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Properties

Label 32.0.15729269092...000.20
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $61.29$
Ramified primes $2, 3, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![33573121, -267857268, 716221644, -481122960, -722556206, 999542892, -790864434, 189665064, 1766268390, 76338312, -92609202, -114096144, -357832945, 25660788, 40514058, 5648256, 18733574, -4926492, 624744, -641568, -95044, -74580, 2994, 984, 3594, -120, 270, -192, 91, -12, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 6*x^30 - 12*x^29 + 91*x^28 - 192*x^27 + 270*x^26 - 120*x^25 + 3594*x^24 + 984*x^23 + 2994*x^22 - 74580*x^21 - 95044*x^20 - 641568*x^19 + 624744*x^18 - 4926492*x^17 + 18733574*x^16 + 5648256*x^15 + 40514058*x^14 + 25660788*x^13 - 357832945*x^12 - 114096144*x^11 - 92609202*x^10 + 76338312*x^9 + 1766268390*x^8 + 189665064*x^7 - 790864434*x^6 + 999542892*x^5 - 722556206*x^4 - 481122960*x^3 + 716221644*x^2 - 267857268*x + 33573121)
 
gp: K = bnfinit(x^32 + 6*x^30 - 12*x^29 + 91*x^28 - 192*x^27 + 270*x^26 - 120*x^25 + 3594*x^24 + 984*x^23 + 2994*x^22 - 74580*x^21 - 95044*x^20 - 641568*x^19 + 624744*x^18 - 4926492*x^17 + 18733574*x^16 + 5648256*x^15 + 40514058*x^14 + 25660788*x^13 - 357832945*x^12 - 114096144*x^11 - 92609202*x^10 + 76338312*x^9 + 1766268390*x^8 + 189665064*x^7 - 790864434*x^6 + 999542892*x^5 - 722556206*x^4 - 481122960*x^3 + 716221644*x^2 - 267857268*x + 33573121, 1)
 

Normalized defining polynomial

\( x^{32} + 6 x^{30} - 12 x^{29} + 91 x^{28} - 192 x^{27} + 270 x^{26} - 120 x^{25} + 3594 x^{24} + 984 x^{23} + 2994 x^{22} - 74580 x^{21} - 95044 x^{20} - 641568 x^{19} + 624744 x^{18} - 4926492 x^{17} + 18733574 x^{16} + 5648256 x^{15} + 40514058 x^{14} + 25660788 x^{13} - 357832945 x^{12} - 114096144 x^{11} - 92609202 x^{10} + 76338312 x^{9} + 1766268390 x^{8} + 189665064 x^{7} - 790864434 x^{6} + 999542892 x^{5} - 722556206 x^{4} - 481122960 x^{3} + 716221644 x^{2} - 267857268 x + 33573121 \)

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:  \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.29$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(769,·)$, $\chi_{840}(659,·)$, $\chi_{840}(533,·)$, $\chi_{840}(797,·)$, $\chi_{840}(799,·)$, $\chi_{840}(673,·)$, $\chi_{840}(419,·)$, $\chi_{840}(293,·)$, $\chi_{840}(169,·)$, $\chi_{840}(391,·)$, $\chi_{840}(559,·)$, $\chi_{840}(433,·)$, $\chi_{840}(827,·)$, $\chi_{840}(701,·)$, $\chi_{840}(323,·)$, $\chi_{840}(197,·)$, $\chi_{840}(587,·)$, $\chi_{840}(461,·)$, $\chi_{840}(463,·)$, $\chi_{840}(337,·)$, $\chi_{840}(83,·)$, $\chi_{840}(727,·)$, $\chi_{840}(601,·)$, $\chi_{840}(223,·)$, $\chi_{840}(97,·)$, $\chi_{840}(491,·)$, $\chi_{840}(29,·)$, $\chi_{840}(629,·)$, $\chi_{840}(631,·)$, $\chi_{840}(251,·)$, $\chi_{840}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{88} a^{20} + \frac{5}{11} a^{19} + \frac{15}{44} a^{18} - \frac{1}{2} a^{17} + \frac{3}{11} a^{16} - \frac{1}{11} a^{15} - \frac{7}{22} a^{14} - \frac{1}{2} a^{13} + \frac{15}{88} a^{12} - \frac{2}{11} a^{11} + \frac{3}{22} a^{10} + \frac{9}{22} a^{9} - \frac{21}{44} a^{8} - \frac{4}{11} a^{7} + \frac{1}{44} a^{6} - \frac{2}{11} a^{5} + \frac{15}{88} a^{4} + \frac{3}{11} a^{3} + \frac{1}{22} a^{2} + \frac{3}{22} a + \frac{9}{88}$, $\frac{1}{88} a^{21} + \frac{7}{44} a^{19} - \frac{3}{22} a^{18} + \frac{3}{11} a^{17} + \frac{7}{22} a^{15} + \frac{5}{22} a^{14} + \frac{15}{88} a^{13} + \frac{9}{22} a^{11} - \frac{1}{22} a^{10} + \frac{7}{44} a^{9} - \frac{3}{11} a^{8} - \frac{19}{44} a^{7} - \frac{1}{11} a^{6} + \frac{39}{88} a^{5} + \frac{5}{11} a^{4} + \frac{3}{22} a^{3} + \frac{7}{22} a^{2} - \frac{31}{88} a - \frac{1}{11}$, $\frac{1}{88} a^{22} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{3}{8} a^{14} + \frac{1}{44} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{8} + \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{88} a^{2} - \frac{19}{44}$, $\frac{1}{88} a^{23} - \frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{3}{8} a^{15} + \frac{1}{44} a^{13} + \frac{1}{4} a^{11} + \frac{1}{4} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{88} a^{3} - \frac{19}{44} a - \frac{1}{2}$, $\frac{1}{3520} a^{24} + \frac{1}{220} a^{23} - \frac{9}{1760} a^{22} + \frac{1}{880} a^{21} - \frac{3}{704} a^{20} + \frac{19}{55} a^{19} - \frac{711}{1760} a^{18} - \frac{26}{55} a^{17} + \frac{203}{704} a^{16} - \frac{59}{440} a^{15} + \frac{3}{22} a^{14} - \frac{27}{440} a^{13} - \frac{723}{3520} a^{12} + \frac{79}{220} a^{11} - \frac{813}{1760} a^{10} + \frac{7}{80} a^{9} + \frac{1029}{3520} a^{8} + \frac{39}{88} a^{7} - \frac{273}{1760} a^{6} - \frac{1}{16} a^{5} + \frac{201}{440} a^{4} + \frac{43}{110} a^{3} + \frac{373}{880} a^{2} - \frac{393}{880} a + \frac{91}{320}$, $\frac{1}{3520} a^{25} + \frac{3}{1760} a^{23} + \frac{3}{880} a^{22} + \frac{1}{3520} a^{21} + \frac{1}{220} a^{20} + \frac{41}{1760} a^{19} - \frac{3}{55} a^{18} - \frac{361}{3520} a^{17} + \frac{191}{440} a^{16} + \frac{29}{440} a^{15} + \frac{16}{55} a^{14} + \frac{973}{3520} a^{13} + \frac{7}{20} a^{12} + \frac{13}{32} a^{11} - \frac{239}{880} a^{10} + \frac{821}{3520} a^{9} - \frac{153}{440} a^{8} - \frac{23}{160} a^{7} - \frac{401}{880} a^{6} + \frac{61}{440} a^{5} - \frac{87}{220} a^{4} + \frac{179}{880} a^{3} - \frac{271}{880} a^{2} - \frac{727}{3520} a + \frac{6}{55}$, $\frac{1}{3520} a^{26} - \frac{1}{880} a^{23} - \frac{1}{320} a^{22} - \frac{1}{440} a^{21} + \frac{3}{880} a^{20} - \frac{49}{110} a^{19} + \frac{1611}{3520} a^{18} + \frac{119}{440} a^{17} - \frac{449}{1760} a^{16} + \frac{23}{110} a^{15} - \frac{507}{3520} a^{14} - \frac{13}{55} a^{13} - \frac{49}{440} a^{12} + \frac{53}{176} a^{11} - \frac{93}{320} a^{10} - \frac{1}{110} a^{9} + \frac{23}{88} a^{8} - \frac{361}{880} a^{7} - \frac{349}{880} a^{6} + \frac{91}{440} a^{5} - \frac{413}{880} a^{4} - \frac{39}{176} a^{3} + \frac{11}{320} a^{2} + \frac{167}{440} a + \frac{317}{1760}$, $\frac{1}{3520} a^{27} + \frac{13}{3520} a^{23} - \frac{3}{880} a^{21} + \frac{3}{880} a^{20} - \frac{129}{704} a^{19} - \frac{51}{220} a^{18} - \frac{67}{160} a^{17} + \frac{39}{880} a^{16} - \frac{247}{704} a^{15} + \frac{63}{220} a^{14} - \frac{1}{20} a^{13} - \frac{107}{220} a^{12} + \frac{113}{3520} a^{11} + \frac{123}{440} a^{10} - \frac{1}{40} a^{9} - \frac{2}{55} a^{8} - \frac{199}{880} a^{7} - \frac{31}{220} a^{6} - \frac{323}{880} a^{5} - \frac{317}{880} a^{4} - \frac{259}{704} a^{3} - \frac{157}{440} a^{2} - \frac{37}{160} a + \frac{51}{880}$, $\frac{1}{28160} a^{28} - \frac{1}{14080} a^{26} - \frac{1}{7040} a^{25} + \frac{1}{14080} a^{24} - \frac{7}{1760} a^{23} - \frac{1}{352} a^{22} + \frac{3}{1408} a^{21} - \frac{1}{3520} a^{20} + \frac{1077}{3520} a^{19} + \frac{3217}{14080} a^{18} - \frac{147}{352} a^{17} + \frac{61}{7040} a^{16} - \frac{203}{704} a^{15} - \frac{2069}{14080} a^{14} + \frac{1709}{7040} a^{13} + \frac{2401}{14080} a^{12} - \frac{1}{64} a^{11} - \frac{843}{7040} a^{10} - \frac{81}{1760} a^{9} - \frac{1059}{28160} a^{8} + \frac{107}{320} a^{7} - \frac{4679}{14080} a^{6} - \frac{9}{40} a^{5} - \frac{10831}{28160} a^{4} + \frac{459}{1760} a^{3} - \frac{1343}{7040} a^{2} + \frac{333}{7040} a - \frac{2171}{5632}$, $\frac{1}{28160} a^{29} - \frac{1}{14080} a^{27} - \frac{1}{7040} a^{26} + \frac{1}{14080} a^{25} + \frac{7}{1760} a^{23} - \frac{9}{7040} a^{22} + \frac{3}{704} a^{21} - \frac{13}{3520} a^{20} + \frac{5713}{14080} a^{19} - \frac{769}{1760} a^{18} - \frac{2691}{7040} a^{17} + \frac{175}{704} a^{16} - \frac{6581}{14080} a^{15} - \frac{2291}{7040} a^{14} + \frac{7}{256} a^{13} + \frac{1743}{3520} a^{12} + \frac{159}{640} a^{11} + \frac{57}{1760} a^{10} - \frac{6243}{28160} a^{9} - \frac{1057}{3520} a^{8} - \frac{4519}{14080} a^{7} - \frac{49}{880} a^{6} - \frac{12751}{28160} a^{5} - \frac{17}{352} a^{4} + \frac{125}{1408} a^{3} - \frac{1891}{7040} a^{2} - \frac{3559}{28160} a - \frac{43}{160}$, $\frac{1}{667687331267506226733534212910517760} a^{30} + \frac{639640472611402493205949668027}{667687331267506226733534212910517760} a^{29} + \frac{6852011160397083953998879773153}{667687331267506226733534212910517760} a^{28} + \frac{43054716001837956704641547187627}{333843665633753113366767106455258880} a^{27} + \frac{9528973194155781326107425761169}{83460916408438278341691776613814720} a^{26} + \frac{43660292337243953291659465237053}{333843665633753113366767106455258880} a^{25} + \frac{4473949306167667481351511099771}{66768733126750622673353421291051776} a^{24} + \frac{100009823749501066187641676976843}{166921832816876556683383553227629440} a^{23} - \frac{20132975944190624522083212082059}{33384366563375311336676710645525888} a^{22} + \frac{365898407511294564082041207901193}{166921832816876556683383553227629440} a^{21} - \frac{294611494500254321256008641976863}{66768733126750622673353421291051776} a^{20} + \frac{62789635443414198049536118768603}{3305382828056961518482842638170880} a^{19} + \frac{7000290488162396047271464846653697}{333843665633753113366767106455258880} a^{18} - \frac{61320905456576569810755865308879851}{166921832816876556683383553227629440} a^{17} + \frac{117987733266637263304543451177262681}{333843665633753113366767106455258880} a^{16} + \frac{128957707310024037792896133533088919}{333843665633753113366767106455258880} a^{15} - \frac{595887950049310047314022310681481}{16692183281687655668338355322762944} a^{14} - \frac{96355968604705194603283279603354643}{333843665633753113366767106455258880} a^{13} - \frac{55063112340461150787554811491999523}{333843665633753113366767106455258880} a^{12} + \frac{73465491067592088841942766869503093}{166921832816876556683383553227629440} a^{11} + \frac{63991508050429809774221296025664861}{133537466253501245346706842582103552} a^{10} + \frac{307065590649235114585449327762658727}{667687331267506226733534212910517760} a^{9} + \frac{283386834831245058423770201309563273}{667687331267506226733534212910517760} a^{8} + \frac{122125869955318016294196893055095}{661076565611392303696568527634176} a^{7} - \frac{263119368742977333651882584916742313}{667687331267506226733534212910517760} a^{6} - \frac{151160230965930474499356632297309221}{667687331267506226733534212910517760} a^{5} - \frac{23606552656588468895043017876661977}{667687331267506226733534212910517760} a^{4} + \frac{5501731141847063401576173140117263}{20865229102109569585422944153453680} a^{3} - \frac{163450316170255660687898758457001543}{667687331267506226733534212910517760} a^{2} + \frac{4196161945048236190909542318300349}{60698848297046020612139473900956160} a - \frac{10400836516545186009382661384184601}{133537466253501245346706842582103552}$, $\frac{1}{5271211407702432777631559435004617112006877304300230933304630658335733104537133003703980068533760} a^{31} + \frac{10617833466331891966427289159719427160593016322802597012571}{658901425962804097203944929375577139000859663037528866663078832291966638067141625462997508566720} a^{30} + \frac{8239866413467637002250095711526312592032063058208178826205063588666697115360282965576819243}{1054242281540486555526311887000923422401375460860046186660926131667146620907426600740796013706752} a^{29} + \frac{1266519997264174892595749535123862274487607646071961162701309066521880005633799708270503297}{239600518531928762619616337954755323273039877468192315150210484469806050206233318350180912206080} a^{28} + \frac{7092951278073017771545458044047370031640017765063069727472846327038956367894231327677460363}{59900129632982190654904084488688830818259969367048078787552621117451512551558329587545228051520} a^{27} + \frac{44538745316945234354968646530257614928181732392543222099687229192559154102850664794953812577}{329450712981402048601972464687788569500429831518764433331539416145983319033570812731498754283360} a^{26} - \frac{24300740928325419565883944194583530909619680239674883434327687619443961348709002616844754719}{2635605703851216388815779717502308556003438652150115466652315329167866552268566501851990034266880} a^{25} + \frac{111962810157829989950990009711762935248713282287145150113983485950716302741754331263177327}{131780285192560819440788985875115427800171932607505773332615766458393327613428325092599501713344} a^{24} + \frac{1755205262679776551031010847025417362417997669685301823174798228514121918634921440634937402443}{329450712981402048601972464687788569500429831518764433331539416145983319033570812731498754283360} a^{23} + \frac{378022144340026267636895585225182995282381967413669934541146218094687468433492427869598592483}{263560570385121638881577971750230855600343865215011546665231532916786655226856650185199003426688} a^{22} + \frac{3086994422378775128842432407119181708621964344412032626015797682994581422666808115049659507261}{2635605703851216388815779717502308556003438652150115466652315329167866552268566501851990034266880} a^{21} - \frac{1304505646145761078423245751359579689973071926252907979989742400800935102411700196913218301659}{329450712981402048601972464687788569500429831518764433331539416145983319033570812731498754283360} a^{20} + \frac{85859767199202053563233427251835746492334154931616943198512377963016124475143516007096439147499}{527121140770243277763155943500461711200687730430023093330463065833573310453713300370398006853376} a^{19} - \frac{60945785266052604261190381609169208289179406515451063772273890874703029975594944800872090686907}{1317802851925608194407889858751154278001719326075057733326157664583933276134283250925995017133440} a^{18} - \frac{307714090018952051034664861005443694455490509918159152038247928457903048219282721248559236119319}{2635605703851216388815779717502308556003438652150115466652315329167866552268566501851990034266880} a^{17} + \frac{34224232115777008486809164314387550997361054857054771352513991112742704777572296924808734787131}{1317802851925608194407889858751154278001719326075057733326157664583933276134283250925995017133440} a^{16} - \frac{69671296712516174235570022728950052509550856581219219370825939029626645830480063034949760499279}{329450712981402048601972464687788569500429831518764433331539416145983319033570812731498754283360} a^{15} - \frac{37584375748782362802039766742927412265425598750241802609798986442395364592439948329914214335903}{329450712981402048601972464687788569500429831518764433331539416145983319033570812731498754283360} a^{14} + \frac{613617065222286311713374128796232086268543815196329610111153529279919922750765173268754271885147}{2635605703851216388815779717502308556003438652150115466652315329167866552268566501851990034266880} a^{13} - \frac{316993798333902995053146081838491801696710669699464543383138425502968504896901081051891233896079}{1317802851925608194407889858751154278001719326075057733326157664583933276134283250925995017133440} a^{12} - \frac{28223612822555896413377341486937014216757433153034912987651635477142323698989247331933939706103}{5271211407702432777631559435004617112006877304300230933304630658335733104537133003703980068533760} a^{11} - \frac{82917523801357779544552810298983495786533294911342352253796017811358400242469408951088184867337}{329450712981402048601972464687788569500429831518764433331539416145983319033570812731498754283360} a^{10} - \frac{118851527300572236115557961547896344990699388960101687530002159837599150922621889060073397472721}{5271211407702432777631559435004617112006877304300230933304630658335733104537133003703980068533760} a^{9} - \frac{1021306309924903043449015022285693050972420101188608560164218188015053183381420961579575503886567}{2635605703851216388815779717502308556003438652150115466652315329167866552268566501851990034266880} a^{8} + \frac{139468434336816230442209090884600107024226980519660315342809612301020479305508619948312264042295}{1054242281540486555526311887000923422401375460860046186660926131667146620907426600740796013706752} a^{7} - \frac{567862153431498081000883614451931372291192569080920066580878342557570310468733358731194924733549}{1317802851925608194407889858751154278001719326075057733326157664583933276134283250925995017133440} a^{6} - \frac{179327352290363042802213934253645350464459564408743066893213213749272481759079540246824590782679}{1054242281540486555526311887000923422401375460860046186660926131667146620907426600740796013706752} a^{5} + \frac{1202640808137284930735716801745143887578946555456256786293783064437653655162555367362728441966423}{2635605703851216388815779717502308556003438652150115466652315329167866552268566501851990034266880} a^{4} + \frac{1612598724761885477712367278152257167953413924794762850123125822423159374733174675866119201265109}{5271211407702432777631559435004617112006877304300230933304630658335733104537133003703980068533760} a^{3} + \frac{86820840606070827577430846606375873946131937202842403655099136050845425776948736052957407267351}{1317802851925608194407889858751154278001719326075057733326157664583933276134283250925995017133440} a^{2} + \frac{2617248758706448386503404157005786983226425959320423152891585713529938134896504083591422975888913}{5271211407702432777631559435004617112006877304300230933304630658335733104537133003703980068533760} a - \frac{212683934308684372201729404919173346889570920417042368513103263619765033917955213578282558430543}{2635605703851216388815779717502308556003438652150115466652315329167866552268566501851990034266880}$

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

Class group and class number

Not computed

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{1379048785597311160144601281719717094222909097970580942874544271108600947160968907}{3271210377088458329510103785524254009250623905480588249323933161330834192764530479980032} a^{31} + \frac{2957243617242623000729110557280239777669442911918066326316085715264570517497394927}{16356051885442291647550518927621270046253119527402941246619665806654170963822652399900160} a^{30} + \frac{42639579327734653828467037656777760407557102445088693895186501267541875780836319319}{16356051885442291647550518927621270046253119527402941246619665806654170963822652399900160} a^{29} - \frac{8064888526658049955636727046847925538536849385656842188742485899289866896255271949}{2044506485680286455943814865952658755781639940925367655827458225831771370477831549987520} a^{28} + \frac{3749388722044577074437952850160574347423638229700285591238069450486662398331815129}{102225324284014322797190743297632937789081997046268382791372911291588568523891577499376} a^{27} - \frac{106700408478920528385072762344055453719465928786883091308701881003392891042204118697}{1635605188544229164755051892762127004625311952740294124661966580665417096382265239990016} a^{26} + \frac{702891699587560357913059936348331327477990688990778530372159096192155553132165188721}{8178025942721145823775259463810635023126559763701470623309832903327085481911326199950080} a^{25} - \frac{7235357976980370539969928926286676973010932106435154645425371745432370417565210373}{511126621420071613985953716488164688945409985231341913956864556457942842619457887496880} a^{24} + \frac{6176778701129324782479855540076406044049088005911699717645629407337405867174920694621}{4089012971360572911887629731905317511563279881850735311654916451663542740955663099975040} a^{23} + \frac{4334983526400548984123460722555478173087091404883478843558641382331118686615192867379}{4089012971360572911887629731905317511563279881850735311654916451663542740955663099975040} a^{22} + \frac{14077890937816622583550254275393050913173334897203186406887208353539433512798214511179}{8178025942721145823775259463810635023126559763701470623309832903327085481911326199950080} a^{21} - \frac{251230162396821241633817856361211987772652346404705200545403425873359233185509318012737}{8178025942721145823775259463810635023126559763701470623309832903327085481911326199950080} a^{20} - \frac{87037301070216411731627338594053309034901377442817074874717239088869223993791816498541}{1635605188544229164755051892762127004625311952740294124661966580665417096382265239990016} a^{19} - \frac{599656853845877566894755003976322775795705771890437552766180944966157091362649230202069}{2044506485680286455943814865952658755781639940925367655827458225831771370477831549987520} a^{18} + \frac{1128389876029418723748417636241848373612079021206798262130326064655684514398532380518243}{8178025942721145823775259463810635023126559763701470623309832903327085481911326199950080} a^{17} - \frac{16501147566266966692645855433189538066565467195847560693426775458320404691852891960482041}{8178025942721145823775259463810635023126559763701470623309832903327085481911326199950080} a^{16} + \frac{7191010098341382454457794550799172855661589051942783502563108715263172082323847298128157}{1022253242840143227971907432976329377890819970462683827913729112915885685238915774993760} a^{15} + \frac{44085704218035649926824803889880649481722818333520567541440567113036697257971157905656731}{8178025942721145823775259463810635023126559763701470623309832903327085481911326199950080} a^{14} + \frac{158770199976962699800266045659777806497195341518005430192139439225568500998954133790602891}{8178025942721145823775259463810635023126559763701470623309832903327085481911326199950080} a^{13} + \frac{9723359725416437839840805747017286523481896893126613303444524565121106750140773873179427}{511126621420071613985953716488164688945409985231341913956864556457942842619457887496880} a^{12} - \frac{466431091579257338201121550108365288206405299312737031594389962685692338523292910713132853}{3271210377088458329510103785524254009250623905480588249323933161330834192764530479980032} a^{11} - \frac{162868247108672423532258845009337587953120887918369801047395193754634723881053788700792919}{1486913807767481058868228993420115458750283593400267386056333255150379178529332036354560} a^{10} - \frac{127286237466977229588915233308396400489973778902608696409526378303316789573973521771692971}{1486913807767481058868228993420115458750283593400267386056333255150379178529332036354560} a^{9} - \frac{1289407637211437649524664223811040283962981267326566865694441069737075851243845590343117}{408901297136057291188762973190531751156327988185073531165491645166354274095566309997504} a^{8} + \frac{12145852492919824504455638804242436641625641448043272490897554749092624948448457937794639217}{16356051885442291647550518927621270046253119527402941246619665806654170963822652399900160} a^{7} + \frac{594332363693616006853132325391469415890184598884941042098540259771690628760674109888404169}{1486913807767481058868228993420115458750283593400267386056333255150379178529332036354560} a^{6} - \frac{2702630662518580957824563754804655234542507070560113635343949922742395171641500217625324959}{16356051885442291647550518927621270046253119527402941246619665806654170963822652399900160} a^{5} + \frac{2820095845136375055965854498453023704241654195729948289724167296934893184663619785671542421}{8178025942721145823775259463810635023126559763701470623309832903327085481911326199950080} a^{4} - \frac{2522167250190201255525601079129340065908149240923151890411922587270732962062151581332953257}{16356051885442291647550518927621270046253119527402941246619665806654170963822652399900160} a^{3} - \frac{4439369335082767819765733115605451346331254601073731590083221256646923595543637438496231789}{16356051885442291647550518927621270046253119527402941246619665806654170963822652399900160} a^{2} + \frac{3017836955697091009604372087184049654333546140418720454440188984706320260257786004076348693}{16356051885442291647550518927621270046253119527402941246619665806654170963822652399900160} a - \frac{50883005711248783967729435922905412125301703454699113289031575159441830207455540333825351}{1635605188544229164755051892762127004625311952740294124661966580665417096382265239990016} \) (order $20$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3\times C_4$ (as 32T34):

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 $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-210}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-30}) \), \(\Q(i, \sqrt{35})\), \(\Q(i, \sqrt{210})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{35})\), \(\Q(\sqrt{6}, \sqrt{35})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{42})\), \(\Q(i, \sqrt{30})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(\sqrt{30}, \sqrt{-35})\), \(\Q(\sqrt{-30}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{-5}, \sqrt{-7})\), \(\Q(\sqrt{-30}, \sqrt{35})\), \(\Q(\sqrt{30}, \sqrt{35})\), \(\Q(\sqrt{5}, \sqrt{-42})\), \(\Q(\sqrt{-5}, \sqrt{42})\), \(\Q(\sqrt{-7}, \sqrt{30})\), \(\Q(\sqrt{7}, \sqrt{-30})\), \(\Q(\sqrt{5}, \sqrt{42})\), \(\Q(\sqrt{-5}, \sqrt{-42})\), \(\Q(\sqrt{-7}, \sqrt{-30})\), \(\Q(\sqrt{7}, \sqrt{30})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{7})\), 4.0.3528000.1, 4.4.3528000.1, 4.4.72000.1, 4.0.72000.2, \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), 4.0.98000.1, 4.4.6125.1, 8.0.7965941760000.54, 8.0.384160000.1, 8.0.7965941760000.50, 8.0.7965941760000.68, 8.0.7965941760000.65, 8.0.3317760000.9, 8.0.12745506816.7, 8.0.497871360000.15, 8.0.7965941760000.31, 8.0.497871360000.9, 8.0.7965941760000.36, 8.0.7965941760000.59, 8.0.7965941760000.56, 8.8.7965941760000.9, 8.0.7965941760000.45, 8.0.199148544000000.177, 8.0.82944000000.7, \(\Q(\zeta_{20})\), 8.0.9604000000.1, 8.0.12446784000000.15, 8.0.12446784000000.10, 8.0.9604000000.2, 8.0.37515625.1, 8.0.199148544000000.157, 8.8.199148544000000.2, 8.8.9604000000.1, 8.0.9604000000.3, 8.0.199148544000000.160, 8.0.12446784000000.19, 8.0.199148544000000.142, 8.0.12446784000000.2, 8.0.12446784000000.20, 8.8.199148544000000.7, 8.8.12446784000000.4, 8.0.199148544000000.140, 8.0.199148544000000.168, 8.8.12446784000000.1, 8.8.82944000000.2, 8.0.5184000000.6, 8.0.12446784000000.6, 8.0.199148544000000.67, 8.0.5184000000.5, 8.0.82944000000.2, 16.0.63456228123711897600000000.13, 16.0.39660142577319936000000000000.41, 16.0.92236816000000000000.1, 16.0.39660142577319936000000000000.66, 16.0.39660142577319936000000000000.63, 16.0.39660142577319936000000000000.22, 16.0.6879707136000000000000.9, 16.0.39660142577319936000000000000.34, 16.0.154922431942656000000000000.10, 16.0.154922431942656000000000000.8, 16.0.39660142577319936000000000000.70, 16.0.39660142577319936000000000000.51, 16.0.39660142577319936000000000000.54, 16.0.39660142577319936000000000000.55, 16.16.39660142577319936000000000000.2

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 R ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
2Data not computed
$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}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$