↑

Properties

Label 32.0.59816430901...0000.6
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $79.30$
Ramified primes $2, 3, 5, 13$
Class number $10240$ (GRH)
Class group $[2, 8, 8, 80]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6599581, -86254878, 508183759, -1529717908, 2331262742, -1037802612, -1690380687, 1855727336, 821617744, -1745550454, 58509126, 897468480, -314737755, -219341242, 177618004, -4100710, -49420166, 20297534, 6681279, -6794274, 219366, 1284032, -293732, -176218, 70123, 20072, -9626, -1842, 790, 118, -38, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 38*x^30 + 118*x^29 + 790*x^28 - 1842*x^27 - 9626*x^26 + 20072*x^25 + 70123*x^24 - 176218*x^23 - 293732*x^22 + 1284032*x^21 + 219366*x^20 - 6794274*x^19 + 6681279*x^18 + 20297534*x^17 - 49420166*x^16 - 4100710*x^15 + 177618004*x^14 - 219341242*x^13 - 314737755*x^12 + 897468480*x^11 + 58509126*x^10 - 1745550454*x^9 + 821617744*x^8 + 1855727336*x^7 - 1690380687*x^6 - 1037802612*x^5 + 2331262742*x^4 - 1529717908*x^3 + 508183759*x^2 - 86254878*x + 6599581)
 
gp: K = bnfinit(x^32 - 4*x^31 - 38*x^30 + 118*x^29 + 790*x^28 - 1842*x^27 - 9626*x^26 + 20072*x^25 + 70123*x^24 - 176218*x^23 - 293732*x^22 + 1284032*x^21 + 219366*x^20 - 6794274*x^19 + 6681279*x^18 + 20297534*x^17 - 49420166*x^16 - 4100710*x^15 + 177618004*x^14 - 219341242*x^13 - 314737755*x^12 + 897468480*x^11 + 58509126*x^10 - 1745550454*x^9 + 821617744*x^8 + 1855727336*x^7 - 1690380687*x^6 - 1037802612*x^5 + 2331262742*x^4 - 1529717908*x^3 + 508183759*x^2 - 86254878*x + 6599581, 1)
 

Normalized defining polynomial

\( x^{32} - 4 x^{31} - 38 x^{30} + 118 x^{29} + 790 x^{28} - 1842 x^{27} - 9626 x^{26} + 20072 x^{25} + 70123 x^{24} - 176218 x^{23} - 293732 x^{22} + 1284032 x^{21} + 219366 x^{20} - 6794274 x^{19} + 6681279 x^{18} + 20297534 x^{17} - 49420166 x^{16} - 4100710 x^{15} + 177618004 x^{14} - 219341242 x^{13} - 314737755 x^{12} + 897468480 x^{11} + 58509126 x^{10} - 1745550454 x^{9} + 821617744 x^{8} + 1855727336 x^{7} - 1690380687 x^{6} - 1037802612 x^{5} + 2331262742 x^{4} - 1529717908 x^{3} + 508183759 x^{2} - 86254878 x + 6599581 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5981643090147991811559885370844487936000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.30$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(389,·)$, $\chi_{780}(521,·)$, $\chi_{780}(343,·)$, $\chi_{780}(109,·)$, $\chi_{780}(281,·)$, $\chi_{780}(541,·)$, $\chi_{780}(287,·)$, $\chi_{780}(161,·)$, $\chi_{780}(547,·)$, $\chi_{780}(421,·)$, $\chi_{780}(47,·)$, $\chi_{780}(307,·)$, $\chi_{780}(181,·)$, $\chi_{780}(649,·)$, $\chi_{780}(443,·)$, $\chi_{780}(701,·)$, $\chi_{780}(703,·)$, $\chi_{780}(707,·)$, $\chi_{780}(203,·)$, $\chi_{780}(463,·)$, $\chi_{780}(209,·)$, $\chi_{780}(83,·)$, $\chi_{780}(469,·)$, $\chi_{780}(727,·)$, $\chi_{780}(187,·)$, $\chi_{780}(229,·)$, $\chi_{780}(103,·)$, $\chi_{780}(749,·)$, $\chi_{780}(623,·)$, $\chi_{780}(467,·)$, $\chi_{780}(629,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{3} a^{20} - \frac{1}{3} a^{16} + \frac{1}{3} a^{10} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{17} + \frac{1}{3} a^{11} + \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{183} a^{22} - \frac{13}{183} a^{21} + \frac{26}{183} a^{20} + \frac{5}{61} a^{19} - \frac{64}{183} a^{18} + \frac{64}{183} a^{17} + \frac{49}{183} a^{16} + \frac{4}{61} a^{15} - \frac{1}{61} a^{14} - \frac{30}{61} a^{13} - \frac{80}{183} a^{12} + \frac{26}{183} a^{11} - \frac{19}{183} a^{10} + \frac{26}{61} a^{9} - \frac{17}{61} a^{8} - \frac{22}{61} a^{7} + \frac{43}{183} a^{6} + \frac{44}{183} a^{5} + \frac{35}{183} a^{4} - \frac{16}{61} a^{3} - \frac{34}{183} a^{2} + \frac{25}{183} a + \frac{22}{183}$, $\frac{1}{183} a^{23} - \frac{7}{61} a^{21} - \frac{13}{183} a^{20} - \frac{52}{183} a^{19} - \frac{12}{61} a^{18} + \frac{9}{61} a^{17} - \frac{83}{183} a^{16} - \frac{10}{61} a^{15} + \frac{18}{61} a^{14} + \frac{31}{183} a^{13} + \frac{28}{61} a^{12} + \frac{25}{61} a^{11} + \frac{14}{183} a^{10} + \frac{16}{61} a^{9} + \frac{1}{61} a^{8} - \frac{83}{183} a^{7} + \frac{18}{61} a^{6} - \frac{1}{61} a^{5} + \frac{41}{183} a^{4} + \frac{74}{183} a^{3} - \frac{17}{61} a^{2} + \frac{14}{61} a - \frac{80}{183}$, $\frac{1}{366} a^{24} - \frac{7}{61} a^{21} + \frac{1}{61} a^{20} + \frac{16}{61} a^{19} + \frac{49}{122} a^{18} + \frac{17}{61} a^{17} - \frac{80}{183} a^{16} - \frac{10}{61} a^{15} - \frac{16}{183} a^{14} + \frac{4}{61} a^{13} - \frac{47}{122} a^{12} + \frac{12}{61} a^{11} + \frac{38}{183} a^{10} - \frac{1}{61} a^{9} - \frac{28}{183} a^{8} + \frac{22}{61} a^{7} - \frac{5}{122} a^{6} - \frac{12}{61} a^{5} + \frac{23}{61} a^{4} - \frac{24}{61} a^{3} + \frac{10}{61} a^{2} + \frac{3}{61} a + \frac{35}{366}$, $\frac{1}{366} a^{25} - \frac{26}{183} a^{21} - \frac{16}{183} a^{20} + \frac{15}{122} a^{19} - \frac{4}{61} a^{18} - \frac{26}{61} a^{17} - \frac{38}{183} a^{16} + \frac{53}{183} a^{15} - \frac{17}{61} a^{14} + \frac{35}{122} a^{13} + \frac{1}{61} a^{12} - \frac{29}{61} a^{11} + \frac{86}{183} a^{10} - \frac{37}{183} a^{9} - \frac{30}{61} a^{8} + \frac{47}{122} a^{7} - \frac{16}{61} a^{6} - \frac{44}{183} a^{5} + \frac{53}{183} a^{4} - \frac{21}{61} a^{3} + \frac{9}{61} a^{2} - \frac{45}{122} a - \frac{26}{183}$, $\frac{1}{366} a^{26} + \frac{4}{61} a^{21} + \frac{55}{366} a^{20} + \frac{4}{61} a^{19} + \frac{88}{183} a^{18} - \frac{7}{61} a^{17} - \frac{5}{61} a^{16} + \frac{26}{61} a^{15} - \frac{17}{122} a^{14} + \frac{14}{61} a^{13} + \frac{29}{183} a^{12} + \frac{10}{61} a^{11} + \frac{79}{183} a^{10} - \frac{25}{61} a^{9} + \frac{17}{122} a^{8} + \frac{22}{61} a^{7} - \frac{8}{61} a^{6} - \frac{28}{61} a^{5} - \frac{7}{183} a^{4} + \frac{20}{61} a^{3} - \frac{73}{366} a^{2} + \frac{25}{61} a - \frac{38}{183}$, $\frac{1}{366} a^{27} + \frac{1}{366} a^{21} + \frac{5}{183} a^{20} + \frac{91}{183} a^{19} + \frac{5}{61} a^{18} - \frac{17}{61} a^{17} - \frac{83}{183} a^{16} + \frac{9}{122} a^{15} + \frac{26}{61} a^{14} + \frac{11}{183} a^{13} + \frac{25}{61} a^{12} - \frac{50}{183} a^{11} - \frac{91}{183} a^{10} + \frac{3}{122} a^{9} - \frac{18}{61} a^{8} + \frac{12}{61} a^{7} - \frac{17}{61} a^{6} + \frac{14}{183} a^{5} - \frac{55}{183} a^{4} - \frac{19}{366} a^{3} - \frac{22}{61} a^{2} + \frac{28}{183} a - \frac{20}{183}$, $\frac{1}{86742} a^{28} + \frac{5}{43371} a^{27} - \frac{32}{43371} a^{26} + \frac{20}{43371} a^{25} - \frac{16}{14457} a^{24} + \frac{62}{43371} a^{23} + \frac{55}{86742} a^{22} - \frac{3326}{43371} a^{21} - \frac{1126}{43371} a^{20} + \frac{5159}{43371} a^{19} - \frac{5866}{43371} a^{18} + \frac{4433}{43371} a^{17} - \frac{4627}{9638} a^{16} + \frac{11312}{43371} a^{15} - \frac{5794}{43371} a^{14} + \frac{2146}{43371} a^{13} + \frac{17930}{43371} a^{12} - \frac{5803}{43371} a^{11} - \frac{31271}{86742} a^{10} - \frac{14926}{43371} a^{9} + \frac{1420}{4819} a^{8} - \frac{20947}{43371} a^{7} - \frac{4069}{43371} a^{6} + \frac{7549}{43371} a^{5} - \frac{377}{1422} a^{4} - \frac{14983}{43371} a^{3} - \frac{2551}{14457} a^{2} - \frac{4699}{14457} a + \frac{137}{549}$, $\frac{1}{21329077122} a^{29} + \frac{1201}{3554846187} a^{28} - \frac{17020307}{21329077122} a^{27} - \frac{16430749}{21329077122} a^{26} - \frac{2283245}{10664538561} a^{25} - \frac{8757935}{21329077122} a^{24} - \frac{1741379}{7109692374} a^{23} + \frac{840626}{10664538561} a^{22} + \frac{2555071805}{21329077122} a^{21} + \frac{202217039}{7109692374} a^{20} + \frac{311887136}{1184948729} a^{19} - \frac{1961336695}{7109692374} a^{18} + \frac{7379174999}{21329077122} a^{17} - \frac{2438255704}{10664538561} a^{16} + \frac{676543035}{2369897458} a^{15} - \frac{7764280643}{21329077122} a^{14} + \frac{899228806}{10664538561} a^{13} - \frac{2620242413}{7109692374} a^{12} - \frac{819827509}{7109692374} a^{11} - \frac{1285983458}{3554846187} a^{10} + \frac{3729353429}{21329077122} a^{9} - \frac{5121311393}{21329077122} a^{8} - \frac{951706853}{3554846187} a^{7} - \frac{5724508979}{21329077122} a^{6} + \frac{9674847077}{21329077122} a^{5} - \frac{1636780355}{3554846187} a^{4} + \frac{8417692691}{21329077122} a^{3} + \frac{1765542241}{7109692374} a^{2} - \frac{505827607}{10664538561} a - \frac{736375}{1942362}$, $\frac{1}{175333373382326350023223413914302344342} a^{30} + \frac{2434222280884599861903165409}{175333373382326350023223413914302344342} a^{29} - \frac{157548907872369171470642646014141}{87666686691163175011611706957151172171} a^{28} - \frac{87917707907937986436626367005462989}{87666686691163175011611706957151172171} a^{27} - \frac{305135962272522677786511059185789}{1007663065415668678294387436289093933} a^{26} - \frac{54190353846138180285140260000766656}{87666686691163175011611706957151172171} a^{25} - \frac{119441837750788155898165644921262001}{175333373382326350023223413914302344342} a^{24} + \frac{109342042266802467648390260330495651}{175333373382326350023223413914302344342} a^{23} + \frac{209273843620260883661013708479292977}{87666686691163175011611706957151172171} a^{22} - \frac{134198142632564816068014035848051592}{3022989196247006034883162308867281799} a^{21} + \frac{12421342481322982185502663431687399728}{87666686691163175011611706957151172171} a^{20} - \frac{42584458965594919555132157681708734744}{87666686691163175011611706957151172171} a^{19} + \frac{25634200311570116141697813298686317651}{58444457794108783341074471304767448114} a^{18} - \frac{69653124766164701800500513534050707181}{175333373382326350023223413914302344342} a^{17} - \frac{746933131226580427884854545929875668}{87666686691163175011611706957151172171} a^{16} - \frac{6227481445043981588861237283900183596}{87666686691163175011611706957151172171} a^{15} + \frac{13835223585536283744353256718527595424}{87666686691163175011611706957151172171} a^{14} - \frac{14656037287361163704338444518230076316}{87666686691163175011611706957151172171} a^{13} + \frac{9662860810007223014374568320591341637}{175333373382326350023223413914302344342} a^{12} + \frac{1307266375837728895182925992323970965}{6045978392494012069766324617734563598} a^{11} - \frac{961080399901331998026374476703250410}{9740742965684797223512411884127908019} a^{10} - \frac{37699009711037961050809343818863687736}{87666686691163175011611706957151172171} a^{9} - \frac{13177858706779737798190366154005000621}{87666686691163175011611706957151172171} a^{8} + \frac{13008004245184952877500410477730240059}{87666686691163175011611706957151172171} a^{7} + \frac{25434996888067376997869767921289555509}{175333373382326350023223413914302344342} a^{6} - \frac{69740602946063629756117715849801744021}{175333373382326350023223413914302344342} a^{5} - \frac{7717234903881628950655846933623293465}{29222228897054391670537235652383724057} a^{4} + \frac{6852946560793974290641811936550838328}{29222228897054391670537235652383724057} a^{3} + \frac{41529056001454668908049407253399663521}{87666686691163175011611706957151172171} a^{2} - \frac{8982997615057210843308690422166398366}{29222228897054391670537235652383724057} a + \frac{105597698782831527411197722022930}{2661162817325780135737841330696997}$, $\frac{1}{4064380662338080409212928016951000187639208164438480547915751931623663836026088180226402} a^{31} + \frac{1658592030455095711918781181289262588333879362775}{1354793554112693469737642672317000062546402721479493515971917310541221278675362726742134} a^{30} + \frac{40263475539591083338187532419533255202539757590435419028023892849435742150192}{2032190331169040204606464008475500093819604082219240273957875965811831918013044090113201} a^{29} + \frac{22398845630335291832427125594130773790174654151443806325787650073897118942849049911}{4064380662338080409212928016951000187639208164438480547915751931623663836026088180226402} a^{28} + \frac{1671726424373303217934291765524706054106391271827214546964597046436428290601215577877}{2032190331169040204606464008475500093819604082219240273957875965811831918013044090113201} a^{27} - \frac{654083698677932351048330848208188612881960621142508689268773451673306196520594970641}{4064380662338080409212928016951000187639208164438480547915751931623663836026088180226402} a^{26} - \frac{188326022634795477757805567016749335982198504582388636299792715732559224947520028563}{451597851370897823245880890772333354182134240493164505323972436847073759558454242247378} a^{25} - \frac{3509072153520961289071000903265214016695305040909121565957018861105140126353219085925}{4064380662338080409212928016951000187639208164438480547915751931623663836026088180226402} a^{24} - \frac{935050561742468625987503999813039285742761571095751060270116695183070712754745452067}{2032190331169040204606464008475500093819604082219240273957875965811831918013044090113201} a^{23} + \frac{627677144818858384908332695413398982298590313173924451508391010301078819599664738585}{1354793554112693469737642672317000062546402721479493515971917310541221278675362726742134} a^{22} - \frac{12256694980544306691542783887899915640952817424671832802605349170327223349798631451798}{677396777056346734868821336158500031273201360739746757985958655270610639337681363371067} a^{21} + \frac{206071210011662163796299912463169446844109258398397872709478269168762860611587584690711}{1354793554112693469737642672317000062546402721479493515971917310541221278675362726742134} a^{20} - \frac{629142757554285391301541896630774643030148323104441900242232468216232890292904914957481}{4064380662338080409212928016951000187639208164438480547915751931623663836026088180226402} a^{19} - \frac{752275179356765619502741368459288219241434160788304222218862623925325555410175554979303}{4064380662338080409212928016951000187639208164438480547915751931623663836026088180226402} a^{18} - \frac{1242350961355283100467200411580102921848557012236626889091593209983658450336136080690}{11104865197645028440472480920631148053659038700651586196491125496239518677666907596247} a^{17} + \frac{394107345634675852466713527529380389448838617237737357671027999322926326680840704268809}{4064380662338080409212928016951000187639208164438480547915751931623663836026088180226402} a^{16} - \frac{640945813132745555763182358803979674175208904898472286187330772727568823462587862719002}{2032190331169040204606464008475500093819604082219240273957875965811831918013044090113201} a^{15} - \frac{167489514790990262541525650584923799931510493750673642530737020005081926775007298856367}{1354793554112693469737642672317000062546402721479493515971917310541221278675362726742134} a^{14} + \frac{1925933594226871994526559261370488668581406768501238512200312145519486715732652885233}{17149285495097385692881552814139241298055730651639158430024269753686345299688135781546} a^{13} - \frac{522171447510978187440446675809219181162773327791655317429510275673844567672581542083667}{1354793554112693469737642672317000062546402721479493515971917310541221278675362726742134} a^{12} + \frac{916017979685988612063745799269249762928721592289147511515128200669232625978067568710790}{2032190331169040204606464008475500093819604082219240273957875965811831918013044090113201} a^{11} - \frac{17867351877018615026425100522760898477700492510748902777385343018658181641583315390331}{140151057322002772731480276446586213366869247049602777514336273504264270207796144145738} a^{10} - \frac{119924764068357292517203299801607401275409528414110359018216841772513780548937774671869}{677396777056346734868821336158500031273201360739746757985958655270610639337681363371067} a^{9} + \frac{1032667224634271108617745285690791909925627188582503385135899067760191316527277690439901}{4064380662338080409212928016951000187639208164438480547915751931623663836026088180226402} a^{8} - \frac{1337576637658113577708451380152381771008968615205075245988207203479074920431401315104729}{4064380662338080409212928016951000187639208164438480547915751931623663836026088180226402} a^{7} - \frac{41451309359545658908271960688199431975905068785521685872128016305775898809152824641941}{1354793554112693469737642672317000062546402721479493515971917310541221278675362726742134} a^{6} - \frac{11566915238390501216336344373499546133731047441773087136361463201103629703005288250020}{70075528661001386365740138223293106683434623524801388757168136752132135103898072072869} a^{5} - \frac{327358704777162653288339726248698436963406272854239992711737973260102740594898493936169}{1354793554112693469737642672317000062546402721479493515971917310541221278675362726742134} a^{4} - \frac{692169641961456996158871845877965585045497153548182871583200668913811756427226721404781}{2032190331169040204606464008475500093819604082219240273957875965811831918013044090113201} a^{3} + \frac{51088612178960708295429605944595154853490781863137210644694963413844826706852720847267}{140151057322002772731480276446586213366869247049602777514336273504264270207796144145738} a^{2} + \frac{240087366223703933742993266982058999019509948155137732064036818899577878391133820849103}{677396777056346734868821336158500031273201360739746757985958655270610639337681363371067} a - \frac{6007703407378431662285319935221456473687335648823798299507706046412217423335873450}{61688077320494193139861700770285040640488239754097692194331905588799803236288258207}$

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

Class group and class number

$C_{2}\times C_{8}\times C_{8}\times C_{80}$, which has order $10240$ (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{19385174692504488717023868075295820795825440}{81307596511916590191493092054594891530840594045673} a^{31} + \frac{20119496990548484637434798331638541654122492}{27102532170638863397164364018198297176946864681891} a^{30} + \frac{786761668559484738106905287628036173341551496}{81307596511916590191493092054594891530840594045673} a^{29} - \frac{526092839570350811041738551470472301976941875}{27102532170638863397164364018198297176946864681891} a^{28} - \frac{5526091488932794030230391682914361681142680256}{27102532170638863397164364018198297176946864681891} a^{27} + \frac{6893259762898672631845678453463707210799534669}{27102532170638863397164364018198297176946864681891} a^{26} + \frac{67355156610490962990966390129067517505103784952}{27102532170638863397164364018198297176946864681891} a^{25} - \frac{68383204048755407756008600823105428285168614571}{27102532170638863397164364018198297176946864681891} a^{24} - \frac{1505566306608125287634972243250061338235259653004}{81307596511916590191493092054594891530840594045673} a^{23} + \frac{676958541876131703984365189896118612928002430971}{27102532170638863397164364018198297176946864681891} a^{22} + \frac{2407849843973540189428048289634913994853419240704}{27102532170638863397164364018198297176946864681891} a^{21} - \frac{6007514052029489694731307670197522331340023886602}{27102532170638863397164364018198297176946864681891} a^{20} - \frac{18938925257078751330172751843384555086957774613144}{81307596511916590191493092054594891530840594045673} a^{19} + \frac{37054576483655419074392322854997242795387795493164}{27102532170638863397164364018198297176946864681891} a^{18} - \frac{11305546780997503825808702249235936690038694802940}{27102532170638863397164364018198297176946864681891} a^{17} - \frac{133801309104001537028767539030606936935840225331412}{27102532170638863397164364018198297176946864681891} a^{16} + \frac{197223445199938836904703385452984493383190724147916}{27102532170638863397164364018198297176946864681891} a^{15} + \frac{2218710356062701305657064338312859010257004918111}{343070027476441308824865367318965787049960312429} a^{14} - \frac{2848989425106609516041678268759342444708337414813544}{81307596511916590191493092054594891530840594045673} a^{13} + \frac{606965023372971253366903698398846033283536756829840}{27102532170638863397164364018198297176946864681891} a^{12} + \frac{7142977879634433283189752234175147553109862413887432}{81307596511916590191493092054594891530840594045673} a^{11} - \frac{3545912015529641554013558210239497929482804853097964}{27102532170638863397164364018198297176946864681891} a^{10} - \frac{3059343612712702759553386913059585812861553207976994}{27102532170638863397164364018198297176946864681891} a^{9} + \frac{7773359042265854787582834091687704958027720560199028}{27102532170638863397164364018198297176946864681891} a^{8} + \frac{3054913888381717101436744091192061219718801623304456}{81307596511916590191493092054594891530840594045673} a^{7} - \frac{9290826271276224585025536129345580738947998798002801}{27102532170638863397164364018198297176946864681891} a^{6} + \frac{8452332662347426235378288739646023323847706954930960}{81307596511916590191493092054594891530840594045673} a^{5} + \frac{6914958304349131190560612257137081095608965563380792}{27102532170638863397164364018198297176946864681891} a^{4} - \frac{8295620990887502528126070382105836269690347703686950}{27102532170638863397164364018198297176946864681891} a^{3} + \frac{4354978154289187833589011749367067682449181226279284}{27102532170638863397164364018198297176946864681891} a^{2} - \frac{3405004050828837289429580168801595702360736288233566}{81307596511916590191493092054594891530840594045673} a + \frac{12554155766438046390130075935560535868201999467}{2468129694075117329675290412366660338488923111} \) (order $6$)
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:  \( 95287215485273.48 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-3}, \sqrt{65})\), 4.0.4394000.2, 4.4.39546000.2, \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), 4.0.4394000.1, 4.4.39546000.1, \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.4.19773.1, 4.0.54925.1, 4.4.494325.1, 4.0.2197.1, 4.0.18000.1, 4.4.338000.1, 4.0.3042000.1, \(\Q(\zeta_{20})^+\), 8.0.1563886116000000.48, 8.0.1445900625.1, 8.0.1563886116000000.39, 8.0.19307236000000.5, 8.0.1563886116000000.66, 8.8.1563886116000000.7, 8.0.1563886116000000.47, 8.0.244357205625.2, 8.0.244357205625.1, 8.0.9253764000000.2, 8.0.9253764000000.5, 8.8.244357205625.1, 8.0.3016755625.1, 8.0.9253764000000.3, 8.8.114244000000.1, 8.0.390971529.1, 8.0.244357205625.3, 8.0.324000000.2, 8.0.9253764000000.9, 16.0.2445739783817565456000000000000.12, 16.0.59710443940858531640625.1, 16.0.85632148167696000000000000.7, 16.0.2445739783817565456000000000000.16, 16.0.372769361959696000000000000.2, 16.16.2445739783817565456000000000000.3, 16.0.2445739783817565456000000000000.6

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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

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

Local algebras for ramified primes

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