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Properties

Label 32.0.67239249502...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $5^{24}\cdot 7^{16}\cdot 17^{24}$
Root discriminant $74.07$
Ramified primes $5, 7, 17$
Class number $3700$ (GRH)
Class group $[5, 740]$ (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![5158686976, -3503287424, -1913391360, 2751530272, 178038704, -817676696, -253102672, 337841718, 295991545, -248528948, 185367884, -73453252, -48019048, 90363958, -28234359, -11616776, 11001816, -2941856, 2273432, -285314, 75369, 41636, -23076, 8592, -4256, 822, -407, -8, 4, -12, 8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + 8*x^30 - 12*x^29 + 4*x^28 - 8*x^27 - 407*x^26 + 822*x^25 - 4256*x^24 + 8592*x^23 - 23076*x^22 + 41636*x^21 + 75369*x^20 - 285314*x^19 + 2273432*x^18 - 2941856*x^17 + 11001816*x^16 - 11616776*x^15 - 28234359*x^14 + 90363958*x^13 - 48019048*x^12 - 73453252*x^11 + 185367884*x^10 - 248528948*x^9 + 295991545*x^8 + 337841718*x^7 - 253102672*x^6 - 817676696*x^5 + 178038704*x^4 + 2751530272*x^3 - 1913391360*x^2 - 3503287424*x + 5158686976)
 
gp: K = bnfinit(x^32 - 2*x^31 + 8*x^30 - 12*x^29 + 4*x^28 - 8*x^27 - 407*x^26 + 822*x^25 - 4256*x^24 + 8592*x^23 - 23076*x^22 + 41636*x^21 + 75369*x^20 - 285314*x^19 + 2273432*x^18 - 2941856*x^17 + 11001816*x^16 - 11616776*x^15 - 28234359*x^14 + 90363958*x^13 - 48019048*x^12 - 73453252*x^11 + 185367884*x^10 - 248528948*x^9 + 295991545*x^8 + 337841718*x^7 - 253102672*x^6 - 817676696*x^5 + 178038704*x^4 + 2751530272*x^3 - 1913391360*x^2 - 3503287424*x + 5158686976, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} + 8 x^{30} - 12 x^{29} + 4 x^{28} - 8 x^{27} - 407 x^{26} + 822 x^{25} - 4256 x^{24} + 8592 x^{23} - 23076 x^{22} + 41636 x^{21} + 75369 x^{20} - 285314 x^{19} + 2273432 x^{18} - 2941856 x^{17} + 11001816 x^{16} - 11616776 x^{15} - 28234359 x^{14} + 90363958 x^{13} - 48019048 x^{12} - 73453252 x^{11} + 185367884 x^{10} - 248528948 x^{9} + 295991545 x^{8} + 337841718 x^{7} - 253102672 x^{6} - 817676696 x^{5} + 178038704 x^{4} + 2751530272 x^{3} - 1913391360 x^{2} - 3503287424 x + 5158686976 \)

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:  \(672392495020735742667824195375358053817672789096832275390625=5^{24}\cdot 7^{16}\cdot 17^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.07$
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:  $5, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(595=5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{595}(1,·)$, $\chi_{595}(132,·)$, $\chi_{595}(13,·)$, $\chi_{595}(526,·)$, $\chi_{595}(531,·)$, $\chi_{595}(302,·)$, $\chi_{595}(407,·)$, $\chi_{595}(412,·)$, $\chi_{595}(288,·)$, $\chi_{595}(293,·)$, $\chi_{595}(169,·)$, $\chi_{595}(426,·)$, $\chi_{595}(174,·)$, $\chi_{595}(307,·)$, $\chi_{595}(183,·)$, $\chi_{595}(188,·)$, $\chi_{595}(64,·)$, $\chi_{595}(69,·)$, $\chi_{595}(582,·)$, $\chi_{595}(463,·)$, $\chi_{595}(594,·)$, $\chi_{595}(344,·)$, $\chi_{595}(477,·)$, $\chi_{595}(421,·)$, $\chi_{595}(356,·)$, $\chi_{595}(358,·)$, $\chi_{595}(489,·)$, $\chi_{595}(106,·)$, $\chi_{595}(237,·)$, $\chi_{595}(239,·)$, $\chi_{595}(118,·)$, $\chi_{595}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{20} + \frac{1}{6} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{21} + \frac{1}{6} a^{16} + \frac{1}{6} a^{6} - \frac{1}{6} a$, $\frac{1}{6} a^{22} + \frac{1}{6} a^{17} + \frac{1}{6} a^{7} - \frac{1}{6} a^{2}$, $\frac{1}{6} a^{23} + \frac{1}{6} a^{18} + \frac{1}{6} a^{8} - \frac{1}{6} a^{3}$, $\frac{1}{6} a^{24} + \frac{1}{6} a^{19} + \frac{1}{6} a^{9} - \frac{1}{6} a^{4}$, $\frac{1}{6} a^{25} - \frac{1}{6} a^{15} - \frac{1}{2} a^{12} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{3} - \frac{1}{3}$, $\frac{1}{12} a^{26} - \frac{1}{12} a^{20} + \frac{1}{6} a^{16} + \frac{1}{6} a^{15} - \frac{1}{4} a^{14} - \frac{1}{2} a^{12} + \frac{1}{3} a^{11} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{24} a^{27} + \frac{1}{24} a^{21} - \frac{1}{6} a^{17} + \frac{1}{6} a^{16} - \frac{1}{8} a^{15} + \frac{1}{6} a^{12} + \frac{3}{8} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{87488976} a^{28} - \frac{45887}{3645374} a^{27} + \frac{550645}{21872244} a^{26} + \frac{480933}{7290748} a^{25} - \frac{341515}{21872244} a^{24} - \frac{891217}{10936122} a^{23} - \frac{1730437}{29162992} a^{22} - \frac{79975}{5468061} a^{21} + \frac{593083}{7290748} a^{20} - \frac{1184866}{5468061} a^{19} - \frac{441111}{7290748} a^{18} - \frac{1807833}{7290748} a^{17} + \frac{6055177}{87488976} a^{16} - \frac{432210}{1822687} a^{15} + \frac{206425}{7290748} a^{14} - \frac{967819}{10936122} a^{13} + \frac{346538}{1822687} a^{12} + \frac{1994692}{5468061} a^{11} - \frac{11435805}{29162992} a^{10} + \frac{2304974}{5468061} a^{9} - \frac{5127559}{21872244} a^{8} + \frac{750481}{7290748} a^{7} - \frac{2736711}{7290748} a^{6} - \frac{376325}{7290748} a^{5} + \frac{26123929}{87488976} a^{4} + \frac{466885}{3645374} a^{3} + \frac{2412697}{7290748} a^{2} - \frac{2031017}{10936122} a + \frac{780123}{1822687}$, $\frac{1}{361734577283733782842096014460176279874385576931936} a^{29} + \frac{315385908606890065787230647765210269499387}{60289096213955630473682669076696046645730929488656} a^{28} - \frac{59897776425633461761758459175516586677296372037}{30144548106977815236841334538348023322865464744328} a^{27} + \frac{119796183623084137303648492812328703775131742849}{30144548106977815236841334538348023322865464744328} a^{26} - \frac{1437544741899751440941808296828511488993495932579}{90433644320933445710524003615044069968596394232984} a^{25} + \frac{544421656943976526554817991517101223795488892269}{15072274053488907618420667269174011661432732372164} a^{24} + \frac{3496224841805815339853460271818008647515901424713}{361734577283733782842096014460176279874385576931936} a^{23} + \frac{14989777498825273978279055148412539076438579046861}{180867288641866891421048007230088139937192788465968} a^{22} + \frac{5352685611027906460185189391207833192984409063075}{90433644320933445710524003615044069968596394232984} a^{21} + \frac{18972103037092733681491810936842860171990938195}{15072274053488907618420667269174011661432732372164} a^{20} + \frac{5524425992191289884293324764407070170665424511891}{90433644320933445710524003615044069968596394232984} a^{19} - \frac{1686895045272500906534813914738865937181860954059}{90433644320933445710524003615044069968596394232984} a^{18} + \frac{87836955143014620968561918017212581815560425192281}{361734577283733782842096014460176279874385576931936} a^{17} + \frac{2076622914784491583159828155783401029928658425243}{9519330981150889022160421433162533680904883603472} a^{16} - \frac{18213643277384663368550251116246924734808132444107}{90433644320933445710524003615044069968596394232984} a^{15} + \frac{1280188815538632330816066159372173500555064658005}{45216822160466722855262001807522034984298197116492} a^{14} - \frac{2467285533278871271131764800155242296098232353373}{15072274053488907618420667269174011661432732372164} a^{13} - \frac{7462440227793293201137899416566784259825666930835}{15072274053488907618420667269174011661432732372164} a^{12} - \frac{6916169695681430569764123389566828657243540590509}{120578192427911260947365338153392093291461858977312} a^{11} - \frac{88791442297415085157714462909503159685511146472219}{180867288641866891421048007230088139937192788465968} a^{10} + \frac{22941265960086687217114882620792570039357644535301}{90433644320933445710524003615044069968596394232984} a^{9} + \frac{3694382307916422905643729836882405948372771621581}{90433644320933445710524003615044069968596394232984} a^{8} - \frac{3084123461889400388813105765727385970175965271701}{90433644320933445710524003615044069968596394232984} a^{7} + \frac{14673503123122543768846466531632872452258062865911}{90433644320933445710524003615044069968596394232984} a^{6} + \frac{2890604322963725365908950712605687522676355265225}{361734577283733782842096014460176279874385576931936} a^{5} - \frac{34578714136288264457857142357840074402780895124955}{180867288641866891421048007230088139937192788465968} a^{4} - \frac{26975191067692930703550083573838651627398517224389}{90433644320933445710524003615044069968596394232984} a^{3} - \frac{1662822441962957286856773366328743878065843215950}{11304205540116680713815500451880508746074549279123} a^{2} - \frac{9000232571718337920510186395860258163773168536787}{22608411080233361427631000903761017492149098558246} a - \frac{64778016704892940060906539536915939584617698456}{168719485673383294236052245550455354419023123569}$, $\frac{1}{48472433356020326900840865937663621503167667308879424} a^{30} - \frac{1}{24236216678010163450420432968831810751583833654439712} a^{29} - \frac{17081763990771380557920053406428354887018301}{3029527084751270431302554121103976343947979206804964} a^{28} - \frac{42037731968821071897147005939808545746052746712025}{4039369446335027241736738828138635125263972275739952} a^{27} + \frac{4425019622889518930540229981017487221449794525953}{212598391912369854828249412007296585540209067144208} a^{26} + \frac{114065425639039579789742611915856399328619145038963}{6059054169502540862605108242207952687895958413609928} a^{25} - \frac{3842854665029358499593870955114258550824299855287095}{48472433356020326900840865937663621503167667308879424} a^{24} - \frac{852824297290291301453077393679423215356136845138725}{24236216678010163450420432968831810751583833654439712} a^{23} + \frac{165378026680503182479549985887296810264126493401147}{2019684723167513620868369414069317562631986137869976} a^{22} - \frac{44519524192312055066801700371516204361192588222915}{1009842361583756810434184707034658781315993068934988} a^{21} + \frac{605374328308387209430165198709037720230287738080431}{12118108339005081725210216484415905375791916827219856} a^{20} - \frac{1781641569845232311149849959733022188302776052646679}{12118108339005081725210216484415905375791916827219856} a^{19} - \frac{146296244385028116896387719727885259826164293545655}{48472433356020326900840865937663621503167667308879424} a^{18} + \frac{1337793123023795991320122522393349386890186703509685}{8078738892670054483473477656277270250527944551479904} a^{17} - \frac{24748336799082176854658356590344950152632956716301}{252460590395939202608546176758664695328998267233747} a^{16} - \frac{344577066966408510341640961360520574736549287758105}{1514763542375635215651277060551988171973989603402482} a^{15} + \frac{1496130214225491178585831051400421589287399629122775}{6059054169502540862605108242207952687895958413609928} a^{14} - \frac{429011095300694061137999690992969567472206880393881}{6059054169502540862605108242207952687895958413609928} a^{13} + \frac{2127838231421859901023627629523897180996169377440419}{16157477785340108966946955312554540501055889102959808} a^{12} + \frac{3447147429368649253320312915583982682980213407237897}{8078738892670054483473477656277270250527944551479904} a^{11} - \frac{94948338421761591406651541702949717243920456268227}{252460590395939202608546176758664695328998267233747} a^{10} + \frac{452286345319906801633080375675499954532445788516533}{4039369446335027241736738828138635125263972275739952} a^{9} - \frac{52795749289090223879512916654519129225847480718375}{4039369446335027241736738828138635125263972275739952} a^{8} + \frac{614104714497633125831898456827910553019719160920665}{4039369446335027241736738828138635125263972275739952} a^{7} + \frac{6981943946260166529542113023845341164015271145797011}{16157477785340108966946955312554540501055889102959808} a^{6} - \frac{2364445474553460322879795296720522035708028555828487}{8078738892670054483473477656277270250527944551479904} a^{5} + \frac{1478860279152106724514117232987333400473411962110659}{6059054169502540862605108242207952687895958413609928} a^{4} - \frac{1286657621563612968362384243229754641306021507943595}{6059054169502540862605108242207952687895958413609928} a^{3} + \frac{338382427726793105338748394355964514940031778071289}{1009842361583756810434184707034658781315993068934988} a^{2} - \frac{767316355141158933352137149254391645955941371439}{7536137026744453809210333634587005830716366186082} a - \frac{65776664102492309198015834397579764573720496967}{168719485673383294236052245550455354419023123569}$, $\frac{1}{6495306069706723804712676035646925281424467419389842816} a^{31} - \frac{1}{3247653034853361902356338017823462640712233709694921408} a^{30} + \frac{1}{811913258713340475589084504455865660178058427423730352} a^{29} - \frac{6943580438691424907950301611656287331082760273}{1623826517426680951178169008911731320356116854847460704} a^{28} - \frac{12170818033552924558043411595063691071103421662085627}{1623826517426680951178169008911731320356116854847460704} a^{27} - \frac{26594529609455464063013056733337238513086878990169781}{811913258713340475589084504455865660178058427423730352} a^{26} + \frac{230779051922579574891118740597899399569943425276853065}{6495306069706723804712676035646925281424467419389842816} a^{25} + \frac{44720080659718226070957108181740221612339833841085881}{1082551011617787300785446005941154213570744569898307136} a^{24} + \frac{2654671130140088948692763147605013307503655774181183}{202978314678335118897271126113966415044514606855932588} a^{23} + \frac{3296128834572451311157313948638934154006370228494497}{42732276774386340820478131813466613693582022495985808} a^{22} - \frac{100727626143837232097249577533233256771372131768896389}{1623826517426680951178169008911731320356116854847460704} a^{21} + \frac{5795407645262452070499632576741381580078122848059851}{85464553548772681640956263626933227387164044991971616} a^{20} - \frac{542956437455138308503503079691287560473811606658997655}{6495306069706723804712676035646925281424467419389842816} a^{19} - \frac{20455952366289064711291046915350538674626353907355051}{170929107097545363281912527253866454774328089983943232} a^{18} - \frac{42687433872022469895937003708231825290094655575369363}{270637752904446825196361501485288553392686142474576784} a^{17} + \frac{46656093526400631143997494691206498863657675371402109}{811913258713340475589084504455865660178058427423730352} a^{16} - \frac{55638471514173490117130065113196324498891792693912849}{270637752904446825196361501485288553392686142474576784} a^{15} - \frac{177516475742920383679604631464125237017395508722316461}{811913258713340475589084504455865660178058427423730352} a^{14} - \frac{278959316780904257129563913703104312832239360964727991}{6495306069706723804712676035646925281424467419389842816} a^{13} - \frac{337802729276921566372688509900300731413467935003265669}{3247653034853361902356338017823462640712233709694921408} a^{12} - \frac{128275505823581151754000728738220748937368804767068605}{811913258713340475589084504455865660178058427423730352} a^{11} + \frac{696844869878144092537495459306148030298674244299868689}{1623826517426680951178169008911731320356116854847460704} a^{10} + \frac{658044767886651796051092764393332855962594339332739431}{1623826517426680951178169008911731320356116854847460704} a^{9} + \frac{225766564554269842812407014296605220793452152084173241}{541275505808893650392723002970577106785372284949153568} a^{8} - \frac{1078087119137130900811941233433509651964364754224327399}{6495306069706723804712676035646925281424467419389842816} a^{7} + \frac{712221216143812280603099194891594406646549907937667179}{3247653034853361902356338017823462640712233709694921408} a^{6} + \frac{6001106861438077073475962981448198014016990587794679}{21366138387193170410239065906733306846791011247992904} a^{5} - \frac{22024597655674220750945893816243457896187684726879653}{405956629356670237794542252227932830089029213711865176} a^{4} + \frac{19592928345853661877700419974138851875788619209522663}{101489157339167559448635563056983207522257303427966294} a^{3} - \frac{21131984298498510034760075816565669037601961834476}{252460590395939202608546176758664695328998267233747} a^{2} - \frac{3953456337871082425050155710183082619773150166437}{11304205540116680713815500451880508746074549279123} a - \frac{77360674987001477767597215897109971349135705459}{168719485673383294236052245550455354419023123569}$

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

Class group and class number

$C_{5}\times C_{740}$, which has order $3700$ (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{10081168764665358607630404860324317457}{53187885090550738443672299125042995866177426304} a^{31} + \frac{456323472224309982409034579109434405}{2216161878772947435153012463543458161090726096} a^{30} - \frac{9607158242258433406050565655580577949}{6648485636318842305459037390630374483272178288} a^{29} + \frac{20358464302760296046268735193258817081}{13296971272637684610918074781260748966544356576} a^{28} - \frac{10085878369701498020662648047074311675}{13296971272637684610918074781260748966544356576} a^{27} + \frac{5642426142935978638733216641047537311}{1108080939386473717576506231771729080545363048} a^{26} + \frac{4119600108843638786804290612421834620039}{53187885090550738443672299125042995866177426304} a^{25} - \frac{88875397673659757195596464393179238585}{1108080939386473717576506231771729080545363048} a^{24} + \frac{2562244476231483749377610596586849549759}{3324242818159421152729518695315187241636089144} a^{23} - \frac{7683746110020093341817543611301725319689}{6648485636318842305459037390630374483272178288} a^{22} + \frac{53407539713443255461855021581884204203601}{13296971272637684610918074781260748966544356576} a^{21} - \frac{89672268965495113342520926346911283863471}{13296971272637684610918074781260748966544356576} a^{20} - \frac{801494237144611251653314802324081214917009}{53187885090550738443672299125042995866177426304} a^{19} + \frac{58985649110839807191134002684306615043985}{2216161878772947435153012463543458161090726096} a^{18} - \frac{13238911538101695068507046769905699413819}{33077042966760409479895708411096390464040688} a^{17} + \frac{541163094074102849182784685996966130212417}{2216161878772947435153012463543458161090726096} a^{16} - \frac{14522308050099216030961818934573185282955903}{6648485636318842305459037390630374483272178288} a^{15} + \frac{8664380918089472896502714465051194255851427}{6648485636318842305459037390630374483272178288} a^{14} + \frac{239701364549571685771613769006535144645399127}{53187885090550738443672299125042995866177426304} a^{13} - \frac{23479610752112010331436887058606229006419047}{3324242818159421152729518695315187241636089144} a^{12} + \frac{12294501525607439644558621957830636582463607}{6648485636318842305459037390630374483272178288} a^{11} + \frac{2459968475075994859902867594491433210529159}{13296971272637684610918074781260748966544356576} a^{10} - \frac{165715608938784826380873798219669406997300785}{13296971272637684610918074781260748966544356576} a^{9} + \frac{470833089691259291325649150254114986906486803}{13296971272637684610918074781260748966544356576} a^{8} + \frac{517687359029422051221344001198888636322209679}{53187885090550738443672299125042995866177426304} a^{7} - \frac{30040905798491001024331905110317700116992529}{554040469693236858788253115885864540272681524} a^{6} - \frac{39620015156754093141750228609824268051048807}{554040469693236858788253115885864540272681524} a^{5} + \frac{100268337128160828468087017523646988226223409}{831060704539855288182379673828796810409022286} a^{4} + \frac{303015240026216049115536071074552897794385}{2067315185422525592493481775693524404002543} a^{3} + \frac{156406399227515279424107141815596782830752545}{554040469693236858788253115885864540272681524} a^{2} - \frac{8851143838704175115116464487759439216812}{92566351586083235484782766075829450925487} a + \frac{30347192934632353322361139299762566855464}{92566351586083235484782766075829450925487} \) (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:  \( 76039275253910.16 \) (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

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-595}) \), \(\Q(\sqrt{-35}, \sqrt{85})\), 4.4.30092125.1, 4.0.614125.2, \(\Q(\sqrt{-7}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-119})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-7}, \sqrt{85})\), 4.4.30092125.2, 4.0.614125.1, \(\Q(\sqrt{17}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), 4.4.122825.1, 4.0.6018425.1, 4.4.4913.1, 4.0.240737.1, 4.4.1770125.1, \(\Q(\zeta_{5})\), 4.4.6125.1, 4.0.36125.1, 8.0.905535987015625.5, 8.0.125333700625.1, 8.0.905535987015625.4, 8.8.905535987015625.1, 8.0.905535987015625.1, 8.0.377149515625.1, 8.0.905535987015625.6, 8.0.36221439480625.6, 8.0.57954303169.1, 8.0.3133342515625.3, 8.0.3133342515625.2, 8.8.15085980625.1, 8.0.36221439480625.8, 8.8.3133342515625.1, 8.0.1305015625.1, 8.0.36221439480625.3, 8.0.36221439480625.7, 8.0.3133342515625.1, 8.0.37515625.1, 16.0.819995423780362168593994140625.1, 16.0.1311992678048579469750390625.1, 16.0.9817835320223203369140625.1, 16.16.819995423780362168593994140625.1, 16.0.819995423780362168593994140625.3, 16.0.142241757136172119140625.1, 16.0.819995423780362168593994140625.2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
$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}$
17Data not computed