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Properties

Label 32.0.24000959919...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $43.34$
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![256, 1536, 6272, 15872, -5632, 461696, 2473504, 9340480, 20240880, -33044576, 18224552, 4718368, -16249472, 14422848, -3050102, -6803856, 8820755, -1917530, 273805, -71630, 22444, -113020, 20765, -380, 1455, 412, 489, -136, -20, -10, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + x^30 - 10*x^29 - 20*x^28 - 136*x^27 + 489*x^26 + 412*x^25 + 1455*x^24 - 380*x^23 + 20765*x^22 - 113020*x^21 + 22444*x^20 - 71630*x^19 + 273805*x^18 - 1917530*x^17 + 8820755*x^16 - 6803856*x^15 - 3050102*x^14 + 14422848*x^13 - 16249472*x^12 + 4718368*x^11 + 18224552*x^10 - 33044576*x^9 + 20240880*x^8 + 9340480*x^7 + 2473504*x^6 + 461696*x^5 - 5632*x^4 + 15872*x^3 + 6272*x^2 + 1536*x + 256)
 
gp: K = bnfinit(x^32 - 2*x^31 + x^30 - 10*x^29 - 20*x^28 - 136*x^27 + 489*x^26 + 412*x^25 + 1455*x^24 - 380*x^23 + 20765*x^22 - 113020*x^21 + 22444*x^20 - 71630*x^19 + 273805*x^18 - 1917530*x^17 + 8820755*x^16 - 6803856*x^15 - 3050102*x^14 + 14422848*x^13 - 16249472*x^12 + 4718368*x^11 + 18224552*x^10 - 33044576*x^9 + 20240880*x^8 + 9340480*x^7 + 2473504*x^6 + 461696*x^5 - 5632*x^4 + 15872*x^3 + 6272*x^2 + 1536*x + 256, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} + x^{30} - 10 x^{29} - 20 x^{28} - 136 x^{27} + 489 x^{26} + 412 x^{25} + 1455 x^{24} - 380 x^{23} + 20765 x^{22} - 113020 x^{21} + 22444 x^{20} - 71630 x^{19} + 273805 x^{18} - 1917530 x^{17} + 8820755 x^{16} - 6803856 x^{15} - 3050102 x^{14} + 14422848 x^{13} - 16249472 x^{12} + 4718368 x^{11} + 18224552 x^{10} - 33044576 x^{9} + 20240880 x^{8} + 9340480 x^{7} + 2473504 x^{6} + 461696 x^{5} - 5632 x^{4} + 15872 x^{3} + 6272 x^{2} + 1536 x + 256 \)

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:  \(24000959919026880122072334336000000000000000000000000=2^{48}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.34$
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$$\chi_{840}(1,·)$, $\chi_{840}(517,·)$, $\chi_{840}(769,·)$, $\chi_{840}(13,·)$, $\chi_{840}(533,·)$, $\chi_{840}(281,·)$, $\chi_{840}(797,·)$, $\chi_{840}(673,·)$, $\chi_{840}(421,·)$, $\chi_{840}(41,·)$, $\chi_{840}(29,·)$, $\chi_{840}(433,·)$, $\chi_{840}(181,·)$, $\chi_{840}(701,·)$, $\chi_{840}(629,·)$, $\chi_{840}(449,·)$, $\chi_{840}(197,·)$, $\chi_{840}(713,·)$, $\chi_{840}(589,·)$, $\chi_{840}(461,·)$, $\chi_{840}(209,·)$, $\chi_{840}(601,·)$, $\chi_{840}(349,·)$, $\chi_{840}(293,·)$, $\chi_{840}(97,·)$, $\chi_{840}(337,·)$, $\chi_{840}(617,·)$, $\chi_{840}(113,·)$, $\chi_{840}(757,·)$, $\chi_{840}(169,·)$, $\chi_{840}(377,·)$, $\chi_{840}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{18} - \frac{1}{2} a^{16} + \frac{1}{4} a^{14} - \frac{1}{2} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{21} - \frac{1}{4} a^{19} - \frac{1}{2} a^{17} + \frac{1}{4} a^{15} - \frac{1}{2} a^{14} + \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{22} - \frac{1}{8} a^{20} - \frac{1}{4} a^{18} - \frac{3}{8} a^{16} - \frac{1}{4} a^{15} + \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{23} - \frac{1}{8} a^{21} - \frac{1}{4} a^{19} - \frac{3}{8} a^{17} - \frac{1}{4} a^{16} + \frac{1}{8} a^{15} - \frac{1}{4} a^{14} - \frac{1}{8} a^{13} + \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{3}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{336} a^{24} + \frac{1}{56} a^{23} + \frac{1}{336} a^{22} - \frac{17}{168} a^{21} + \frac{1}{21} a^{19} - \frac{19}{336} a^{18} - \frac{17}{84} a^{17} - \frac{113}{336} a^{16} + \frac{11}{84} a^{15} + \frac{1}{336} a^{14} + \frac{5}{28} a^{13} + \frac{5}{42} a^{12} - \frac{1}{24} a^{11} + \frac{27}{112} a^{10} - \frac{1}{168} a^{9} - \frac{15}{112} a^{8} - \frac{1}{21} a^{7} + \frac{3}{8} a^{6} - \frac{3}{14} a^{5} - \frac{13}{84} a^{4} + \frac{3}{7} a^{3} - \frac{4}{21} a + \frac{1}{21}$, $\frac{1}{336} a^{25} + \frac{1}{48} a^{23} + \frac{1}{168} a^{22} - \frac{1}{56} a^{21} - \frac{13}{168} a^{20} - \frac{31}{336} a^{19} - \frac{19}{168} a^{18} - \frac{167}{336} a^{17} - \frac{10}{21} a^{16} - \frac{137}{336} a^{15} + \frac{1}{28} a^{14} + \frac{29}{168} a^{13} + \frac{31}{84} a^{12} + \frac{55}{112} a^{11} - \frac{19}{42} a^{10} + \frac{31}{112} a^{9} - \frac{5}{42} a^{8} - \frac{3}{14} a^{7} + \frac{23}{56} a^{6} - \frac{5}{42} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{4}{21} a^{2} + \frac{4}{21} a - \frac{2}{7}$, $\frac{1}{8555232} a^{26} - \frac{379}{1069404} a^{25} + \frac{9139}{8555232} a^{24} - \frac{19281}{356468} a^{23} + \frac{202837}{4277616} a^{22} - \frac{15367}{1069404} a^{21} - \frac{244801}{2851744} a^{20} - \frac{166055}{1425872} a^{19} + \frac{270317}{8555232} a^{18} - \frac{659415}{1425872} a^{17} - \frac{425519}{2851744} a^{16} - \frac{1110559}{4277616} a^{15} - \frac{808657}{4277616} a^{14} - \frac{933689}{4277616} a^{13} - \frac{981059}{8555232} a^{12} - \frac{395105}{2138808} a^{11} - \frac{3022619}{8555232} a^{10} - \frac{421313}{4277616} a^{9} + \frac{230263}{1069404} a^{8} - \frac{13919}{89117} a^{7} + \frac{448585}{1069404} a^{6} - \frac{10763}{267351} a^{5} - \frac{1343}{89117} a^{4} + \frac{5230}{267351} a^{3} + \frac{7930}{89117} a^{2} + \frac{55931}{267351} a + \frac{627}{89117}$, $\frac{1}{8555232} a^{27} + \frac{7897}{8555232} a^{25} + \frac{197}{712936} a^{24} - \frac{26207}{712936} a^{23} - \frac{4277}{152772} a^{22} - \frac{22819}{295008} a^{21} - \frac{323467}{4277616} a^{20} - \frac{681733}{8555232} a^{19} + \frac{73859}{4277616} a^{18} + \frac{2402585}{8555232} a^{17} - \frac{2048293}{4277616} a^{16} - \frac{37648}{89117} a^{15} - \frac{1885979}{4277616} a^{14} + \frac{213835}{2851744} a^{13} - \frac{598351}{2138808} a^{12} + \frac{1293571}{8555232} a^{11} - \frac{341039}{4277616} a^{10} - \frac{535489}{4277616} a^{9} - \frac{165115}{712936} a^{8} + \frac{54245}{267351} a^{7} - \frac{308983}{712936} a^{6} + \frac{24565}{50924} a^{5} + \frac{187627}{534702} a^{4} + \frac{23917}{534702} a^{3} + \frac{22032}{89117} a^{2} - \frac{30440}{267351} a + \frac{63359}{267351}$, $\frac{1}{17880434880} a^{28} + \frac{211}{8940217440} a^{27} - \frac{11}{232213440} a^{26} + \frac{5553337}{8940217440} a^{25} - \frac{614671}{447010872} a^{24} - \frac{6246727}{127717392} a^{23} - \frac{187532801}{5960144960} a^{22} - \frac{4820043}{78422960} a^{21} + \frac{673921309}{5960144960} a^{20} + \frac{12089465}{223505436} a^{19} - \frac{863782915}{3576086976} a^{18} + \frac{414498767}{1117527180} a^{17} - \frac{22992538}{55876359} a^{16} - \frac{55063451}{812747040} a^{15} + \frac{285526855}{1192028992} a^{14} + \frac{260431565}{596014496} a^{13} + \frac{2250470049}{5960144960} a^{12} + \frac{736765027}{1490036240} a^{11} - \frac{902218109}{8940217440} a^{10} - \frac{426886445}{894021744} a^{9} - \frac{404586643}{894021744} a^{8} - \frac{9824039}{50796690} a^{7} + \frac{1022203}{319293480} a^{6} - \frac{40143434}{279381795} a^{5} - \frac{9155921}{38535420} a^{4} - \frac{23811454}{55876359} a^{3} + \frac{81393091}{186254530} a^{2} + \frac{3394115}{55876359} a + \frac{1992179}{13303895}$, $\frac{1}{5164304785068194153704857673809600} a^{29} + \frac{217390513579358576101}{12719962524798507767745954861600} a^{28} + \frac{17582498244626722532863991}{469482253188017650336805243073600} a^{27} + \frac{3867198668956995691626611}{92219728304789181316158172746600} a^{26} - \frac{308902194111935534268041650849}{322769049066762134606553604613100} a^{25} - \frac{199771164436298126418223552881}{172143492835606471790161922460320} a^{24} + \frac{8606529593702409172255431865937}{5164304785068194153704857673809600} a^{23} - \frac{32622330509970543757646149868669}{1291076196267048538426214418452400} a^{22} + \frac{176163506419981961875743856364891}{5164304785068194153704857673809600} a^{21} + \frac{198046479952147168461808180418959}{2582152392534097076852428836904800} a^{20} - \frac{36934351765966778360282204523187}{206572191402727766148194306952384} a^{19} - \frac{404298993033349484920842746559259}{2582152392534097076852428836904800} a^{18} + \frac{327622822941902474552756057435137}{1291076196267048538426214418452400} a^{17} - \frac{492906075824054331082793925857}{4047260803344979744282803819600} a^{16} + \frac{793500155621469910210298010522379}{1721434928356064717901619224603200} a^{15} + \frac{4182149417962481492054235250923}{24591927547943781684308846065760} a^{14} + \frac{1249669835516787666269369556330227}{5164304785068194153704857673809600} a^{13} - \frac{384973277261333614530212209990133}{860717464178032358950809612301600} a^{12} + \frac{98344128572847908861802989717357}{368878913219156725264632690986400} a^{11} + \frac{5924054820121194613202992057707}{45300919167264860997411032226400} a^{10} - \frac{80988722273090477814878907246017}{258215239253409707685242883690480} a^{9} - \frac{24588089729953557841456144033877}{117370563297004412584201310768400} a^{8} - \frac{144331487968091020668134706775969}{322769049066762134606553604613100} a^{7} - \frac{7066031732602085725254069017027}{46109864152394590658079086373300} a^{6} - \frac{3044113478189382249387887746994}{26897420755563511217212800384425} a^{5} + \frac{7392051960849255907983282379659}{53794841511127022434425600768850} a^{4} - \frac{12057406529751270917400058732126}{80692262266690533651638401153275} a^{3} - \frac{62415782607326988065034080732503}{161384524533381067303276802306550} a^{2} - \frac{1898902722207843356680884377116}{80692262266690533651638401153275} a - \frac{3095565270019416682845364832224}{7335660206062775786512581923025}$, $\frac{1}{10328609570136388307409715347619200} a^{30} - \frac{15735235634367839771133}{688573971342425887160647689841280} a^{28} - \frac{67690622543430729567523}{9221972830478918131615817274660} a^{27} + \frac{1269459000840261558507937}{86071746417803235895080961230160} a^{26} - \frac{1439249982116773704878253596927}{1291076196267048538426214418452400} a^{25} - \frac{6958027411826394797692149230983}{10328609570136388307409715347619200} a^{24} - \frac{297824346386994583033061955346049}{5164304785068194153704857673809600} a^{23} - \frac{249754302230711516395648330260473}{10328609570136388307409715347619200} a^{22} + \frac{91686399504441243968602862591527}{1721434928356064717901619224603200} a^{21} + \frac{1254753706502091528084033710588597}{10328609570136388307409715347619200} a^{20} - \frac{1154802358629236196594444729909709}{5164304785068194153704857673809600} a^{19} - \frac{140400939248140003956098363677411}{2582152392534097076852428836904800} a^{18} - \frac{156755767118996836641404999556259}{1032860957013638830740971534761920} a^{17} - \frac{63982370333220117806278890795323}{134137786625147900096230069449600} a^{16} + \frac{178305968734991029143104878375183}{645538098133524269213107209226200} a^{15} - \frac{1379799020784523490661976060118651}{3442869856712129435803238449206400} a^{14} + \frac{14255702558003599324544651281369}{93896450637603530067361048614720} a^{13} - \frac{1080915860344578017137088164287467}{5164304785068194153704857673809600} a^{12} - \frac{16387564850971771030456377150349}{516430478506819415370485767380960} a^{11} + \frac{249214464504423684780530513727947}{645538098133524269213107209226200} a^{10} - \frac{55853447634955853684994192615557}{161384524533381067303276802306550} a^{9} + \frac{194556913528337629286508101571203}{1291076196267048538426214418452400} a^{8} - \frac{1856470877347224513807572764973}{3912352109900147086140043692280} a^{7} - \frac{74396762648690532731395699833283}{215179366044508089737702403075400} a^{6} + \frac{6468339745811252014294629291554}{80692262266690533651638401153275} a^{5} - \frac{26694554743391167898632008964349}{53794841511127022434425600768850} a^{4} + \frac{14923756790273152319254737981271}{80692262266690533651638401153275} a^{3} - \frac{19125995997203608158910625030447}{161384524533381067303276802306550} a^{2} - \frac{30242704260507092349199168786909}{80692262266690533651638401153275} a - \frac{1620622703222053743442046392304}{11527466038098647664519771593325}$, $\frac{1}{10328609570136388307409715347619200} a^{31} - \frac{1}{10328609570136388307409715347619200} a^{29} + \frac{7559610264912318967991}{737757826438313450529265381972800} a^{28} - \frac{96569832611092112705713463}{5164304785068194153704857673809600} a^{27} + \frac{57350619873850226086142033}{1721434928356064717901619224603200} a^{26} - \frac{838918830143111276173240531457}{1475515652876626901058530763945600} a^{25} - \frac{570956652618788414280982189469}{469482253188017650336805243073600} a^{24} - \frac{67451040630618985256823287025979}{2065721914027277661481943069523840} a^{23} - \frac{53817872136831151726949639860549}{1291076196267048538426214418452400} a^{22} + \frac{962421559853171879601017240244671}{10328609570136388307409715347619200} a^{21} + \frac{45181754695961994771870987963913}{368878913219156725264632690986400} a^{20} + \frac{90188044945711713168772922773253}{5164304785068194153704857673809600} a^{19} - \frac{34262463546588843505554890669369}{368878913219156725264632690986400} a^{18} + \frac{1696538717837690291757680407558927}{3442869856712129435803238449206400} a^{17} + \frac{4319067537519614590121468461409}{129107619626704853842621441845240} a^{16} + \frac{711531935479156496846926189375097}{2065721914027277661481943069523840} a^{15} + \frac{48411305051433706037515609898957}{129107619626704853842621441845240} a^{14} + \frac{101652379568619004783749985135923}{215179366044508089737702403075400} a^{13} + \frac{76891675223615168086957708204559}{469482253188017650336805243073600} a^{12} + \frac{100931159636003975759500658616353}{215179366044508089737702403075400} a^{11} - \frac{111077123801655521334923297225237}{1291076196267048538426214418452400} a^{10} - \frac{14839078300784308366605579392289}{430358732089016179475404806150800} a^{9} + \frac{146368952543198176046522821238117}{430358732089016179475404806150800} a^{8} - \frac{50608290190599381727229109653449}{129107619626704853842621441845240} a^{7} - \frac{70047011994711249736853170732237}{322769049066762134606553604613100} a^{6} - \frac{4536987200486520347357480571301}{16138452453338106730327680230655} a^{5} - \frac{4102270825181330977882932946387}{23054932076197295329039543186650} a^{4} + \frac{2465791340650367923762906920553}{53794841511127022434425600768850} a^{3} - \frac{916534184908881972873066896513}{2782491802299673574194427625975} a^{2} + \frac{191163811960891399103146059412}{2305493207619729532903954318665} a - \frac{18400671885311529434010754470443}{80692262266690533651638401153275}$

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{82175595185965938672707651061}{11325229791816215249352758056600} a^{31} - \frac{26701819999859913810235209629}{1698784468772432287402913708490} a^{30} + \frac{13706791363226770543279549359}{1415653723977026906169094757075} a^{29} - \frac{1671152306197306775677274851841}{22650459583632430498705516113200} a^{28} - \frac{1506943496369743649854791473343}{11325229791816215249352758056600} a^{27} - \frac{16356534095865417746439594112931}{16987844687724322874029137084900} a^{26} + \frac{18015989745671620884900856714651}{4853669910778377964008324881400} a^{25} + \frac{40725804854207092887382951058213}{16987844687724322874029137084900} a^{24} + \frac{113991246225722764379634139412731}{11325229791816215249352758056600} a^{23} - \frac{5097729093982118295010770760363}{1132522979181621524935275805660} a^{22} + \frac{61140042844982344495276602880473}{404472492564864830334027073450} a^{21} - \frac{14356461068927968888762883763803543}{16987844687724322874029137084900} a^{20} + \frac{10174948495183650040658189021999363}{33975689375448645748058274169800} a^{19} - \frac{37244919594318353754509517131809061}{67951378750897291496116548339600} a^{18} + \frac{1280019767873655751586519199830651}{617739806826339013601059530360} a^{17} - \frac{22000380562174585061581977727602199}{1544349517065847534002648825900} a^{16} + \frac{1126630771778838593691159169621089799}{16987844687724322874029137084900} a^{15} - \frac{4635542506904882232781517606789149}{77217475853292376700132441295} a^{14} - \frac{23486210697672924247031233563400291}{1698784468772432287402913708490} a^{13} + \frac{460102146078922789207275207650823929}{4246961171931080718507284271225} a^{12} - \frac{574791037378212255886983800795259817}{4246961171931080718507284271225} a^{11} + \frac{76278049075806349923722190381186046}{1415653723977026906169094757075} a^{10} + \frac{16283810845399857035078901961888778}{128695793088820627833554068825} a^{9} - \frac{2540981192431039565516625508570012423}{9707339821556755928016649762800} a^{8} + \frac{793366949813010457078359422801356028}{4246961171931080718507284271225} a^{7} + \frac{183857073164236699454522670426163424}{4246961171931080718507284271225} a^{6} + \frac{28247335603402843542673245984163616}{4246961171931080718507284271225} a^{5} + \frac{1523561486501767148210046857652176}{4246961171931080718507284271225} a^{4} - \frac{225468321952735216276132948492996}{4246961171931080718507284271225} a^{3} + \frac{172658858323415880000941410095072}{1415653723977026906169094757075} a^{2} + \frac{37248650865183486278795687381536}{1415653723977026906169094757075} a + \frac{15118514187288634740492822019328}{4246961171931080718507284271225} \) (order $30$)
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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{-35})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{2}, \sqrt{105})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{-70})\), \(\Q(\sqrt{-6}, \sqrt{-70})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{-30}, \sqrt{-35})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{-15}, \sqrt{42})\), \(\Q(\sqrt{21}, \sqrt{-30})\), \(\Q(\sqrt{5}, \sqrt{42})\), \(\Q(\sqrt{-7}, \sqrt{-30})\), \(\Q(\sqrt{10}, \sqrt{21})\), \(\Q(\sqrt{-14}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{10}, \sqrt{42})\), \(\Q(\sqrt{-14}, \sqrt{-30})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), 4.4.6125.1, \(\Q(\zeta_{5})\), 4.4.392000.1, 4.0.8000.2, 4.4.72000.1, 4.0.3528000.1, \(\Q(\zeta_{15})^+\), 4.0.55125.1, 8.0.497871360000.14, 8.0.6146560000.2, 8.0.497871360000.1, 8.0.497871360000.9, 8.0.497871360000.5, 8.0.121550625.1, 8.0.497871360000.11, 8.8.497871360000.1, 8.0.497871360000.10, 8.0.207360000.1, 8.0.796594176.2, 8.0.497871360000.19, 8.0.497871360000.16, 8.0.497871360000.6, 8.0.497871360000.7, 8.0.37515625.1, 8.0.153664000000.2, 8.0.12446784000000.15, 8.0.3038765625.2, 8.8.153664000000.1, 8.0.64000000.2, 8.8.5184000000.1, 8.0.12446784000000.14, 8.0.153664000000.1, 8.0.153664000000.5, 8.0.12446784000000.3, 8.0.12446784000000.16, 8.8.12446784000000.4, 8.0.12446784000000.20, 8.8.12446784000000.6, 8.0.12446784000000.11, 8.0.12446784000000.6, 8.0.5184000000.5, 8.0.12446784000000.8, 8.0.5184000000.1, 8.8.3038765625.1, 8.0.3038765625.3, 8.8.12446784000000.3, 8.0.12446784000000.5, 8.0.3038765625.1, \(\Q(\zeta_{15})\), 8.0.12446784000000.18, 8.0.5184000000.3, 16.0.247875891108249600000000.1, 16.0.23612624896000000000000.2, 16.0.154922431942656000000000000.13, 16.0.154922431942656000000000000.8, 16.0.154922431942656000000000000.18, 16.0.9234096523681640625.1, 16.0.154922431942656000000000000.14, 16.16.154922431942656000000000000.1, 16.0.154922431942656000000000000.20, 16.0.154922431942656000000000000.7, 16.0.26873856000000000000.2, 16.0.154922431942656000000000000.11, 16.0.154922431942656000000000000.2, 16.0.154922431942656000000000000.4, 16.0.154922431942656000000000000.5

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{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}$
$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}$