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Properties

Label 32.0.84762290704...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{88}\cdot 5^{24}\cdot 11^{16}$
Root discriminant $74.60$
Ramified primes $2, 5, 11$
Class number $5400$ (GRH)
Class group $[3, 15, 120]$ (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![6872111, 34978664, 246059100, -1470536872, 4079279422, -7670495384, 10863979076, -12412466704, 11993265010, -10407112256, 8765771372, -7714898672, 7182394274, -6655144784, 5769439004, -4516634096, 3158187595, -1969257808, 1098447236, -550035696, 248067450, -100999328, 37160956, -12348736, 3698828, -993304, 237760, -50008, 9140, -1400, 176, -16, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^31 + 176*x^30 - 1400*x^29 + 9140*x^28 - 50008*x^27 + 237760*x^26 - 993304*x^25 + 3698828*x^24 - 12348736*x^23 + 37160956*x^22 - 100999328*x^21 + 248067450*x^20 - 550035696*x^19 + 1098447236*x^18 - 1969257808*x^17 + 3158187595*x^16 - 4516634096*x^15 + 5769439004*x^14 - 6655144784*x^13 + 7182394274*x^12 - 7714898672*x^11 + 8765771372*x^10 - 10407112256*x^9 + 11993265010*x^8 - 12412466704*x^7 + 10863979076*x^6 - 7670495384*x^5 + 4079279422*x^4 - 1470536872*x^3 + 246059100*x^2 + 34978664*x + 6872111)
 
gp: K = bnfinit(x^32 - 16*x^31 + 176*x^30 - 1400*x^29 + 9140*x^28 - 50008*x^27 + 237760*x^26 - 993304*x^25 + 3698828*x^24 - 12348736*x^23 + 37160956*x^22 - 100999328*x^21 + 248067450*x^20 - 550035696*x^19 + 1098447236*x^18 - 1969257808*x^17 + 3158187595*x^16 - 4516634096*x^15 + 5769439004*x^14 - 6655144784*x^13 + 7182394274*x^12 - 7714898672*x^11 + 8765771372*x^10 - 10407112256*x^9 + 11993265010*x^8 - 12412466704*x^7 + 10863979076*x^6 - 7670495384*x^5 + 4079279422*x^4 - 1470536872*x^3 + 246059100*x^2 + 34978664*x + 6872111, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{31} + 176 x^{30} - 1400 x^{29} + 9140 x^{28} - 50008 x^{27} + 237760 x^{26} - 993304 x^{25} + 3698828 x^{24} - 12348736 x^{23} + 37160956 x^{22} - 100999328 x^{21} + 248067450 x^{20} - 550035696 x^{19} + 1098447236 x^{18} - 1969257808 x^{17} + 3158187595 x^{16} - 4516634096 x^{15} + 5769439004 x^{14} - 6655144784 x^{13} + 7182394274 x^{12} - 7714898672 x^{11} + 8765771372 x^{10} - 10407112256 x^{9} + 11993265010 x^{8} - 12412466704 x^{7} + 10863979076 x^{6} - 7670495384 x^{5} + 4079279422 x^{4} - 1470536872 x^{3} + 246059100 x^{2} + 34978664 x + 6872111 \)

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:  \(847622907049404564614012839370162176000000000000000000000000=2^{88}\cdot 5^{24}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.60$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(880=2^{4}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{880}(1,·)$, $\chi_{880}(133,·)$, $\chi_{880}(769,·)$, $\chi_{880}(397,·)$, $\chi_{880}(109,·)$, $\chi_{880}(529,·)$, $\chi_{880}(21,·)$, $\chi_{880}(793,·)$, $\chi_{880}(153,·)$, $\chi_{880}(837,·)$, $\chi_{880}(417,·)$, $\chi_{880}(549,·)$, $\chi_{880}(681,·)$, $\chi_{880}(813,·)$, $\chi_{880}(177,·)$, $\chi_{880}(309,·)$, $\chi_{880}(441,·)$, $\chi_{880}(573,·)$, $\chi_{880}(197,·)$, $\chi_{880}(329,·)$, $\chi_{880}(461,·)$, $\chi_{880}(593,·)$, $\chi_{880}(857,·)$, $\chi_{880}(221,·)$, $\chi_{880}(353,·)$, $\chi_{880}(617,·)$, $\chi_{880}(749,·)$, $\chi_{880}(241,·)$, $\chi_{880}(373,·)$, $\chi_{880}(89,·)$, $\chi_{880}(637,·)$, $\chi_{880}(661,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a$, $\frac{1}{164} a^{18} - \frac{9}{164} a^{17} + \frac{2}{41} a^{16} - \frac{6}{41} a^{15} + \frac{9}{82} a^{14} - \frac{5}{82} a^{13} - \frac{8}{41} a^{12} - \frac{1}{41} a^{11} - \frac{17}{164} a^{10} + \frac{15}{164} a^{9} + \frac{9}{41} a^{8} - \frac{4}{41} a^{7} - \frac{29}{82} a^{6} - \frac{5}{41} a^{5} + \frac{6}{41} a^{4} + \frac{13}{41} a^{3} + \frac{13}{164} a^{2} - \frac{59}{164} a + \frac{17}{41}$, $\frac{1}{164} a^{19} + \frac{9}{164} a^{17} + \frac{7}{164} a^{16} - \frac{17}{82} a^{15} - \frac{3}{41} a^{14} - \frac{10}{41} a^{13} + \frac{9}{41} a^{12} + \frac{29}{164} a^{11} + \frac{13}{82} a^{10} + \frac{7}{164} a^{9} + \frac{21}{164} a^{8} + \frac{11}{41} a^{7} - \frac{25}{82} a^{6} - \frac{37}{82} a^{5} + \frac{11}{82} a^{4} + \frac{71}{164} a^{3} + \frac{29}{82} a^{2} - \frac{53}{164} a - \frac{3}{164}$, $\frac{1}{164} a^{20} + \frac{3}{82} a^{17} + \frac{17}{164} a^{16} + \frac{10}{41} a^{15} - \frac{19}{82} a^{14} - \frac{19}{82} a^{13} - \frac{11}{164} a^{12} - \frac{5}{41} a^{11} - \frac{1}{41} a^{10} - \frac{8}{41} a^{9} + \frac{7}{164} a^{8} + \frac{3}{41} a^{7} - \frac{11}{41} a^{6} + \frac{19}{82} a^{5} + \frac{19}{164} a^{4} - \frac{3}{82} a^{2} - \frac{23}{82} a + \frac{3}{164}$, $\frac{1}{164} a^{21} - \frac{11}{164} a^{17} - \frac{2}{41} a^{16} + \frac{6}{41} a^{15} + \frac{9}{82} a^{14} - \frac{33}{164} a^{13} + \frac{2}{41} a^{12} + \frac{5}{41} a^{11} - \frac{3}{41} a^{10} - \frac{1}{164} a^{9} - \frac{10}{41} a^{8} + \frac{13}{41} a^{7} - \frac{6}{41} a^{6} - \frac{25}{164} a^{5} + \frac{5}{41} a^{4} - \frac{18}{41} a^{3} - \frac{21}{82} a^{2} + \frac{29}{164} a - \frac{20}{41}$, $\frac{1}{164} a^{22} + \frac{4}{41} a^{17} - \frac{11}{164} a^{16} + \frac{1}{164} a^{14} - \frac{5}{41} a^{13} - \frac{1}{41} a^{12} + \frac{13}{82} a^{11} - \frac{6}{41} a^{10} + \frac{1}{82} a^{9} - \frac{3}{164} a^{8} + \frac{23}{82} a^{7} - \frac{7}{164} a^{6} + \frac{23}{82} a^{5} + \frac{7}{41} a^{4} + \frac{19}{82} a^{3} + \frac{2}{41} a^{2} + \frac{25}{82} a - \frac{31}{164}$, $\frac{1}{5084} a^{23} + \frac{1}{1271} a^{22} - \frac{3}{1271} a^{21} + \frac{3}{5084} a^{20} + \frac{3}{5084} a^{19} - \frac{11}{5084} a^{18} + \frac{257}{2542} a^{17} + \frac{605}{5084} a^{16} + \frac{543}{5084} a^{15} - \frac{147}{2542} a^{14} + \frac{87}{1271} a^{13} - \frac{869}{5084} a^{12} + \frac{1041}{5084} a^{11} - \frac{491}{5084} a^{10} + \frac{202}{1271} a^{9} - \frac{333}{5084} a^{8} + \frac{1711}{5084} a^{7} + \frac{213}{1271} a^{6} + \frac{841}{2542} a^{5} - \frac{23}{124} a^{4} + \frac{2539}{5084} a^{3} + \frac{1539}{5084} a^{2} - \frac{264}{1271} a - \frac{23}{164}$, $\frac{1}{10168} a^{24} - \frac{7}{2542} a^{22} + \frac{5}{2542} a^{21} - \frac{9}{10168} a^{20} + \frac{1}{1271} a^{19} + \frac{189}{5084} a^{17} - \frac{99}{1271} a^{16} - \frac{130}{1271} a^{15} - \frac{228}{1271} a^{14} + \frac{70}{1271} a^{13} + \frac{363}{10168} a^{12} - \frac{268}{1271} a^{11} - \frac{227}{1271} a^{10} + \frac{37}{164} a^{9} - \frac{611}{2542} a^{8} + \frac{336}{1271} a^{7} + \frac{1103}{2542} a^{6} + \frac{564}{1271} a^{5} - \frac{4477}{10168} a^{4} - \frac{337}{1271} a^{3} - \frac{935}{2542} a^{2} + \frac{2329}{5084} a - \frac{71}{328}$, $\frac{1}{10168} a^{25} + \frac{1}{1271} a^{22} + \frac{27}{10168} a^{21} + \frac{15}{5084} a^{20} + \frac{11}{5084} a^{19} + \frac{1}{1271} a^{18} - \frac{237}{5084} a^{17} - \frac{203}{5084} a^{16} - \frac{499}{2542} a^{15} + \frac{221}{2542} a^{14} + \frac{683}{10168} a^{13} - \frac{1117}{5084} a^{12} + \frac{367}{5084} a^{11} + \frac{250}{1271} a^{10} + \frac{1193}{5084} a^{9} - \frac{1117}{5084} a^{8} + \frac{309}{1271} a^{7} + \frac{539}{1271} a^{6} + \frac{583}{10168} a^{5} - \frac{1871}{5084} a^{4} + \frac{1405}{5084} a^{3} + \frac{421}{2542} a^{2} + \frac{409}{10168} a + \frac{53}{164}$, $\frac{1}{10168} a^{26} - \frac{5}{10168} a^{22} + \frac{1}{5084} a^{21} - \frac{1}{5084} a^{20} - \frac{2}{1271} a^{19} - \frac{7}{5084} a^{18} + \frac{281}{2542} a^{17} - \frac{163}{5084} a^{16} - \frac{14}{1271} a^{15} + \frac{2415}{10168} a^{14} + \frac{219}{5084} a^{13} - \frac{63}{5084} a^{12} - \frac{16}{1271} a^{11} + \frac{739}{5084} a^{10} - \frac{113}{2542} a^{9} + \frac{305}{5084} a^{8} - \frac{180}{1271} a^{7} - \frac{4497}{10168} a^{6} + \frac{1941}{5084} a^{5} + \frac{25}{164} a^{4} + \frac{570}{1271} a^{3} + \frac{3225}{10168} a^{2} + \frac{498}{1271} a + \frac{45}{164}$, $\frac{1}{10168} a^{27} - \frac{1}{10168} a^{23} + \frac{9}{5084} a^{22} + \frac{3}{2542} a^{21} - \frac{1}{2542} a^{20} - \frac{1}{5084} a^{19} + \frac{13}{5084} a^{18} + \frac{183}{5084} a^{17} + \frac{503}{5084} a^{16} + \frac{867}{10168} a^{15} + \frac{871}{5084} a^{14} - \frac{51}{1271} a^{13} + \frac{29}{2542} a^{12} + \frac{15}{164} a^{11} - \frac{247}{5084} a^{10} - \frac{931}{5084} a^{9} + \frac{9}{5084} a^{8} + \frac{2099}{10168} a^{7} + \frac{421}{5084} a^{6} - \frac{337}{1271} a^{5} + \frac{269}{1271} a^{4} + \frac{4949}{10168} a^{3} - \frac{541}{5084} a^{2} + \frac{771}{5084} a + \frac{71}{164}$, $\frac{1}{10168} a^{28} - \frac{13}{5084} a^{22} - \frac{2}{1271} a^{21} - \frac{3}{10168} a^{20} - \frac{5}{2542} a^{19} + \frac{3}{5084} a^{18} + \frac{623}{5084} a^{17} - \frac{213}{10168} a^{16} + \frac{106}{1271} a^{15} + \frac{755}{5084} a^{14} - \frac{63}{1271} a^{13} + \frac{1373}{10168} a^{12} + \frac{211}{1271} a^{11} + \frac{317}{5084} a^{10} - \frac{939}{5084} a^{9} - \frac{1729}{10168} a^{8} - \frac{555}{2542} a^{7} - \frac{1943}{5084} a^{6} - \frac{261}{2542} a^{5} + \frac{251}{1271} a^{4} - \frac{295}{1271} a^{3} + \frac{1945}{5084} a^{2} - \frac{939}{5084} a + \frac{113}{328}$, $\frac{1}{10168} a^{29} + \frac{13}{5084} a^{22} - \frac{5}{10168} a^{21} - \frac{1}{2542} a^{20} + \frac{11}{5084} a^{19} + \frac{15}{5084} a^{18} + \frac{937}{10168} a^{17} + \frac{96}{1271} a^{16} - \frac{185}{2542} a^{15} - \frac{509}{5084} a^{14} + \frac{315}{10168} a^{13} - \frac{17}{1271} a^{12} - \frac{69}{5084} a^{11} - \frac{1215}{5084} a^{10} + \frac{1361}{10168} a^{9} - \frac{298}{1271} a^{8} - \frac{102}{1271} a^{7} + \frac{1347}{5084} a^{6} - \frac{2519}{5084} a^{5} + \frac{891}{2542} a^{4} + \frac{1317}{5084} a^{3} + \frac{2111}{5084} a^{2} + \frac{2273}{10168} a + \frac{17}{82}$, $\frac{1}{46799778634125534153460054018790294431681178168791432} a^{30} - \frac{15}{46799778634125534153460054018790294431681178168791432} a^{29} + \frac{197906135855483038316720802403793398169486356005}{11699944658531383538365013504697573607920294542197858} a^{28} - \frac{1877437031564498187935410170884992368364332282867}{46799778634125534153460054018790294431681178168791432} a^{27} + \frac{511315588544872027568275454970128117792810808329}{11699944658531383538365013504697573607920294542197858} a^{26} + \frac{196948277553415470231289232389225030972571815577}{23399889317062767076730027009395147215840589084395716} a^{25} + \frac{1085339192175061674837643980662446075273744934401}{46799778634125534153460054018790294431681178168791432} a^{24} + \frac{1233147363460163746121834963882473788932643166897}{46799778634125534153460054018790294431681178168791432} a^{23} - \frac{130731608373078152472686002769965826022605965887719}{46799778634125534153460054018790294431681178168791432} a^{22} + \frac{49642799624821441601604927439924343747723476687005}{46799778634125534153460054018790294431681178168791432} a^{21} - \frac{102779751236243967773917983030465739927439593344993}{46799778634125534153460054018790294431681178168791432} a^{20} + \frac{62519874525435373213318284710979629312869530268589}{23399889317062767076730027009395147215840589084395716} a^{19} + \frac{47419987837828480236881151061039640080336843182279}{46799778634125534153460054018790294431681178168791432} a^{18} - \frac{1112743373223113621393722751921374958280176496680007}{46799778634125534153460054018790294431681178168791432} a^{17} - \frac{605972031003339267014971729504415670968756691600229}{5849972329265691769182506752348786803960147271098929} a^{16} - \frac{1696207265437103089720276774185405111570351868023839}{46799778634125534153460054018790294431681178168791432} a^{15} + \frac{5894079342488781575014036923548889505950424798523861}{46799778634125534153460054018790294431681178168791432} a^{14} - \frac{4171274977601268389399150718340528702781602508505111}{46799778634125534153460054018790294431681178168791432} a^{13} + \frac{1511345480922581082650484978768386731341151471360887}{46799778634125534153460054018790294431681178168791432} a^{12} - \frac{2771636154091398703305147734783488499265998736208381}{23399889317062767076730027009395147215840589084395716} a^{11} - \frac{10455072386792936343981447323650629592174447497105561}{46799778634125534153460054018790294431681178168791432} a^{10} - \frac{94024264908544912735095180559940526764780812007049}{1509670278520178521079356581251299820376812198993272} a^{9} + \frac{818187921216864672600122089985961787164489943380331}{11699944658531383538365013504697573607920294542197858} a^{8} + \frac{7193280361763597743441336531380784858787849933990045}{46799778634125534153460054018790294431681178168791432} a^{7} - \frac{26953183772512500543256505345012219870896359827928}{188708784815022315134919572656412477547101524874159} a^{6} - \frac{4046407804981307132312177270802631865730084635501621}{11699944658531383538365013504697573607920294542197858} a^{5} - \frac{11852635376347457709156951157622913210432446214955385}{46799778634125534153460054018790294431681178168791432} a^{4} - \frac{7064640858033865102200674305121704460321865533592287}{46799778634125534153460054018790294431681178168791432} a^{3} - \frac{491470583849985883773695992181035006079231299322313}{1141458015466476442767318390702202303211736052897352} a^{2} - \frac{13505344087474113385015051546616229224168587931762621}{46799778634125534153460054018790294431681178168791432} a + \frac{722494635446167399579208677521624565562032091528805}{1509670278520178521079356581251299820376812198993272}$, $\frac{1}{470301786243191975706509532107302009301977877770342090988792} a^{31} + \frac{628075}{58787723280398996963313691513412751162747234721292761373599} a^{30} + \frac{589749868258877730332542051573257777813722830055470765}{235150893121595987853254766053651004650988938885171045494396} a^{29} + \frac{5000712912211084547985224563204767954675899896203130431}{235150893121595987853254766053651004650988938885171045494396} a^{28} + \frac{9133955682668391664702415681537386895416572811050842857}{470301786243191975706509532107302009301977877770342090988792} a^{27} + \frac{4534105293469061398919873452543916481737957394984433955}{470301786243191975706509532107302009301977877770342090988792} a^{26} + \frac{284981294869339372732001568959565593348971884506800613}{11470775274224194529427061758714683153706777506593709536312} a^{25} - \frac{6653880464256344998847547822693457958455676882963500585}{470301786243191975706509532107302009301977877770342090988792} a^{24} + \frac{21635225956624569023143143912184875393701726674926354055}{235150893121595987853254766053651004650988938885171045494396} a^{23} - \frac{1293469474765404929288670094045463810767560332697795523861}{470301786243191975706509532107302009301977877770342090988792} a^{22} - \frac{300546285543542596116122802839291026507882751820789192707}{470301786243191975706509532107302009301977877770342090988792} a^{21} + \frac{91382456450458704103677077985458346045216610535745317425}{470301786243191975706509532107302009301977877770342090988792} a^{20} - \frac{567480343058856446913765827381541910907351804397750167931}{470301786243191975706509532107302009301977877770342090988792} a^{19} + \frac{180539085024058180279505453922263070808550919677750402543}{235150893121595987853254766053651004650988938885171045494396} a^{18} + \frac{9895265350860969782264131996308002573175221670414990247935}{117575446560797993926627383026825502325494469442585522747198} a^{17} - \frac{3523686129369925140545562665181169478129071257529355396939}{58787723280398996963313691513412751162747234721292761373599} a^{16} + \frac{1035294087867988703067192809723950158620446341507101176873}{235150893121595987853254766053651004650988938885171045494396} a^{15} + \frac{95207585642208678997639983832417704574674048510271365772347}{470301786243191975706509532107302009301977877770342090988792} a^{14} + \frac{39619917142004322310169040189728533950068350789536433605409}{470301786243191975706509532107302009301977877770342090988792} a^{13} + \frac{2604185631832329239088449186952624981932279792060650588313}{11470775274224194529427061758714683153706777506593709536312} a^{12} - \frac{65333553713580573063270735666156178707296681745504151038387}{470301786243191975706509532107302009301977877770342090988792} a^{11} + \frac{10955483305178238552573109932019064707513822650041258883071}{235150893121595987853254766053651004650988938885171045494396} a^{10} - \frac{10797430729583557197817691428850976455369875411233360253478}{58787723280398996963313691513412751162747234721292761373599} a^{9} + \frac{175356642438542634842023077555434877946737789094326464477}{117575446560797993926627383026825502325494469442585522747198} a^{8} + \frac{3592511956334783295903379823042585063323697023350668041115}{15171025362683612119564823616364580945225092831301357773832} a^{7} + \frac{185094023164916275628383551538770805116953982722024861129835}{470301786243191975706509532107302009301977877770342090988792} a^{6} + \frac{197557380627886128490096756478147287802177387879511858498359}{470301786243191975706509532107302009301977877770342090988792} a^{5} + \frac{128990480655858647928311731433681346131749932486293453709959}{470301786243191975706509532107302009301977877770342090988792} a^{4} - \frac{24529989168456018129622541588487546964986495137047483237618}{58787723280398996963313691513412751162747234721292761373599} a^{3} + \frac{135860786609531025621367047450425460839066560412403782184419}{470301786243191975706509532107302009301977877770342090988792} a^{2} + \frac{119024392291523480321737998918783485002601140694095973860647}{470301786243191975706509532107302009301977877770342090988792} a - \frac{2200181948810421962155108478803582552558601001108596881581}{15171025362683612119564823616364580945225092831301357773832}$

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

Class group and class number

$C_{3}\times C_{15}\times C_{120}$, which has order $5400$ (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{194192354925125763461675690502500983}{44811166263654260723540957054673173758510129} a^{30} - \frac{2912885323876886451925135357537514745}{44811166263654260723540957054673173758510129} a^{29} + \frac{63418558545115333649646449473867041401}{89622332527308521447081914109346347517020258} a^{28} - \frac{246824669566804685633924320457030792062}{44811166263654260723540957054673173758510129} a^{27} + \frac{3197900573584593536532627272873381157315}{89622332527308521447081914109346347517020258} a^{26} - \frac{8653241869555114881866634189503126610537}{44811166263654260723540957054673173758510129} a^{25} + \frac{40929906181041317775639016525631995633084}{44811166263654260723540957054673173758510129} a^{24} - \frac{169944479378273224538908935406041338660478}{44811166263654260723540957054673173758510129} a^{23} + \frac{1261591442224541897958287955662865880123819}{89622332527308521447081914109346347517020258} a^{22} - \frac{2098710235622159753657010492630630804059108}{44811166263654260723540957054673173758510129} a^{21} + \frac{12608078666262685005961960569144459567433021}{89622332527308521447081914109346347517020258} a^{20} - \frac{17105201628100992000656734876638628826252928}{44811166263654260723540957054673173758510129} a^{19} + \frac{41980187720024796604884937936927628725010938}{44811166263654260723540957054673173758510129} a^{18} - \frac{93039242826795624780277600998670191053531937}{44811166263654260723540957054673173758510129} a^{17} + \frac{743131472676813843223698794118884171668964261}{179244665054617042894163828218692695034040516} a^{16} - \frac{333017152962633564409110791536594044682302092}{44811166263654260723540957054673173758510129} a^{15} + \frac{1067542949423722293362987916773003971449478869}{89622332527308521447081914109346347517020258} a^{14} - \frac{761850714459592974621100696042877878340593012}{44811166263654260723540957054673173758510129} a^{13} + \frac{969687921800753698797625934948768786502556446}{44811166263654260723540957054673173758510129} a^{12} - \frac{1109910088239409380810781092803747703985733414}{44811166263654260723540957054673173758510129} a^{11} + \frac{1189102861909915305362161236789960867931746504}{44811166263654260723540957054673173758510129} a^{10} - \frac{1266803229648437062155176836196760397427721103}{44811166263654260723540957054673173758510129} a^{9} + \frac{5800165338555899821606173760952783827412793979}{179244665054617042894163828218692695034040516} a^{8} - \frac{1730503360177402191287600875136514186936077852}{44811166263654260723540957054673173758510129} a^{7} + \frac{4027765820903374965896999145881065091562327057}{89622332527308521447081914109346347517020258} a^{6} - \frac{2053006011294027581172358197574043212817569657}{44811166263654260723540957054673173758510129} a^{5} + \frac{3553418530721639258088457492210739964297200595}{89622332527308521447081914109346347517020258} a^{4} - \frac{39221731504424017134823173294591479543171326}{1445521492375943894307772808215263669629359} a^{3} + \frac{28961543606794317164527232665167150664083551}{2185910549446549303587363758764545061390738} a^{2} - \frac{178261628948738270877236344179507444298251318}{44811166263654260723540957054673173758510129} a + \frac{3720393733609086151030377223700930254368235}{5782085969503775577231091232861054678517436} \) (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:  \( 122816592296668.77 \) (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

\(\Q(\sqrt{-22}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{2}, \sqrt{-11})\), 4.0.247808.2, \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{5}, \sqrt{-22})\), \(\Q(\sqrt{10}, \sqrt{-22})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-55})\), 4.0.6195200.5, 4.4.51200.1, \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{10}, \sqrt{-11})\), 4.4.968000.2, \(\Q(\zeta_{5})\), 4.4.15125.1, 4.0.8000.2, 4.0.256000.4, 4.4.30976000.2, 4.0.256000.2, 4.4.30976000.1, 8.0.61408804864.2, 8.0.37480960000.9, 8.0.38380503040000.48, 8.0.38380503040000.63, 8.0.38380503040000.23, 8.8.2621440000.1, 8.0.38380503040000.59, 8.0.937024000000.5, 8.0.937024000000.1, 8.0.959512576000000.38, 8.0.959512576000000.36, 8.8.937024000000.1, 8.0.64000000.2, 8.0.65536000000.1, 8.8.959512576000000.2, 8.0.937024000000.2, 8.0.228765625.1, 8.0.959512576000000.12, 8.0.959512576000000.19, 16.0.1473063013603449241600000000.1, 16.0.878013976576000000000000.1, 16.0.920664383502155776000000000000.8, 16.0.920664383502155776000000000000.5, 16.0.920664383502155776000000000000.6, 16.16.920664383502155776000000000000.2, 16.0.4294967296000000000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
2Data not computed
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}$