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Properties

Label 32.0.14866321544...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $5^{24}\cdot 11^{16}\cdot 13^{24}$
Root discriminant $75.92$
Ramified primes $5, 11, 13$
Class number Not computed
Class group Not computed
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![43046721, 81310473, 167935356, 324119961, 581698260, 391599846, 552237912, 453150045, 39044106, -773200890, 484557723, -183754935, -171658424, 132433057, 14501232, -50339403, 35958729, -2749375, -5738943, 2546494, -232848, -65208, 99151, -42162, -2856, 6853, 111, -778, 117, 24, -12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 12*x^30 + 24*x^29 + 117*x^28 - 778*x^27 + 111*x^26 + 6853*x^25 - 2856*x^24 - 42162*x^23 + 99151*x^22 - 65208*x^21 - 232848*x^20 + 2546494*x^19 - 5738943*x^18 - 2749375*x^17 + 35958729*x^16 - 50339403*x^15 + 14501232*x^14 + 132433057*x^13 - 171658424*x^12 - 183754935*x^11 + 484557723*x^10 - 773200890*x^9 + 39044106*x^8 + 453150045*x^7 + 552237912*x^6 + 391599846*x^5 + 581698260*x^4 + 324119961*x^3 + 167935356*x^2 + 81310473*x + 43046721)
 
gp: K = bnfinit(x^32 - 2*x^31 - 12*x^30 + 24*x^29 + 117*x^28 - 778*x^27 + 111*x^26 + 6853*x^25 - 2856*x^24 - 42162*x^23 + 99151*x^22 - 65208*x^21 - 232848*x^20 + 2546494*x^19 - 5738943*x^18 - 2749375*x^17 + 35958729*x^16 - 50339403*x^15 + 14501232*x^14 + 132433057*x^13 - 171658424*x^12 - 183754935*x^11 + 484557723*x^10 - 773200890*x^9 + 39044106*x^8 + 453150045*x^7 + 552237912*x^6 + 391599846*x^5 + 581698260*x^4 + 324119961*x^3 + 167935356*x^2 + 81310473*x + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} - 12 x^{30} + 24 x^{29} + 117 x^{28} - 778 x^{27} + 111 x^{26} + 6853 x^{25} - 2856 x^{24} - 42162 x^{23} + 99151 x^{22} - 65208 x^{21} - 232848 x^{20} + 2546494 x^{19} - 5738943 x^{18} - 2749375 x^{17} + 35958729 x^{16} - 50339403 x^{15} + 14501232 x^{14} + 132433057 x^{13} - 171658424 x^{12} - 183754935 x^{11} + 484557723 x^{10} - 773200890 x^{9} + 39044106 x^{8} + 453150045 x^{7} + 552237912 x^{6} + 391599846 x^{5} + 581698260 x^{4} + 324119961 x^{3} + 167935356 x^{2} + 81310473 x + 43046721 \)

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:  \(1486632154491018526083574799564592499515123426914215087890625=5^{24}\cdot 11^{16}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.92$
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:  $5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(715=5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{715}(1,·)$, $\chi_{715}(131,·)$, $\chi_{715}(12,·)$, $\chi_{715}(142,·)$, $\chi_{715}(144,·)$, $\chi_{715}(274,·)$, $\chi_{715}(21,·)$, $\chi_{715}(408,·)$, $\chi_{715}(538,·)$, $\chi_{715}(287,·)$, $\chi_{715}(417,·)$, $\chi_{715}(34,·)$, $\chi_{715}(164,·)$, $\chi_{715}(551,·)$, $\chi_{715}(681,·)$, $\chi_{715}(298,·)$, $\chi_{715}(428,·)$, $\chi_{715}(177,·)$, $\chi_{715}(307,·)$, $\chi_{715}(694,·)$, $\chi_{715}(441,·)$, $\chi_{715}(571,·)$, $\chi_{715}(573,·)$, $\chi_{715}(703,·)$, $\chi_{715}(584,·)$, $\chi_{715}(714,·)$, $\chi_{715}(463,·)$, $\chi_{715}(593,·)$, $\chi_{715}(606,·)$, $\chi_{715}(109,·)$, $\chi_{715}(122,·)$, $\chi_{715}(252,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{1}{9} a^{7} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{18} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{16} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{8} + \frac{1}{3} a^{6} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{17} + \frac{1}{9} a^{12} + \frac{4}{9} a^{7} + \frac{7}{18} a^{2}$, $\frac{1}{18} a^{18} + \frac{1}{9} a^{13} + \frac{4}{9} a^{8} + \frac{7}{18} a^{3}$, $\frac{1}{54} a^{19} - \frac{1}{54} a^{17} - \frac{4}{27} a^{13} + \frac{4}{27} a^{11} - \frac{1}{9} a^{10} - \frac{1}{27} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{13}{27} a^{5} + \frac{7}{18} a^{4} - \frac{10}{27} a^{3} + \frac{7}{18} a^{2} + \frac{1}{3} a$, $\frac{1}{54} a^{20} - \frac{1}{54} a^{18} - \frac{1}{27} a^{14} + \frac{1}{9} a^{13} + \frac{1}{27} a^{12} + \frac{1}{9} a^{11} - \frac{1}{27} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{13}{27} a^{6} - \frac{7}{18} a^{5} - \frac{13}{27} a^{4} - \frac{1}{2} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{54} a^{21} - \frac{1}{54} a^{17} + \frac{1}{54} a^{15} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{27} a^{9} + \frac{1}{9} a^{8} + \frac{13}{27} a^{7} - \frac{7}{18} a^{6} + \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{27} a^{3} - \frac{5}{18} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{54} a^{22} - \frac{1}{54} a^{18} + \frac{1}{54} a^{16} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{27} a^{10} + \frac{1}{9} a^{9} + \frac{13}{27} a^{8} - \frac{7}{18} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} - \frac{1}{27} a^{4} - \frac{5}{18} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{54} a^{23} + \frac{2}{27} a^{13} - \frac{1}{2} a^{8} + \frac{11}{27} a^{3}$, $\frac{1}{162} a^{24} + \frac{1}{162} a^{23} - \frac{1}{162} a^{19} - \frac{1}{81} a^{17} - \frac{1}{54} a^{15} - \frac{1}{81} a^{14} + \frac{2}{27} a^{13} - \frac{7}{81} a^{11} - \frac{1}{9} a^{10} - \frac{13}{162} a^{9} + \frac{7}{18} a^{8} - \frac{2}{27} a^{7} - \frac{1}{9} a^{6} + \frac{35}{81} a^{5} - \frac{17}{162} a^{4} + \frac{1}{3} a^{3} - \frac{1}{9} a^{2} - \frac{1}{2}$, $\frac{1}{486} a^{25} + \frac{1}{486} a^{24} - \frac{1}{162} a^{22} - \frac{2}{243} a^{20} + \frac{2}{243} a^{18} + \frac{1}{54} a^{17} + \frac{2}{81} a^{16} - \frac{11}{486} a^{15} - \frac{1}{27} a^{14} + \frac{4}{27} a^{13} - \frac{28}{243} a^{12} + \frac{1}{9} a^{11} - \frac{1}{486} a^{10} - \frac{1}{18} a^{9} - \frac{5}{27} a^{8} + \frac{5}{54} a^{7} + \frac{113}{243} a^{6} + \frac{113}{243} a^{5} + \frac{2}{81} a^{4} - \frac{10}{27} a^{3} + \frac{5}{18} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{2916} a^{26} + \frac{1}{2916} a^{25} - \frac{1}{324} a^{24} - \frac{1}{972} a^{23} + \frac{5}{2916} a^{21} + \frac{1}{162} a^{20} - \frac{5}{2916} a^{19} - \frac{7}{324} a^{18} + \frac{11}{486} a^{17} + \frac{4}{729} a^{16} - \frac{7}{324} a^{15} - \frac{1}{27} a^{14} + \frac{197}{1458} a^{13} + \frac{17}{162} a^{12} - \frac{19}{2916} a^{11} - \frac{31}{324} a^{10} + \frac{17}{324} a^{9} - \frac{67}{324} a^{8} - \frac{20}{729} a^{7} + \frac{937}{2916} a^{6} - \frac{77}{243} a^{5} + \frac{29}{324} a^{4} - \frac{13}{36} a^{3} - \frac{1}{9} a^{2} - \frac{1}{4}$, $\frac{1}{8748} a^{27} + \frac{1}{8748} a^{26} - \frac{1}{972} a^{25} - \frac{1}{2916} a^{24} - \frac{1}{162} a^{23} - \frac{49}{8748} a^{22} - \frac{1}{243} a^{21} - \frac{59}{8748} a^{20} + \frac{5}{972} a^{19} - \frac{25}{1458} a^{18} + \frac{31}{2187} a^{17} + \frac{23}{972} a^{16} - \frac{1}{54} a^{15} - \frac{73}{4374} a^{14} + \frac{29}{486} a^{13} + \frac{521}{8748} a^{12} - \frac{79}{972} a^{11} + \frac{77}{972} a^{10} + \frac{29}{972} a^{9} + \frac{1769}{4374} a^{8} + \frac{3259}{8748} a^{7} - \frac{361}{1458} a^{6} + \frac{143}{972} a^{5} + \frac{53}{324} a^{4} - \frac{4}{9} a^{3} + \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{2965572} a^{28} - \frac{35}{2965572} a^{27} + \frac{1}{54918} a^{26} + \frac{35}{247131} a^{25} - \frac{25}{109836} a^{24} - \frac{6445}{741393} a^{23} - \frac{29}{27459} a^{22} + \frac{7427}{1482786} a^{21} + \frac{907}{329508} a^{20} - \frac{3779}{988524} a^{19} + \frac{71377}{2965572} a^{18} - \frac{6623}{329508} a^{17} - \frac{289}{82377} a^{16} - \frac{60815}{2965572} a^{15} + \frac{575}{54918} a^{14} - \frac{29809}{2965572} a^{13} + \frac{14069}{109836} a^{12} + \frac{4576}{27459} a^{11} + \frac{10490}{82377} a^{10} - \frac{172313}{2965572} a^{9} + \frac{232648}{741393} a^{8} + \frac{58631}{494262} a^{7} - \frac{6611}{82377} a^{6} - \frac{46007}{109836} a^{5} + \frac{5467}{36612} a^{4} + \frac{1177}{12204} a^{3} - \frac{931}{4068} a^{2} - \frac{100}{339} a + \frac{169}{452}$, $\frac{1}{1002228464704385566364613711396157594205796348} a^{29} + \frac{15689103487944999381378733758250218371}{501114232352192783182306855698078797102898174} a^{28} - \frac{2332632450992106618652385687434071131077}{334076154901461855454871237132052531401932116} a^{27} + \frac{2248957232389733288618365774056736633093}{167038077450730927727435618566026265700966058} a^{26} + \frac{9330529803968426474609542749736284524305}{111358718300487285151623745710684177133977372} a^{25} + \frac{2945444571613297629959213057819008885918259}{1002228464704385566364613711396157594205796348} a^{24} + \frac{187723931880011869569028015906024959127361}{167038077450730927727435618566026265700966058} a^{23} - \frac{686906197400923737465106022909280853619681}{250557116176096391591153427849039398551449087} a^{22} - \frac{1253454797903346174929085913299781545074305}{334076154901461855454871237132052531401932116} a^{21} + \frac{278446150002500769013896405974065273867726}{83519038725365463863717809283013132850483029} a^{20} + \frac{1021172021193788598400154769935395674865165}{250557116176096391591153427849039398551449087} a^{19} + \frac{1851304954151748258723323800430410758811441}{167038077450730927727435618566026265700966058} a^{18} + \frac{1021471427450542332890882595064841519549957}{37119572766829095050541248570228059044659124} a^{17} + \frac{2951899413346300271156847974238219123053947}{1002228464704385566364613711396157594205796348} a^{16} - \frac{7120039341026747681630342575414645780878235}{334076154901461855454871237132052531401932116} a^{15} - \frac{24265846985637950321448110468702700189176731}{1002228464704385566364613711396157594205796348} a^{14} - \frac{16372293546976475877550916567537362310623081}{167038077450730927727435618566026265700966058} a^{13} + \frac{5291068398365605697934836492923067152722281}{111358718300487285151623745710684177133977372} a^{12} + \frac{3579093098670100396940115071739851180858876}{27839679575121821287905936427671044283494343} a^{11} - \frac{161125901459940324232878914140169296310610641}{1002228464704385566364613711396157594205796348} a^{10} + \frac{12782453079740845816857320451110208138337285}{1002228464704385566364613711396157594205796348} a^{9} + \frac{9017922081798848086933887866336710437207424}{27839679575121821287905936427671044283494343} a^{8} + \frac{5073048129096363090965065665799785286792507}{27839679575121821287905936427671044283494343} a^{7} + \frac{1526906230299375089837724733660377997014157}{4124396974092121672282360952247562116073236} a^{6} - \frac{872002937506122064235106499214941361288783}{3093297730569091254211770714185671587054927} a^{5} + \frac{270327407480488725078734379435252439709437}{687399495682020278713726825374593686012206} a^{4} + \frac{29007617983393776773736428837660056146661}{343699747841010139356863412687296843006103} a^{3} + \frac{14726397042126665899447668346626492701851}{152755443484893395269717072305465263558268} a^{2} + \frac{53313493471327873507171229606268459022975}{152755443484893395269717072305465263558268} a - \frac{7665888820057794249563257240617578470867}{16972827053877043918857452478385029284252}$, $\frac{1}{9020056182339470097281523402565418347852167132} a^{30} - \frac{1}{4510028091169735048640761701282709173926083566} a^{29} - \frac{251024350598146306040870041873678813655}{1503342697056578349546920567094236391308694522} a^{28} + \frac{19259257512900010191865383226421408716457}{3006685394113156699093841134188472782617389044} a^{27} - \frac{10161911935553112863474308370961698465321}{1002228464704385566364613711396157594205796348} a^{26} - \frac{3771712027668875418624365379348474452609263}{9020056182339470097281523402565418347852167132} a^{25} + \frac{2431636649415195331422756472414002293233537}{3006685394113156699093841134188472782617389044} a^{24} - \frac{14368266055662031098941167743115455379385756}{2255014045584867524320380850641354586963041783} a^{23} - \frac{15157291117241958358428769464710696401498573}{3006685394113156699093841134188472782617389044} a^{22} - \frac{26474915065126423989871438033449990617399}{5407707543368986868873814989547612918376599} a^{21} - \frac{332328997994948528450162388164829780499819}{47225425038426545011945148704530986114409252} a^{20} - \frac{20564027821447157173671071521879406502368773}{3006685394113156699093841134188472782617389044} a^{19} - \frac{487536993396948976748801047437477589694389}{167038077450730927727435618566026265700966058} a^{18} - \frac{10352973041595447582495335683157656707873974}{2255014045584867524320380850641354586963041783} a^{17} - \frac{59464710630877708509504866323327775103800017}{3006685394113156699093841134188472782617389044} a^{16} + \frac{5722129168817225970315980068887734389130539}{4510028091169735048640761701282709173926083566} a^{15} + \frac{8386426321703477598965113164654150686893489}{751671348528289174773460283547118195654347261} a^{14} + \frac{28699308865773119620170525754397050526099018}{250557116176096391591153427849039398551449087} a^{13} - \frac{94207860567128718651698466837168602063988625}{1002228464704385566364613711396157594205796348} a^{12} - \frac{1108402321492350226144605563772046349401022381}{9020056182339470097281523402565418347852167132} a^{11} + \frac{1461789331856019805933695017012071580744206435}{9020056182339470097281523402565418347852167132} a^{10} - \frac{58039692028330539370156768296431231010021157}{1002228464704385566364613711396157594205796348} a^{9} - \frac{33006473362095436134655545973583002409840360}{250557116176096391591153427849039398551449087} a^{8} + \frac{2110192855201200276823212641303393275577701}{111358718300487285151623745710684177133977372} a^{7} - \frac{25646942092689801792195979105942266643008163}{55679359150243642575811872855342088566988686} a^{6} - \frac{2840105707802006839205559349052060328392347}{12373190922276365016847082856742686348219708} a^{5} - \frac{5644389970871034079696518235365351524362817}{12373190922276365016847082856742686348219708} a^{4} + \frac{43966130050192423257495689349970212640687}{343699747841010139356863412687296843006103} a^{3} + \frac{30096815306505332629424906084543640536317}{343699747841010139356863412687296843006103} a^{2} - \frac{48123005017330194656053225186045622944045}{152755443484893395269717072305465263558268} a - \frac{2016001338191571466020611758687928182767}{16972827053877043918857452478385029284252}$, $\frac{1}{162361011282110461751067421246177530261339008376} a^{31} + \frac{7}{162361011282110461751067421246177530261339008376} a^{30} + \frac{17}{54120337094036820583689140415392510087113002792} a^{29} - \frac{4376248667163987190638261136210185496759}{54120337094036820583689140415392510087113002792} a^{28} + \frac{47731528844545456230176332201452782233399}{4510028091169735048640761701282709173926083566} a^{27} - \frac{1576247224496969128784817741790953788944903}{40590252820527615437766855311544382565334752094} a^{26} - \frac{14056223920424987484485552197802909365515899}{54120337094036820583689140415392510087113002792} a^{25} - \frac{29480879696496781778517844745857832162717645}{40590252820527615437766855311544382565334752094} a^{24} + \frac{21726382306306274737214585037731345330020309}{6765042136754602572961142551924063760889125349} a^{23} + \frac{23635511995749095193248545401929210403442277}{6765042136754602572961142551924063760889125349} a^{22} + \frac{723006278981795613080257751664466729022196925}{162361011282110461751067421246177530261339008376} a^{21} - \frac{14673937248469731213054751401926652933621295}{54120337094036820583689140415392510087113002792} a^{20} + \frac{2713855076585581659497498721849048856020163}{668152309802923710909742474264105062803864232} a^{19} + \frac{3257915932309248120416330685150066063076002331}{162361011282110461751067421246177530261339008376} a^{18} - \frac{503310557831476999971030846054545352611654933}{27060168547018410291844570207696255043556501396} a^{17} + \frac{4098754994476452584241537933940139174668867247}{162361011282110461751067421246177530261339008376} a^{16} - \frac{514373470383448523130542880248892208707380473}{27060168547018410291844570207696255043556501396} a^{15} - \frac{194373129230096317578517726554063507574661}{11939187534532720181709494907432717866118024} a^{14} + \frac{1651049248764312927082519458407857590594575513}{18040112364678940194563046805130836695704334264} a^{13} - \frac{569064607434089648739788169517641664345079113}{81180505641055230875533710623088765130669504188} a^{12} - \frac{1194380644724980077248872536944436362729556971}{40590252820527615437766855311544382565334752094} a^{11} - \frac{192339864484514408620150483042862161182471197}{6013370788226313398187682268376945565234778088} a^{10} - \frac{326424030857772020485851819844981959553081666}{2255014045584867524320380850641354586963041783} a^{9} + \frac{328158397480353363217234769128680494855809903}{1002228464704385566364613711396157594205796348} a^{8} + \frac{70226226635351679225268507662595335081018353}{1002228464704385566364613711396157594205796348} a^{7} + \frac{35660016074895647996077843125933236101285847}{74239145533658190101082497140456118089318248} a^{6} + \frac{56881469663661546264796160221916926777828951}{222717436600974570303247491421368354267954744} a^{5} - \frac{11051736113691909379055167169898994167471277}{24746381844552730033694165713485372696439416} a^{4} - \frac{1179862803704905141804305759231372278648477}{24746381844552730033694165713485372696439416} a^{3} + \frac{28612359805262284937402750485003075468369}{152755443484893395269717072305465263558268} a^{2} + \frac{1449717980076793388352122605221633747419}{152755443484893395269717072305465263558268} a + \frac{11840227644486952624664380694466641905197}{33945654107754087837714904956770058568504}$

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{16687278200040824703051973688658382021}{1436823108691243024345729391559093188153442552} a^{31} + \frac{41837940666818662168675525765655806775}{1436823108691243024345729391559093188153442552} a^{30} + \frac{60828743548118043846413405148720137305}{478941036230414341448576463853031062717814184} a^{29} - \frac{166917613491678665721354307804971759425}{478941036230414341448576463853031062717814184} a^{28} - \frac{96574105241880492179554861888854780365}{79823506038402390241429410642171843786302364} a^{27} + \frac{6978991404949029169079086593381273509825}{718411554345621512172864695779546594076721276} a^{26} - \frac{2849766368780655465155528284654203919825}{478941036230414341448576463853031062717814184} a^{25} - \frac{14102442039019433889623179738938422546018}{179602888586405378043216173944886648519180319} a^{24} + \frac{17653415506158657674482869993333040765709}{239470518115207170724288231926515531358907092} a^{23} + \frac{28066454426631301718222784619820050339786}{59867629528801792681072057981628882839726773} a^{22} - \frac{2012549060584111592080180900522759020665509}{1436823108691243024345729391559093188153442552} a^{21} + \frac{655299822985736259052497319240592105326171}{478941036230414341448576463853031062717814184} a^{20} + \frac{40426009540304170873159947909478757964745}{17738556897422753386984313476038187508067192} a^{19} - \frac{44467145691618422283712523920913324975472205}{1436823108691243024345729391559093188153442552} a^{18} + \frac{4895992093269513263209765597549756074914501}{59867629528801792681072057981628882839726773} a^{17} - \frac{4580551233303694362162996791277881575286645}{1436823108691243024345729391559093188153442552} a^{16} - \frac{103274011406696067793580928190045593229533149}{239470518115207170724288231926515531358907092} a^{15} + \frac{127667303483283598230589209576146461388450089}{159647012076804780482858821284343687572604728} a^{14} - \frac{77235287260641328707698145147673871594078381}{159647012076804780482858821284343687572604728} a^{13} - \frac{258546018681207758708833960270511861159375231}{179602888586405378043216173944886648519180319} a^{12} + \frac{2000045088677682434366942133285266256460545259}{718411554345621512172864695779546594076721276} a^{11} + \frac{56964256235508698147565205417624874131263893}{53215670692268260160952940428114562524201576} a^{10} - \frac{267716330208466516778121041743398701651339629}{39911753019201195120714705321085921893151182} a^{9} + \frac{53315895539453094406830585468189530008412767}{4434639224355688346746078369009546877016798} a^{8} - \frac{45698693006019130895717708192954572637474549}{8869278448711376693492156738019093754033596} a^{7} - \frac{3264630850074172230561947314257574083923315}{656983588793435310629048647260673611409896} a^{6} - \frac{7069530038170896329507442196719794250993847}{1970950766380305931887145941782020834229688} a^{5} - \frac{248106341589223683562512249331287593116049}{218994529597811770209682882420224537136632} a^{4} - \frac{1274007884464698400909348547260063735612021}{218994529597811770209682882420224537136632} a^{3} - \frac{123957735375087566224105911560871326285}{675909041968554846326181735864890546718} a^{2} + \frac{1630365992574368561360065293403227001}{37550502331586380351454540881382808151} a + \frac{26435368969259810688103697502171830433}{300404018652691042811636327051062465208} \) (order $10$)
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

$C_2\times C_4^2$ (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

\(\Q(\sqrt{-715}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{-11}, \sqrt{65})\), 4.4.274625.1, 4.0.33229625.2, \(\Q(\sqrt{13}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{-143})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-55}, \sqrt{65})\), 4.4.274625.2, 4.0.33229625.1, \(\Q(\sqrt{-11}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.0.54925.1, 4.4.265837.1, 4.0.2197.1, 4.4.6645925.1, 4.0.21125.1, 4.4.15125.1, \(\Q(\zeta_{5})\), 4.4.2556125.1, 8.0.1104207977640625.5, 8.0.261351000625.1, 8.0.1104207977640625.4, 8.8.75418890625.1, 8.0.1104207977640625.8, 8.0.1104207977640625.11, 8.0.1104207977640625.10, 8.0.44168319105625.11, 8.0.44168319105625.5, 8.0.6533775015625.2, 8.0.6533775015625.3, 8.0.3016755625.1, 8.8.44168319105625.1, 8.0.446265625.1, 8.8.6533775015625.1, 8.0.44168319105625.2, 8.0.70669310569.1, 8.0.6533775015625.1, 8.0.228765625.1, 16.0.1219275257885198999941650390625.1, 16.0.1950840412616318399906640625.1, 16.0.42690215954805468994140625.1, 16.0.5688009063105712890625.1, 16.16.1219275257885198999941650390625.1, 16.0.1219275257885198999941650390625.2, 16.0.1219275257885198999941650390625.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ R 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
5Data not computed
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed