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Properties

Label 32.0.91066857695...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $64.75$
Ramified primes $2, 5, 13$
Class number $680$ (GRH)
Class group $[2, 340]$ (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![20321401, -37413980, -47686656, 63301378, 125824932, -152351964, -97288993, 208696342, 131454091, -233530804, -30870498, 137098280, -2899755, -34901464, 3553038, -2098242, 2436831, 5298792, -1980711, -1996144, 670872, 416024, -153033, -63352, 27148, 9520, -3949, -1140, 465, 76, -33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 33*x^30 + 76*x^29 + 465*x^28 - 1140*x^27 - 3949*x^26 + 9520*x^25 + 27148*x^24 - 63352*x^23 - 153033*x^22 + 416024*x^21 + 670872*x^20 - 1996144*x^19 - 1980711*x^18 + 5298792*x^17 + 2436831*x^16 - 2098242*x^15 + 3553038*x^14 - 34901464*x^13 - 2899755*x^12 + 137098280*x^11 - 30870498*x^10 - 233530804*x^9 + 131454091*x^8 + 208696342*x^7 - 97288993*x^6 - 152351964*x^5 + 125824932*x^4 + 63301378*x^3 - 47686656*x^2 - 37413980*x + 20321401)
 
gp: K = bnfinit(x^32 - 2*x^31 - 33*x^30 + 76*x^29 + 465*x^28 - 1140*x^27 - 3949*x^26 + 9520*x^25 + 27148*x^24 - 63352*x^23 - 153033*x^22 + 416024*x^21 + 670872*x^20 - 1996144*x^19 - 1980711*x^18 + 5298792*x^17 + 2436831*x^16 - 2098242*x^15 + 3553038*x^14 - 34901464*x^13 - 2899755*x^12 + 137098280*x^11 - 30870498*x^10 - 233530804*x^9 + 131454091*x^8 + 208696342*x^7 - 97288993*x^6 - 152351964*x^5 + 125824932*x^4 + 63301378*x^3 - 47686656*x^2 - 37413980*x + 20321401, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} - 33 x^{30} + 76 x^{29} + 465 x^{28} - 1140 x^{27} - 3949 x^{26} + 9520 x^{25} + 27148 x^{24} - 63352 x^{23} - 153033 x^{22} + 416024 x^{21} + 670872 x^{20} - 1996144 x^{19} - 1980711 x^{18} + 5298792 x^{17} + 2436831 x^{16} - 2098242 x^{15} + 3553038 x^{14} - 34901464 x^{13} - 2899755 x^{12} + 137098280 x^{11} - 30870498 x^{10} - 233530804 x^{9} + 131454091 x^{8} + 208696342 x^{7} - 97288993 x^{6} - 152351964 x^{5} + 125824932 x^{4} + 63301378 x^{3} - 47686656 x^{2} - 37413980 x + 20321401 \)

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:  \(9106685769537214956799814036094976000000000000000000000000=2^{48}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.75$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(520=2^{3}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{520}(1,·)$, $\chi_{520}(261,·)$, $\chi_{520}(129,·)$, $\chi_{520}(57,·)$, $\chi_{520}(109,·)$, $\chi_{520}(21,·)$, $\chi_{520}(281,·)$, $\chi_{520}(157,·)$, $\chi_{520}(389,·)$, $\chi_{520}(161,·)$, $\chi_{520}(421,·)$, $\chi_{520}(177,·)$, $\chi_{520}(181,·)$, $\chi_{520}(313,·)$, $\chi_{520}(317,·)$, $\chi_{520}(53,·)$, $\chi_{520}(417,·)$, $\chi_{520}(73,·)$, $\chi_{520}(77,·)$, $\chi_{520}(333,·)$, $\chi_{520}(337,·)$, $\chi_{520}(213,·)$, $\chi_{520}(441,·)$, $\chi_{520}(473,·)$, $\chi_{520}(229,·)$, $\chi_{520}(209,·)$, $\chi_{520}(233,·)$, $\chi_{520}(493,·)$, $\chi_{520}(437,·)$, $\chi_{520}(369,·)$, $\chi_{520}(489,·)$, $\chi_{520}(469,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{16} + \frac{1}{3} a^{12} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{17} + \frac{1}{3} a^{13} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{12} a^{20} - \frac{1}{3} a^{16} - \frac{1}{2} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{4} a^{10} - \frac{1}{6} a^{8} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{12}$, $\frac{1}{12} a^{21} - \frac{1}{3} a^{17} - \frac{1}{2} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{4} a^{11} - \frac{1}{6} a^{9} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} + \frac{1}{12} a$, $\frac{1}{12} a^{22} - \frac{1}{2} a^{17} + \frac{1}{3} a^{14} - \frac{5}{12} a^{12} - \frac{1}{6} a^{10} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{23} - \frac{1}{6} a^{18} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{5}{12} a^{13} + \frac{1}{3} a^{12} - \frac{1}{6} a^{11} + \frac{1}{3} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{6} - \frac{1}{4} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{24} - \frac{1}{6} a^{19} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{5}{12} a^{14} + \frac{1}{3} a^{13} - \frac{1}{6} a^{12} + \frac{1}{3} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{7} - \frac{1}{4} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{25} + \frac{1}{3} a^{17} - \frac{5}{12} a^{15} - \frac{1}{6} a^{13} + \frac{1}{3} a^{11} - \frac{1}{4} a^{5} + \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{663972} a^{26} + \frac{3178}{165993} a^{25} - \frac{181}{331986} a^{24} + \frac{15577}{663972} a^{23} - \frac{4873}{165993} a^{22} - \frac{4401}{110662} a^{21} - \frac{13111}{331986} a^{20} - \frac{6863}{55331} a^{19} - \frac{32003}{331986} a^{18} + \frac{17659}{165993} a^{17} - \frac{28243}{221324} a^{16} - \frac{56212}{165993} a^{15} - \frac{41296}{165993} a^{14} + \frac{290711}{663972} a^{13} + \frac{23649}{55331} a^{12} - \frac{20851}{165993} a^{11} + \frac{21899}{110662} a^{10} - \frac{43349}{165993} a^{9} + \frac{52111}{110662} a^{8} + \frac{1070}{55331} a^{7} - \frac{285955}{663972} a^{6} + \frac{20254}{55331} a^{5} + \frac{21857}{110662} a^{4} - \frac{292895}{663972} a^{3} + \frac{6204}{55331} a^{2} - \frac{26693}{165993} a - \frac{17197}{331986}$, $\frac{1}{663972} a^{27} + \frac{26545}{663972} a^{25} + \frac{6212}{165993} a^{24} - \frac{4667}{663972} a^{23} - \frac{1360}{55331} a^{22} + \frac{2251}{165993} a^{21} - \frac{6905}{663972} a^{20} + \frac{23402}{165993} a^{19} + \frac{8611}{331986} a^{18} + \frac{41077}{221324} a^{17} - \frac{28096}{165993} a^{16} - \frac{20213}{663972} a^{15} + \frac{11023}{55331} a^{14} - \frac{285595}{663972} a^{13} - \frac{59146}{165993} a^{12} - \frac{110783}{331986} a^{11} + \frac{44297}{663972} a^{10} - \frac{49102}{165993} a^{9} - \frac{70571}{165993} a^{8} - \frac{170923}{663972} a^{7} + \frac{23136}{55331} a^{6} + \frac{44225}{221324} a^{5} + \frac{35087}{165993} a^{4} - \frac{84873}{221324} a^{3} + \frac{37277}{110662} a^{2} - \frac{19957}{55331} a + \frac{89207}{221324}$, $\frac{1}{1991916} a^{28} + \frac{1}{1991916} a^{27} + \frac{1}{1991916} a^{26} - \frac{5707}{497979} a^{25} + \frac{505}{663972} a^{24} - \frac{2078}{497979} a^{23} - \frac{11849}{1991916} a^{22} + \frac{7531}{331986} a^{21} + \frac{1040}{497979} a^{20} - \frac{145651}{995958} a^{19} - \frac{99293}{1991916} a^{18} + \frac{97825}{221324} a^{17} - \frac{69847}{1991916} a^{16} - \frac{66463}{995958} a^{15} + \frac{172999}{1991916} a^{14} + \frac{6400}{497979} a^{13} + \frac{44159}{1991916} a^{12} + \frac{122527}{331986} a^{11} - \frac{263755}{995958} a^{10} + \frac{132731}{995958} a^{9} - \frac{652159}{1991916} a^{8} + \frac{74891}{221324} a^{7} - \frac{945359}{1991916} a^{6} + \frac{86545}{497979} a^{5} + \frac{92479}{221324} a^{4} + \frac{422539}{995958} a^{3} - \frac{570179}{1991916} a^{2} - \frac{8927}{497979} a - \frac{31715}{497979}$, $\frac{1}{1991916} a^{29} + \frac{1}{1991916} a^{26} - \frac{41227}{995958} a^{25} + \frac{25363}{1991916} a^{24} + \frac{20789}{663972} a^{23} + \frac{20477}{497979} a^{22} - \frac{1715}{995958} a^{21} - \frac{20489}{995958} a^{20} - \frac{96913}{663972} a^{19} - \frac{101417}{995958} a^{18} + \frac{270775}{995958} a^{17} - \frac{275713}{1991916} a^{16} - \frac{161389}{331986} a^{15} + \frac{131609}{663972} a^{14} + \frac{550549}{1991916} a^{13} + \frac{183137}{995958} a^{12} + \frac{221905}{497979} a^{11} + \frac{60085}{331986} a^{10} - \frac{283265}{1991916} a^{9} - \frac{431965}{995958} a^{8} + \frac{91525}{497979} a^{7} - \frac{24039}{221324} a^{6} + \frac{477973}{995958} a^{5} - \frac{197107}{1991916} a^{4} + \frac{313835}{1991916} a^{3} - \frac{2333}{55331} a^{2} - \frac{76556}{165993} a - \frac{302621}{995958}$, $\frac{1}{173296692} a^{30} - \frac{7}{86648346} a^{29} + \frac{8}{43324173} a^{28} + \frac{5}{9627594} a^{27} + \frac{1}{43324173} a^{26} - \frac{35037}{1604599} a^{25} - \frac{2633083}{86648346} a^{24} + \frac{1934291}{86648346} a^{23} - \frac{1907219}{57765564} a^{22} - \frac{2262041}{173296692} a^{21} - \frac{206854}{14441391} a^{20} + \frac{8248925}{86648346} a^{19} - \frac{4185091}{28882782} a^{18} - \frac{19757438}{43324173} a^{17} - \frac{2796367}{28882782} a^{16} + \frac{21022753}{86648346} a^{15} + \frac{373721}{3209198} a^{14} - \frac{10912831}{86648346} a^{13} - \frac{7009633}{57765564} a^{12} + \frac{13871401}{173296692} a^{11} + \frac{995653}{3209198} a^{10} + \frac{18663280}{43324173} a^{9} - \frac{4670329}{28882782} a^{8} - \frac{15187670}{43324173} a^{7} - \frac{33175789}{86648346} a^{6} - \frac{4707953}{14441391} a^{5} + \frac{18217787}{86648346} a^{4} - \frac{720667}{28882782} a^{3} - \frac{55799761}{173296692} a^{2} - \frac{18977261}{173296692} a - \frac{75560651}{173296692}$, $\frac{1}{93174089202168523087793484854108763726924741878809980303701297815595697550902475411456655669695480865072415746892} a^{31} + \frac{76057917754130028029815295284180910627438728493316438748235992459180847005684616293239395685103281723215}{46587044601084261543896742427054381863462370939404990151850648907797848775451237705728327834847740432536207873446} a^{30} - \frac{14073313759123625982335078574416608391627643240540179717510212369961224007758133001115034022793610173974043}{93174089202168523087793484854108763726924741878809980303701297815595697550902475411456655669695480865072415746892} a^{29} - \frac{755494661938311481854514177713848801810131149079933072191002831669237632756286380060583203691511926212367}{31058029734056174362597828284702921242308247292936660101233765938531899183634158470485551889898493621690805248964} a^{28} - \frac{9399094754231642909827047384290297529337173902543672754320859850593036455014542249664564576038895422560723}{23293522300542130771948371213527190931731185469702495075925324453898924387725618852864163917423870216268103936723} a^{27} - \frac{2255525403949654429581258118369101644320011695406754830411019048499498631406051528005067314652536120534187}{15529014867028087181298914142351460621154123646468330050616882969265949591817079235242775944949246810845402624482} a^{26} + \frac{3230961352214958593729433967322088022522364681557150437311581052923802242469503876974747980468812038249066172441}{93174089202168523087793484854108763726924741878809980303701297815595697550902475411456655669695480865072415746892} a^{25} - \frac{709889603302619585372249481983979911411230348399935570440485162271291196346821721085952651331416870235680490589}{23293522300542130771948371213527190931731185469702495075925324453898924387725618852864163917423870216268103936723} a^{24} - \frac{89009093505353148692407652408821229391347077838908325271471850846322916456317244823898586178565564679469635763}{10352676578018724787532609428234307080769415764312220033744588646177299727878052823495183963299497873896935082988} a^{23} - \frac{802471489085303345709434043273796415916779804206745570722856743811977627585565405224671740840319218048931438074}{23293522300542130771948371213527190931731185469702495075925324453898924387725618852864163917423870216268103936723} a^{22} + \frac{47597867759021859425828308306484023910457749561269506320088344015232771933264638868074651534722719163262826335}{7764507433514043590649457071175730310577061823234165025308441484632974795908539617621387972474623405422701312241} a^{21} - \frac{982380542565335239639011600547722445698723967057906133824377207072577619595528749827238225613342261299880259999}{46587044601084261543896742427054381863462370939404990151850648907797848775451237705728327834847740432536207873446} a^{20} - \frac{1049429094522137304101029725519958869154932492314102728856351756531105964936919278170982596142739856177312832031}{10352676578018724787532609428234307080769415764312220033744588646177299727878052823495183963299497873896935082988} a^{19} - \frac{636212512901314137359107806460663698737836386654499603874724257697714274665746482319121236218596211858197050405}{93174089202168523087793484854108763726924741878809980303701297815595697550902475411456655669695480865072415746892} a^{18} - \frac{2567731546516845167281443158686111009040422219552485313377599323844918640934185107132034031209778024044420093649}{15529014867028087181298914142351460621154123646468330050616882969265949591817079235242775944949246810845402624482} a^{17} + \frac{16890425490855193580059640193382078905515636418480235904067163618229387785366497708850549857474020842433995621861}{46587044601084261543896742427054381863462370939404990151850648907797848775451237705728327834847740432536207873446} a^{16} - \frac{240404072797799761780534308377860492591068239623113338952439074974945497587050988317547356989559116686296926099}{1070966542553661184917166492575962801458905079066781382801164342707996523573591671396053513444775642127269146516} a^{15} - \frac{5450215685514658487534600575088183267169467653040172608602138246268035649296743125228943725058111443073213399095}{23293522300542130771948371213527190931731185469702495075925324453898924387725618852864163917423870216268103936723} a^{14} + \frac{1684372385921402676025581201685507168994320530247381856628799710241728554969490929073515626033549378318405114251}{3450892192672908262510869809411435693589805254770740011248196215392433242626017607831727987766499291298978360996} a^{13} + \frac{2231074312538656929112160023640261590508163619909601491361826750247187851523026735320409728479437857299239043290}{23293522300542130771948371213527190931731185469702495075925324453898924387725618852864163917423870216268103936723} a^{12} + \frac{7546956576420824156150510386109100110137765780347145856506646462684506301523446999572009866399683901181036816115}{15529014867028087181298914142351460621154123646468330050616882969265949591817079235242775944949246810845402624482} a^{11} + \frac{5309501897297528400567626036371300556826746653544495863505493248597183519980558520575354568530253542993355538983}{46587044601084261543896742427054381863462370939404990151850648907797848775451237705728327834847740432536207873446} a^{10} + \frac{5879824827894781336158429563741815262008911018504709808315720913620330431797935643735889954538870491394401778453}{31058029734056174362597828284702921242308247292936660101233765938531899183634158470485551889898493621690805248964} a^{9} - \frac{703708755989058679648386423502471392870554201872254223028686985329620736996756323185782511198209491284986222343}{93174089202168523087793484854108763726924741878809980303701297815595697550902475411456655669695480865072415746892} a^{8} - \frac{11130227015454461906996186827732341448539583884386098149682559428587993137377574258123590351206876952783332962195}{23293522300542130771948371213527190931731185469702495075925324453898924387725618852864163917423870216268103936723} a^{7} - \frac{1529047822990177842028186064791988781415034544332093583340299114295704455268335480339016219878816880802876753541}{5176338289009362393766304714117153540384707882156110016872294323088649863939026411747591981649748936948467541494} a^{6} + \frac{40158539061467923872070707447708536262202025259665458191581704806367305262263920217163869236644340505828850838905}{93174089202168523087793484854108763726924741878809980303701297815595697550902475411456655669695480865072415746892} a^{5} + \frac{1891341405400592233933243896394070924732110246745385168659608308741168961875731612086977973757116305692831879705}{15529014867028087181298914142351460621154123646468330050616882969265949591817079235242775944949246810845402624482} a^{4} + \frac{3983584178620301078036740167728048847358570174278372005974466970344114480147632186052287203652726102506214539059}{93174089202168523087793484854108763726924741878809980303701297815595697550902475411456655669695480865072415746892} a^{3} + \frac{9122971334642273191543597373583454614660690461331090110881981064286283751358805953630611780043550136259085310308}{23293522300542130771948371213527190931731185469702495075925324453898924387725618852864163917423870216268103936723} a^{2} + \frac{6164273314970062542781767609106552694227347858054003944856447684381290608195823842029225018193648719118117210081}{93174089202168523087793484854108763726924741878809980303701297815595697550902475411456655669695480865072415746892} a - \frac{638246425658795110234182975564932761197273683370592992828439427286224023843166769157256665179264024997926927175}{1725446096336454131255434904705717846794902627385370005624098107696216621313008803915863993883249645649489180498}$

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

Class group and class number

$C_{2}\times C_{340}$, which has order $680$ (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{721211675231999948008733973794946280338856539873761943269596776688759037773497367279}{3773896742914846417263964090151490643624339450432800551960583953227057360491290077120033932} a^{31} - \frac{520717217674098601136310297621075290678993822327971186216719307219539819829416026965}{1886948371457423208631982045075745321812169725216400275980291976613528680245645038560016966} a^{30} - \frac{24377822042334134629677921880922129694100933871390908376968030027645091230524976504219}{3773896742914846417263964090151490643624339450432800551960583953227057360491290077120033932} a^{29} + \frac{13727003361495327947151847735199306985957998579822190450398601240135344411215557581623}{1257965580971615472421321363383830214541446483477600183986861317742352453497096692373344644} a^{28} + \frac{358380131822234587271662736005205675378049784077404784904706024820935378068134558070927}{3773896742914846417263964090151490643624339450432800551960583953227057360491290077120033932} a^{27} - \frac{51695832173150313492871084488271734012059140356427386917049345744476715259628694281837}{314491395242903868105330340845957553635361620869400045996715329435588113374274173093336161} a^{26} - \frac{799604746176411182873271602634786736993020805249059193284500976416039966982214775959967}{943474185728711604315991022537872660906084862608200137990145988306764340122822519280008483} a^{25} + \frac{2525538054120926421483592590450420586786389707890043838237215243054524831120037483086667}{1886948371457423208631982045075745321812169725216400275980291976613528680245645038560016966} a^{24} + \frac{208080341460699518174549330679940555446982102335061756663797528068053845859297059473111}{34943488360322652011703371205106394848373513429933338444079481048398679263808241454815129} a^{23} - \frac{8220840451153920920194177582723171341106637355633551868190944308728973579293430506219205}{943474185728711604315991022537872660906084862608200137990145988306764340122822519280008483} a^{22} - \frac{10780304919773171973689623344319865653386823309547779556962822129919943629846188859210476}{314491395242903868105330340845957553635361620869400045996715329435588113374274173093336161} a^{21} + \frac{56491989791184863987492800633805920167289132192736902531871871316615664162098974265501342}{943474185728711604315991022537872660906084862608200137990145988306764340122822519280008483} a^{20} + \frac{68240867189146769763505833339834096673126242307199507783419671283621997252411962769873389}{419321860323871824140440454461276738180482161159200061328953772580784151165698897457781548} a^{19} - \frac{1086034984268051230567948855803041727721246788816892681500857526449651370260979970218735773}{3773896742914846417263964090151490643624339450432800551960583953227057360491290077120033932} a^{18} - \frac{687640206367460637626170527462680409456027022633254363967378138803879526257225013960601891}{1257965580971615472421321363383830214541446483477600183986861317742352453497096692373344644} a^{17} + \frac{655475093202059859287616699261640343595683804263905953453894957374682989916799524369781791}{943474185728711604315991022537872660906084862608200137990145988306764340122822519280008483} a^{16} + \frac{560563244010020571322429358616470584956227575291719905185308155634498027341268424231324547}{628982790485807736210660681691915107270723241738800091993430658871176226748548346186672322} a^{15} + \frac{256247985138186971967134472940442422735566724246312691443874775667514011985176816503543935}{1886948371457423208631982045075745321812169725216400275980291976613528680245645038560016966} a^{14} + \frac{22887983442315804584632050515515132499883937911759876376824938906637520726750756789223465}{34943488360322652011703371205106394848373513429933338444079481048398679263808241454815129} a^{13} - \frac{5995059601916483921302091402560258332448443151463759672103701418381225506042647140272349230}{943474185728711604315991022537872660906084862608200137990145988306764340122822519280008483} a^{12} - \frac{1290801080784763364436087660485365678494840236332981649591531951084499044413113649503520379}{314491395242903868105330340845957553635361620869400045996715329435588113374274173093336161} a^{11} + \frac{22460086849000254618538332180295114536168760086700608494605275420301054888266104617559554885}{943474185728711604315991022537872660906084862608200137990145988306764340122822519280008483} a^{10} + \frac{10348367971473640016558436778514782463911320057891963692488297137313932000343119320881247393}{1257965580971615472421321363383830214541446483477600183986861317742352453497096692373344644} a^{9} - \frac{150322870321155483480399709354610090593260633055042428657178580263900085624434225538563957259}{3773896742914846417263964090151490643624339450432800551960583953227057360491290077120033932} a^{8} + \frac{857434869179656264780722083014678552687277804215488581636376943738936017392013332398857379}{3773896742914846417263964090151490643624339450432800551960583953227057360491290077120033932} a^{7} + \frac{4255088492094820312143630751308217706824131889152458523276614314725817250295766973860492067}{104830465080967956035110113615319184545120540289800015332238443145196037791424724364445387} a^{6} + \frac{7253774989878402042257397184272859582474886733549609430321128753987086469779465469650553085}{943474185728711604315991022537872660906084862608200137990145988306764340122822519280008483} a^{5} - \frac{8500907968999855213214229240835497323235435099941521936988408887877761839634555896834346277}{314491395242903868105330340845957553635361620869400045996715329435588113374274173093336161} a^{4} + \frac{15488101761903620328184210142272043348054228061618754657859279430495791165988281600184031861}{1886948371457423208631982045075745321812169725216400275980291976613528680245645038560016966} a^{3} + \frac{32753243097747865069607038971082181118666938712426464670509987988400873995953908210446727169}{1886948371457423208631982045075745321812169725216400275980291976613528680245645038560016966} a^{2} + \frac{3612937078539685011999640933174920803914465776721629694828670177445120956536274371216923115}{3773896742914846417263964090151490643624339450432800551960583953227057360491290077120033932} a - \frac{481046753951646482089769475464314414945341394981312961949500810289801922926222287058207631}{69886976720645304023406742410212789696747026859866676888158962096797358527616482909630258} \) (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:  \( 38543990058698.72 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{130}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{2}, \sqrt{65})\), 4.4.274625.1, 4.4.17576000.2, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{10}, \sqrt{26})\), 4.4.274625.2, 4.4.17576000.1, \(\Q(\sqrt{5}, \sqrt{26})\), \(\Q(\sqrt{10}, \sqrt{13})\), \(\Q(\zeta_{5})\), 4.0.8000.2, 4.0.21125.1, 4.0.1352000.1, 4.0.54925.1, 4.0.3515200.1, 4.0.2197.1, 4.0.140608.2, 8.8.308915776000000.3, 8.8.73116160000.2, 8.8.308915776000000.1, 8.8.75418890625.1, 8.8.308915776000000.4, 8.8.308915776000000.5, 8.8.308915776000000.7, 8.0.64000000.2, 8.0.1827904000000.5, 8.0.12356631040000.2, 8.0.19770609664.2, 8.0.446265625.1, 8.0.1827904000000.2, 8.0.3016755625.1, 8.0.12356631040000.5, 8.0.1827904000000.7, 8.0.1827904000000.3, 8.0.12356631040000.4, 8.0.12356631040000.7, 16.16.95428956661682176000000000000.1, 16.0.3341233033216000000000000.1, 16.0.152686330658691481600000000.2, 16.0.5688009063105712890625.1, 16.0.95428956661682176000000000000.6, 16.0.95428956661682176000000000000.4, 16.0.95428956661682176000000000000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ 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$2.8.12.1$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}$
5Data not computed
13Data not computed