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Properties

Label 32.0.32119747989...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 5^{24}\cdot 29^{24}$
Root discriminant $72.37$
Ramified primes $3, 5, 29$
Class number $9984$ (GRH)
Class group $[2, 2, 2, 2, 2, 312]$ (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![78310985281, -68096508940, 13619301788, 22771781534, -19357441493, 4118821508, 2730384303, -1973323893, 481110280, -319900184, -79724373, 326655828, -282513946, 52874360, 72973719, -57696127, 11338251, 5123635, -2094690, 347291, 616564, -135703, 13885, 19174, -3930, -1873, 958, -158, -26, 29, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + 4*x^30 + 29*x^29 - 26*x^28 - 158*x^27 + 958*x^26 - 1873*x^25 - 3930*x^24 + 19174*x^23 + 13885*x^22 - 135703*x^21 + 616564*x^20 + 347291*x^19 - 2094690*x^18 + 5123635*x^17 + 11338251*x^16 - 57696127*x^15 + 72973719*x^14 + 52874360*x^13 - 282513946*x^12 + 326655828*x^11 - 79724373*x^10 - 319900184*x^9 + 481110280*x^8 - 1973323893*x^7 + 2730384303*x^6 + 4118821508*x^5 - 19357441493*x^4 + 22771781534*x^3 + 13619301788*x^2 - 68096508940*x + 78310985281)
 
gp: K = bnfinit(x^32 - x^31 + 4*x^30 + 29*x^29 - 26*x^28 - 158*x^27 + 958*x^26 - 1873*x^25 - 3930*x^24 + 19174*x^23 + 13885*x^22 - 135703*x^21 + 616564*x^20 + 347291*x^19 - 2094690*x^18 + 5123635*x^17 + 11338251*x^16 - 57696127*x^15 + 72973719*x^14 + 52874360*x^13 - 282513946*x^12 + 326655828*x^11 - 79724373*x^10 - 319900184*x^9 + 481110280*x^8 - 1973323893*x^7 + 2730384303*x^6 + 4118821508*x^5 - 19357441493*x^4 + 22771781534*x^3 + 13619301788*x^2 - 68096508940*x + 78310985281, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} + 4 x^{30} + 29 x^{29} - 26 x^{28} - 158 x^{27} + 958 x^{26} - 1873 x^{25} - 3930 x^{24} + 19174 x^{23} + 13885 x^{22} - 135703 x^{21} + 616564 x^{20} + 347291 x^{19} - 2094690 x^{18} + 5123635 x^{17} + 11338251 x^{16} - 57696127 x^{15} + 72973719 x^{14} + 52874360 x^{13} - 282513946 x^{12} + 326655828 x^{11} - 79724373 x^{10} - 319900184 x^{9} + 481110280 x^{8} - 1973323893 x^{7} + 2730384303 x^{6} + 4118821508 x^{5} - 19357441493 x^{4} + 22771781534 x^{3} + 13619301788 x^{2} - 68096508940 x + 78310985281 \)

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:  \(321197479890852368431467961209258857295274794101715087890625=3^{16}\cdot 5^{24}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.37$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(128,·)$, $\chi_{435}(1,·)$, $\chi_{435}(133,·)$, $\chi_{435}(262,·)$, $\chi_{435}(394,·)$, $\chi_{435}(17,·)$, $\chi_{435}(146,·)$, $\chi_{435}(46,·)$, $\chi_{435}(278,·)$, $\chi_{435}(407,·)$, $\chi_{435}(28,·)$, $\chi_{435}(157,·)$, $\chi_{435}(389,·)$, $\chi_{435}(289,·)$, $\chi_{435}(418,·)$, $\chi_{435}(41,·)$, $\chi_{435}(173,·)$, $\chi_{435}(302,·)$, $\chi_{435}(434,·)$, $\chi_{435}(307,·)$, $\chi_{435}(59,·)$, $\chi_{435}(191,·)$, $\chi_{435}(202,·)$, $\chi_{435}(331,·)$, $\chi_{435}(86,·)$, $\chi_{435}(88,·)$, $\chi_{435}(347,·)$, $\chi_{435}(349,·)$, $\chi_{435}(104,·)$, $\chi_{435}(233,·)$, $\chi_{435}(244,·)$, $\chi_{435}(376,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{14} a^{20} + \frac{1}{14} a^{19} - \frac{1}{14} a^{15} - \frac{3}{7} a^{13} - \frac{3}{7} a^{10} - \frac{3}{7} a^{7} - \frac{1}{14} a^{5} - \frac{1}{2} a^{4} - \frac{3}{7} a + \frac{1}{14}$, $\frac{1}{14} a^{21} - \frac{1}{14} a^{19} - \frac{1}{14} a^{16} + \frac{1}{14} a^{15} - \frac{3}{7} a^{14} + \frac{3}{7} a^{13} - \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{14} a^{6} - \frac{3}{7} a^{5} - \frac{1}{2} a^{4} - \frac{3}{7} a^{2} - \frac{1}{2} a - \frac{1}{14}$, $\frac{1}{14} a^{22} + \frac{1}{14} a^{19} - \frac{1}{14} a^{17} + \frac{1}{14} a^{16} + \frac{3}{7} a^{14} - \frac{3}{7} a^{13} - \frac{3}{7} a^{12} + \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{2} a^{7} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} - \frac{1}{2} a^{4} - \frac{3}{7} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{3}{7}$, $\frac{1}{14} a^{23} - \frac{1}{14} a^{19} - \frac{1}{14} a^{18} + \frac{1}{14} a^{17} - \frac{3}{7} a^{14} + \frac{3}{7} a^{12} - \frac{3}{7} a^{11} + \frac{3}{7} a^{9} - \frac{1}{2} a^{8} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{14} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{3}{7}$, $\frac{1}{14} a^{24} + \frac{1}{14} a^{18} - \frac{3}{7} a^{12} - \frac{1}{2} a^{9} - \frac{3}{7} a^{6} - \frac{1}{2} a^{3} - \frac{3}{7}$, $\frac{1}{322} a^{25} - \frac{1}{322} a^{24} + \frac{2}{161} a^{23} + \frac{3}{161} a^{22} - \frac{3}{322} a^{21} + \frac{3}{322} a^{20} - \frac{27}{161} a^{19} - \frac{33}{322} a^{18} + \frac{13}{161} a^{17} - \frac{27}{161} a^{16} - \frac{38}{161} a^{15} - \frac{13}{161} a^{14} + \frac{59}{161} a^{13} + \frac{53}{161} a^{12} + \frac{29}{161} a^{11} - \frac{121}{322} a^{10} + \frac{79}{322} a^{9} + \frac{20}{161} a^{8} - \frac{1}{23} a^{7} - \frac{87}{322} a^{6} - \frac{1}{322} a^{5} - \frac{54}{161} a^{4} - \frac{71}{322} a^{3} - \frac{19}{161} a^{2} + \frac{72}{161} a$, $\frac{1}{18232730248852} a^{26} + \frac{438279409}{9116365124426} a^{25} - \frac{275063002699}{9116365124426} a^{24} + \frac{54499658823}{2604675749836} a^{23} - \frac{437835280699}{18232730248852} a^{22} + \frac{28399038369}{1302337874918} a^{21} + \frac{938838947}{18232730248852} a^{20} - \frac{290052861061}{1302337874918} a^{19} - \frac{903064660467}{18232730248852} a^{18} - \frac{1710306293677}{18232730248852} a^{17} - \frac{3125446477585}{18232730248852} a^{16} - \frac{2742523248293}{18232730248852} a^{15} - \frac{731337624861}{9116365124426} a^{14} - \frac{84427920622}{651168937459} a^{13} - \frac{3180044616309}{9116365124426} a^{12} - \frac{7908689525067}{18232730248852} a^{11} - \frac{269769287750}{4558182562213} a^{10} - \frac{213799708881}{4558182562213} a^{9} + \frac{394910983627}{18232730248852} a^{8} - \frac{7262666559941}{18232730248852} a^{7} + \frac{294138800143}{9116365124426} a^{6} - \frac{5765957982629}{18232730248852} a^{5} + \frac{434600737579}{1302337874918} a^{4} - \frac{5313898757445}{18232730248852} a^{3} + \frac{4678841739083}{18232730248852} a^{2} - \frac{41621522083}{792727402124} a - \frac{13900677215}{34466408788}$, $\frac{1}{419352795723596} a^{27} - \frac{1}{419352795723596} a^{26} - \frac{296294638}{14976885561557} a^{25} - \frac{13773281294559}{419352795723596} a^{24} - \frac{44882040117}{209676397861798} a^{23} + \frac{825770485}{330980896388} a^{22} + \frac{11836440297015}{419352795723596} a^{21} + \frac{13916537354605}{419352795723596} a^{20} + \frac{15268314639071}{419352795723596} a^{19} - \frac{2636247469528}{14976885561557} a^{18} - \frac{1404960638695}{209676397861798} a^{17} + \frac{22646267348459}{104838198930899} a^{16} - \frac{8987723715721}{59907542246228} a^{15} - \frac{5949401986431}{29953771123114} a^{14} + \frac{80045921376341}{209676397861798} a^{13} + \frac{16911403381433}{59907542246228} a^{12} + \frac{127842388737225}{419352795723596} a^{11} - \frac{31269012288190}{104838198930899} a^{10} + \frac{31134656645845}{419352795723596} a^{9} + \frac{98685117019459}{209676397861798} a^{8} + \frac{61058031348357}{419352795723596} a^{7} - \frac{74936928177465}{419352795723596} a^{6} + \frac{157972127493645}{419352795723596} a^{5} - \frac{171020158471739}{419352795723596} a^{4} - \frac{62487423472673}{209676397861798} a^{3} - \frac{3683999415167}{9116365124426} a^{2} - \frac{131522416335}{396363701062} a + \frac{7202301611}{34466408788}$, $\frac{1}{33516772198208410300} a^{28} + \frac{3093}{16758386099104205150} a^{27} - \frac{6183}{33516772198208410300} a^{26} - \frac{839521600161873}{33516772198208410300} a^{25} - \frac{410983111451372053}{33516772198208410300} a^{24} - \frac{879004505325967877}{33516772198208410300} a^{23} + \frac{3139130384314553}{478811031402977290} a^{22} - \frac{229106409807794299}{8379193049552102575} a^{21} + \frac{56359174641616454}{1675838609910420515} a^{20} + \frac{1523637983732806547}{6703354439641682060} a^{19} + \frac{630058950820388231}{16758386099104205150} a^{18} - \frac{1521803773126076889}{16758386099104205150} a^{17} - \frac{46909358093229885}{191524412561190916} a^{16} - \frac{4982066559283365151}{33516772198208410300} a^{15} - \frac{2998930403735207467}{8379193049552102575} a^{14} + \frac{12221861261660751921}{33516772198208410300} a^{13} - \frac{8157464526638458929}{16758386099104205150} a^{12} + \frac{2989237592494626887}{6703354439641682060} a^{11} - \frac{3409568077275172173}{33516772198208410300} a^{10} - \frac{487819492177589123}{4788110314029772900} a^{9} - \frac{8147627361350239323}{33516772198208410300} a^{8} + \frac{1585628262024012333}{8379193049552102575} a^{7} - \frac{4005844313632242208}{8379193049552102575} a^{6} - \frac{4090305542878458987}{16758386099104205150} a^{5} + \frac{7322982993118659599}{33516772198208410300} a^{4} - \frac{100972999559244991}{364312741284874025} a^{3} - \frac{96872220208282}{15839684403690175} a^{2} + \frac{379102752833529}{2754727722380900} a - \frac{58193666922969}{119770770538300}$, $\frac{1}{56054453226947729164989498755595882645500} a^{29} - \frac{180108288848797642143}{14013613306736932291247374688898970661375} a^{28} + \frac{58323962340877748089807829}{56054453226947729164989498755595882645500} a^{27} - \frac{58326123640343933661513509}{56054453226947729164989498755595882645500} a^{26} - \frac{1614087871247732321105429872337934569}{56054453226947729164989498755595882645500} a^{25} + \frac{227101003130949506220926956855312165621}{8007779032421104166427071250799411806500} a^{24} - \frac{34820952427434991450298593369255812256}{14013613306736932291247374688898970661375} a^{23} - \frac{311843320518122639162697473817721412219}{14013613306736932291247374688898970661375} a^{22} + \frac{980470521227573964231743684674983474999}{28027226613473864582494749377797941322750} a^{21} + \frac{33593425567265414534359325005454577799}{11210890645389545832997899751119176529100} a^{20} + \frac{1171531464165699948450058112980500427491}{28027226613473864582494749377797941322750} a^{19} + \frac{4341973725414923109987599786110852873563}{28027226613473864582494749377797941322750} a^{18} - \frac{1704889551970809884283083688704840596843}{8007779032421104166427071250799411806500} a^{17} + \frac{547525679558462100504068261607192126407}{8007779032421104166427071250799411806500} a^{16} - \frac{1351328416302654960094315360195036956081}{5605445322694772916498949875559588264550} a^{15} - \frac{3429244090516797459228572052182719563947}{11210890645389545832997899751119176529100} a^{14} - \frac{66814366241187244322251744546117798669}{14013613306736932291247374688898970661375} a^{13} - \frac{25941886496113990063854838797001702939401}{56054453226947729164989498755595882645500} a^{12} - \frac{11088547081865044054506957440324221213253}{56054453226947729164989498755595882645500} a^{11} - \frac{3786256623099723189830868179862087531111}{8007779032421104166427071250799411806500} a^{10} - \frac{240082238261451089763128225805019881847}{11210890645389545832997899751119176529100} a^{9} + \frac{576266368903312568460593288180200149683}{28027226613473864582494749377797941322750} a^{8} + \frac{2244430333176460910709853298923401230703}{14013613306736932291247374688898970661375} a^{7} - \frac{724025034617119643974266298161332036081}{2001944758105276041606767812699852951625} a^{6} - \frac{11358470174802226222747250375171977422309}{56054453226947729164989498755595882645500} a^{5} - \frac{949800237979560939959011947096553756}{87041076439359827895946426639124041375} a^{4} + \frac{6544149071934084843420642008486795979}{26490762394587773707461955933646447375} a^{3} + \frac{1459189879971596510321235901697927267}{4607089112102221514341209727590686500} a^{2} + \frac{79717186543395825744012842387884647}{200308222265313978884400422938725500} a + \frac{275249276334135928811110676124741}{622075224426440928212423673722750}$, $\frac{1}{1289252424219797770794758471378705300846500} a^{30} - \frac{1}{1289252424219797770794758471378705300846500} a^{29} - \frac{7873309750580343388083}{1289252424219797770794758471378705300846500} a^{28} - \frac{7997293385416640984098892}{12892524242197977707947584713787053008465} a^{27} + \frac{399852859306206178689862471}{644626212109898885397379235689352650423250} a^{26} - \frac{302220757067395366666480957331920981451}{644626212109898885397379235689352650423250} a^{25} - \frac{13795621460024346899183914932852588476487}{1289252424219797770794758471378705300846500} a^{24} - \frac{4166508218971235789118079936890904326473}{128925242421979777079475847137870530084650} a^{23} + \frac{11135884578340027191630823854268469699763}{322313106054949442698689617844676325211625} a^{22} - \frac{37874866214895004949736871323947982577597}{1289252424219797770794758471378705300846500} a^{21} + \frac{3584237359883370928340209758436449574411}{184178917745685395827822638768386471549500} a^{20} - \frac{81616473308761419604379388159442546703951}{644626212109898885397379235689352650423250} a^{19} - \frac{49485416848499709396409920985828804173061}{257850484843959554158951694275741060169300} a^{18} + \frac{110430692187218359931547103333616722001139}{644626212109898885397379235689352650423250} a^{17} + \frac{134534810778496716176527192870384238569719}{1289252424219797770794758471378705300846500} a^{16} + \frac{29186215019859425711240918242435815613861}{257850484843959554158951694275741060169300} a^{15} - \frac{274741344021812438029826107824543917807461}{1289252424219797770794758471378705300846500} a^{14} + \frac{129381560358571385667271567575320186094303}{1289252424219797770794758471378705300846500} a^{13} - \frac{244818983968898424136534318783338000972037}{644626212109898885397379235689352650423250} a^{12} + \frac{1545054945201296705788336935638619336051}{128925242421979777079475847137870530084650} a^{11} - \frac{130157400879002846961084752586572019109938}{322313106054949442698689617844676325211625} a^{10} - \frac{300342462081259512542185147640520680168519}{1289252424219797770794758471378705300846500} a^{9} - \frac{44706756238901613565556844075963629753513}{322313106054949442698689617844676325211625} a^{8} + \frac{246747705874682881719251880422752909378017}{644626212109898885397379235689352650423250} a^{7} - \frac{586247110341839875513818787628383672855987}{1289252424219797770794758471378705300846500} a^{6} - \frac{19163675218850407143200312594086083482211}{56054453226947729164989498755595882645500} a^{5} - \frac{53102584910877102678107076001301589437}{243715014030207518108649994589547315850} a^{4} + \frac{2015073950635924393243942645244867449}{105963049578351094829847823734585789500} a^{3} + \frac{818153938323053307583903566369886253}{2303544556051110757170604863795343250} a^{2} + \frac{6650716121756930374879830679143549}{28615460323616282697771488991246500} a - \frac{985412266626502825076365973906438}{2177263285492543248743482858029625}$, $\frac{1}{59305611514110697456558889683420443838939000} a^{31} + \frac{11}{29652805757055348728279444841710221919469500} a^{30} + \frac{51}{5930561151411069745655888968342044383893900} a^{29} + \frac{294893584548573018033849}{59305611514110697456558889683420443838939000} a^{28} + \frac{4027160032608794224662541041}{59305611514110697456558889683420443838939000} a^{27} + \frac{218546588528161617879210205123}{11861122302822139491311777936684088767787800} a^{26} - \frac{26640206873476178240525745764262722529867}{59305611514110697456558889683420443838939000} a^{25} - \frac{291344112302754208830129521936952415675371}{14826402878527674364139722420855110959734750} a^{24} - \frac{56793861483119906371924873364574033585473}{14826402878527674364139722420855110959734750} a^{23} + \frac{112257836511273601128442147536181143215229}{29652805757055348728279444841710221919469500} a^{22} + \frac{853914028347956293683775406845694544737}{46807901747522255293258792173181092217000} a^{21} - \frac{140476825527718600357231428907413538231443}{29652805757055348728279444841710221919469500} a^{20} - \frac{1294362286303165105991247774873234817821583}{7413201439263837182069861210427555479867375} a^{19} - \frac{644362394497746896935548767735962203428901}{59305611514110697456558889683420443838939000} a^{18} + \frac{10821811867177051283329741020323940972504667}{59305611514110697456558889683420443838939000} a^{17} - \frac{839771061208618689244698698889014195035941}{4236115108150764104039920691672888845638500} a^{16} - \frac{908248896426794366812291688373076729138763}{8472230216301528208079841383345777691277000} a^{15} - \frac{2272742650055084270383256617396804941184489}{5930561151411069745655888968342044383893900} a^{14} + \frac{3018151950418251573496550346925749528192739}{59305611514110697456558889683420443838939000} a^{13} - \frac{7262461137523537949911669205570581928637063}{59305611514110697456558889683420443838939000} a^{12} + \frac{3423962177141070003303050234888720700243771}{11861122302822139491311777936684088767787800} a^{11} - \frac{3784588665011131071161655784088955968463011}{8472230216301528208079841383345777691277000} a^{10} + \frac{771865835312464622398430293276589342227319}{4236115108150764104039920691672888845638500} a^{9} + \frac{4336977169789303798201124295792167251102791}{14826402878527674364139722420855110959734750} a^{8} + \frac{3479034525925996447471865111059667945506504}{7413201439263837182069861210427555479867375} a^{7} - \frac{669349220447557008933625262357803295158019}{2578504848439595541589516942757410601693000} a^{6} + \frac{13252799698744455048496389088969293220987}{28027226613473864582494749377797941322750} a^{5} + \frac{2563396450744186352530128198478801043}{348164305757439311583785706556496165500} a^{4} - \frac{98477872569779424375274956713660434151}{211926099156702189659695647469171579000} a^{3} - \frac{392951908446184290081445247218014839}{1316311174886349004097488493597339000} a^{2} - \frac{24983835569934424608458534338976193}{57230920647232565395542977982493000} a - \frac{749629503634263812243318631353543}{2488300897705763712849694694891000}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{312}$, which has order $9984$ (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{219912952522483031183920381256783}{14826402878527674364139722420855110959734750} a^{31} - \frac{81419170769426747260599058486891}{4236115108150764104039920691672888845638500} a^{30} + \frac{2055666548238216612176835093887881}{29652805757055348728279444841710221919469500} a^{29} + \frac{3388719072655721674053461633635533}{7413201439263837182069861210427555479867375} a^{28} - \frac{13820792187421172376596991685088351}{29652805757055348728279444841710221919469500} a^{27} - \frac{55089023373796775909080393752396331}{29652805757055348728279444841710221919469500} a^{26} + \frac{487009715399611318126828665819103353}{29652805757055348728279444841710221919469500} a^{25} - \frac{939242420012785245869462356761390317}{29652805757055348728279444841710221919469500} a^{24} - \frac{15460750708151958311145952629215646}{296528057570553487282794448417102219194695} a^{23} + \frac{4821513306414906167889285368441303759}{14826402878527674364139722420855110959734750} a^{22} + \frac{954642231900141630081278639666441459}{14826402878527674364139722420855110959734750} a^{21} - \frac{65893387809288854249108177874071498439}{29652805757055348728279444841710221919469500} a^{20} + \frac{76549577512093489635886514806482201432}{7413201439263837182069861210427555479867375} a^{19} + \frac{51693017528884848481797129952741563833}{14826402878527674364139722420855110959734750} a^{18} - \frac{987220743122132542838027297730477344747}{29652805757055348728279444841710221919469500} a^{17} + \frac{461805004534138904728440835672763247149}{4236115108150764104039920691672888845638500} a^{16} + \frac{2756633894207077026181245965332850576357}{14826402878527674364139722420855110959734750} a^{15} - \frac{26765432105957254221371531850543969726751}{29652805757055348728279444841710221919469500} a^{14} + \frac{2265335567644622673373035742808125514057}{1482640287852767436413972242085511095973475} a^{13} + \frac{27153960795969159391857221225111052892151}{29652805757055348728279444841710221919469500} a^{12} - \frac{23802687705025538822390021701199708953433}{4236115108150764104039920691672888845638500} a^{11} + \frac{198409826729479162464493840643312325673237}{29652805757055348728279444841710221919469500} a^{10} - \frac{35378451931302667149381949865610574065163}{29652805757055348728279444841710221919469500} a^{9} - \frac{96444462575217325801860458738018701518853}{14826402878527674364139722420855110959734750} a^{8} + \frac{4670684800312805664862313581199753209531}{593056115141106974565588896834204438389390} a^{7} - \frac{2385527628660015767957985160135140071103}{64462621210989888539737923568935265042325} a^{6} + \frac{141172444327266124825171703690243707923}{2437150140302075181086499945895473158500} a^{5} + \frac{85316436289918722100916871112825261839}{1218575070151037590543249972947736579250} a^{4} - \frac{80619159880642905952570137821792358}{211926099156702189659695647469171579} a^{3} + \frac{12530346603131050590628523928342199}{26326223497726980081949769871946780} a^{2} + \frac{2053251979839208370798678289334817}{8709053141970172994973931432118500} a - \frac{11933573545788387197915540180953943}{8709053141970172994973931432118500} \) (order $30$)
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:  \( 67401598701182.28 \) (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

\(\Q(\sqrt{-435}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-3}, \sqrt{145})\), 4.4.3048625.2, 4.0.27437625.2, \(\Q(\sqrt{-15}, \sqrt{29})\), \(\Q(\sqrt{5}, \sqrt{-87})\), \(\Q(\sqrt{5}, \sqrt{29})\), \(\Q(\sqrt{-15}, \sqrt{-87})\), 4.4.3048625.1, 4.0.27437625.1, \(\Q(\sqrt{-3}, \sqrt{29})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.4.219501.1, 4.0.609725.2, 4.4.5487525.2, 4.0.24389.1, \(\Q(\zeta_{15})^+\), 4.0.105125.2, 4.4.946125.1, \(\Q(\zeta_{5})\), 8.0.752823265640625.5, 8.0.35806100625.4, 8.0.752823265640625.4, 8.8.9294114390625.2, 8.0.752823265640625.11, 8.0.752823265640625.9, 8.0.752823265640625.10, 8.0.30112930625625.8, 8.0.30112930625625.5, 8.0.895152515625.1, 8.0.895152515625.3, 8.8.30112930625625.2, 8.0.371764575625.1, 8.8.895152515625.1, 8.0.11051265625.1, 8.0.48180689001.1, 8.0.30112930625625.7, \(\Q(\zeta_{15})\), 8.0.895152515625.2, 16.0.566742869289815033691650390625.1, 16.0.906788590863704053906640625.1, 16.0.801298026229765869140625.1, 16.16.566742869289815033691650390625.1, 16.0.86380562306022715087890625.1, 16.0.566742869289815033691650390625.3, 16.0.566742869289815033691650390625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$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}$
$29$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$