702608504} a^{7} + \frac{701392690287167753714003425280879304193236734579598033806573477517997989748357203779703130085355805223603602827931030291}{1964667796253597671102205345264560696339536288531856356501481346458273408425239822595783700808550922675325557388230104669} a^{6} - \frac{1714718283008710833365889576915629392904535053062387667918965375926622017019942909180217286660820732122757674578262000713}{23576013555043172053226464143174728356074435462382276278017776157499280901102877871149404409702611072103906688658761256028} a^{5} - \frac{2543011494187434380986334339548196043985198934724799642000501873373455677863831853133950612240758260457954130439760183143}{11788006777521586026613232071587364178037217731191138139008888078749640450551438935574702204851305536051953344329380628014} a^{4} - \frac{2263139271184245878525490503759301369185574143013172790204972686321231397467063646505229453951485169226225959906605835235}{15717342370028781368817642762116485570716290308254850852011850771666187267401918580766269606468407381402604459105840837352} a^{3} + \frac{1432246267335247272313251685985934455188780698475755019666341915119797534841035404213808114077744846876183746089183992547}{15717342370028781368817642762116485570716290308254850852011850771666187267401918580766269606468407381402604459105840837352} a^{2} - \frac{29749105547358545147867800517915079414812974559027935983690138508517463921429626929620011641275185164920849227207674867}{15717342370028781368817642762116485570716290308254850852011850771666187267401918580766269606468407381402604459105840837352} a - \frac{13707130092900475899055474841692937664629046447641216571054194716163144866118081147538681704245230131678893932232918901}{42943558388056779696223067656055971504689317782117078830633472053732752096726553499361392367400020167766678850015958572}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{80}$, which has order $5120$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{23690573539829165086326814770473220733983402983535547623858187481207424413517897283741001107273}{4852366342992556089593331584014674501973942411619775157611272188790481646282950201155434636446807939612} a^{31} - \frac{54104532676209950998061454474556260413484109793561168384861273775688972922134973026737683907301}{9704732685985112179186663168029349003947884823239550315222544377580963292565900402310869272893615879224} a^{30} - \frac{2368344484951842764967746022688826764850836529935855019983181072793932734088167003769434786803623}{9704732685985112179186663168029349003947884823239550315222544377580963292565900402310869272893615879224} a^{29} + \frac{2992230582493791318033018289009635019399236941734836041167989169911953620909213999196423731921931}{9704732685985112179186663168029349003947884823239550315222544377580963292565900402310869272893615879224} a^{28} + \frac{6695212197801992154859311737474735717933167850904072568245535753324494399713111530843840655164364}{1213091585748139022398332896003668625493485602904943789402818047197620411570737550288858659111701984903} a^{27} - \frac{7318038513348256809335364917270284515010955713081692077122328876580537996996795445238597250463379}{1078303631776123575465184796447705444883098313693283368358060486397884810285100044701207696988179542136} a^{26} - \frac{745501566606053858483846191255675095916759061567993042936635020965997956854984274919875728196804231}{9704732685985112179186663168029349003947884823239550315222544377580963292565900402310869272893615879224} a^{25} + \frac{133700778640199142518472703577643934974331431737437575317008748639473471122513111785733685307002279}{1617455447664185363197777194671558167324647470539925052537090729596827215427650067051811545482269313204} a^{24} + \frac{2494346850413277329950728908467498685470048424314766011789615022123006800083729708654102524192977139}{3234910895328370726395554389343116334649294941079850105074181459193654430855300134103623090964538626408} a^{23} - \frac{1128195213282226690948960738530505806453887851403969748909545943400421625462753904158901378693086629}{1617455447664185363197777194671558167324647470539925052537090729596827215427650067051811545482269313204} a^{22} - \frac{58060951273375916753620178059131841629175705535205250842059524971760616115895559686696169154669718979}{9704732685985112179186663168029349003947884823239550315222544377580963292565900402310869272893615879224} a^{21} + \frac{47401031833348119323926679586616628453628176456062357415952493202545784998004794966737781605961442885}{9704732685985112179186663168029349003947884823239550315222544377580963292565900402310869272893615879224} a^{20} + \frac{179257593145317012367579780448997595012056791271242288388790261248790764739900578709100323380759880963}{4852366342992556089593331584014674501973942411619775157611272188790481646282950201155434636446807939612} a^{19} - \frac{60151561799717601157047815137056947188672424200100158597202310911902909696750180209883343963328329653}{2426183171496278044796665792007337250986971205809887578805636094395240823141475100577717318223403969806} a^{18} - \frac{213729529152613128477652150598183571949908443903025103657460410409123069586892913448477347486510357851}{1213091585748139022398332896003668625493485602904943789402818047197620411570737550288858659111701984903} a^{17} + \frac{43638547518942806311376001442195991370351700521994743467133912687606422804108179187314598718959472931}{808727723832092681598888597335779083662323735269962526268545364798413607713825033525905772741134656602} a^{16} + \frac{969212991586209579514341159628176941849179326135248464035519361004992049983922097595517998732921156357}{1617455447664185363197777194671558167324647470539925052537090729596827215427650067051811545482269313204} a^{15} + \frac{91890014618569820078287319665251906532557629113598370829858458976805965038332179259688957880960623548}{404363861916046340799444298667889541831161867634981263134272682399206803856912516762952886370567328301} a^{14} - \frac{6413732272079313545302050910252873446837977943610794200643071391033134686739048120017414335121240076179}{4852366342992556089593331584014674501973942411619775157611272188790481646282950201155434636446807939612} a^{13} - \frac{7205603780747229603228194327309008636095004852396174309800545424562702720788776514165674938231739081069}{3234910895328370726395554389343116334649294941079850105074181459193654430855300134103623090964538626408} a^{12} + \frac{17463073663333624744044616955939743258151287990160891300624620796134253071092897826675189981934947551961}{9704732685985112179186663168029349003947884823239550315222544377580963292565900402310869272893615879224} a^{11} + \frac{80831034932412867891432196786340318575089778724843295190740279036582104416974155019924865029984036493001}{9704732685985112179186663168029349003947884823239550315222544377580963292565900402310869272893615879224} a^{10} - \frac{1352991710790602857534810711764388965079603507611619806718638636725689698823489397502418074151936631677}{1213091585748139022398332896003668625493485602904943789402818047197620411570737550288858659111701984903} a^{9} - \frac{151089265445125764503482594022516722467658503013872059649942764560153265484541022024461656797671708036051}{9704732685985112179186663168029349003947884823239550315222544377580963292565900402310869272893615879224} a^{8} + \frac{7026101177974658576703705794125003690005063317789140137866423122643334816620773637580391157026648777499}{9704732685985112179186663168029349003947884823239550315222544377580963292565900402310869272893615879224} a^{7} + \frac{13748660125901988326679614381823809256778542831567194702379784409659441010450085266201822507929960343963}{1078303631776123575465184796447705444883098313693283368358060486397884810285100044701207696988179542136} a^{6} + \frac{1047990429523291911966020867138590727427823335391789229456215719501437526435990772333454651781121749295}{539151815888061787732592398223852722441549156846641684179030243198942405142550022350603848494089771068} a^{5} + \frac{7242739421511216977844265327004871586891267112287354472550321096378350239205534158150751188774430604}{4992146443407979516042522205776414096681010711542978557213242992582800047616203910653739337908238621} a^{4} + \frac{199208527721431905686356704387856571908834940694167050913052933272482489964339723483633072948835073720}{44929317990671815644382699851987726870129096403886807014919186933245200428545835195883654041174147589} a^{3} - \frac{1401578611602353449147993344102061992375591842029530753276573764246883149783882848289800611371606249789}{359434543925374525155061598815901814961032771231094456119353495465961603428366681567069232329393180712} a^{2} - \frac{943448591222269959122626181553882245934462403851911707559597281876241510608601632656203606950589201021}{359434543925374525155061598815901814961032771231094456119353495465961603428366681567069232329393180712} a - \frac{1732562116071505935307356588269261399734053941662724734803006304043172305456367482615160421110016869}{654707730283013707021970125347726438908985011349898827175507277715777055425075922708687126283047688} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 252395239518990.62 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{195}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}, \sqrt{65})\), 4.4.274625.1, 4.4.39546000.2, \(\Q(\sqrt{5}, \sqrt{39})\), \(\Q(\sqrt{13}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{15}, \sqrt{39})\), 4.4.274625.2, 4.4.39546000.1, \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{13})\), 4.0.18000.1, 4.0.21125.1, 4.0.3042000.1, \(\Q(\zeta_{5})\), 4.0.7909200.1, 4.0.2197.1, 4.0.316368.2, 4.0.54925.1, 8.8.1563886116000000.5, 8.8.370150560000.1, 8.8.1563886116000000.2, 8.8.75418890625.1, 8.8.1563886116000000.3, 8.8.1563886116000000.7, 8.8.1563886116000000.1, 8.0.9253764000000.6, 8.0.9253764000000.10, 8.0.62555444640000.66, 8.0.62555444640000.61, 8.0.9253764000000.3, 8.0.446265625.1, 8.0.62555444640000.63, 8.0.3016755625.1, 8.0.324000000.3, 8.0.9253764000000.4, 8.0.62555444640000.41, 8.0.100088711424.3, 16.16.2445739783817565456000000000000.2, 16.0.85632148167696000000000000.6, 16.0.3913183654108104729600000000.2, 16.0.2445739783817565456000000000000.13, 16.0.5688009063105712890625.1, 16.0.2445739783817565456000000000000.6, 16.0.2445739783817565456000000000000.14

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
13Data not computed