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Properties

Label 32.0.61303936503...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{88}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $59.51$
Ramified primes $2, 5, 7$
Class number $640$ (GRH)
Class group $[2, 8, 40]$ (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![1225456, -5903488, 22597024, -64335936, 145683560, -258848640, 358649568, -376049408, 268061612, -53416448, -176472000, 302963088, -263767222, 97892432, 95660524, -229422992, 271203819, -239583744, 174237344, -108563512, 59189458, -28585976, 12315372, -4749552, 1639376, -504816, 137688, -32872, 6756, -1160, 160, -16, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^31 + 160*x^30 - 1160*x^29 + 6756*x^28 - 32872*x^27 + 137688*x^26 - 504816*x^25 + 1639376*x^24 - 4749552*x^23 + 12315372*x^22 - 28585976*x^21 + 59189458*x^20 - 108563512*x^19 + 174237344*x^18 - 239583744*x^17 + 271203819*x^16 - 229422992*x^15 + 95660524*x^14 + 97892432*x^13 - 263767222*x^12 + 302963088*x^11 - 176472000*x^10 - 53416448*x^9 + 268061612*x^8 - 376049408*x^7 + 358649568*x^6 - 258848640*x^5 + 145683560*x^4 - 64335936*x^3 + 22597024*x^2 - 5903488*x + 1225456)
 
gp: K = bnfinit(x^32 - 16*x^31 + 160*x^30 - 1160*x^29 + 6756*x^28 - 32872*x^27 + 137688*x^26 - 504816*x^25 + 1639376*x^24 - 4749552*x^23 + 12315372*x^22 - 28585976*x^21 + 59189458*x^20 - 108563512*x^19 + 174237344*x^18 - 239583744*x^17 + 271203819*x^16 - 229422992*x^15 + 95660524*x^14 + 97892432*x^13 - 263767222*x^12 + 302963088*x^11 - 176472000*x^10 - 53416448*x^9 + 268061612*x^8 - 376049408*x^7 + 358649568*x^6 - 258848640*x^5 + 145683560*x^4 - 64335936*x^3 + 22597024*x^2 - 5903488*x + 1225456, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{31} + 160 x^{30} - 1160 x^{29} + 6756 x^{28} - 32872 x^{27} + 137688 x^{26} - 504816 x^{25} + 1639376 x^{24} - 4749552 x^{23} + 12315372 x^{22} - 28585976 x^{21} + 59189458 x^{20} - 108563512 x^{19} + 174237344 x^{18} - 239583744 x^{17} + 271203819 x^{16} - 229422992 x^{15} + 95660524 x^{14} + 97892432 x^{13} - 263767222 x^{12} + 302963088 x^{11} - 176472000 x^{10} - 53416448 x^{9} + 268061612 x^{8} - 376049408 x^{7} + 358649568 x^{6} - 258848640 x^{5} + 145683560 x^{4} - 64335936 x^{3} + 22597024 x^{2} - 5903488 x + 1225456 \)

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:  \(613039365036788240314949190025216000000000000000000000000=2^{88}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.51$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(517,·)$, $\chi_{560}(321,·)$, $\chi_{560}(393,·)$, $\chi_{560}(13,·)$, $\chi_{560}(533,·)$, $\chi_{560}(281,·)$, $\chi_{560}(153,·)$, $\chi_{560}(29,·)$, $\chi_{560}(197,·)$, $\chi_{560}(293,·)$, $\chi_{560}(41,·)$, $\chi_{560}(349,·)$, $\chi_{560}(433,·)$, $\chi_{560}(309,·)$, $\chi_{560}(57,·)$, $\chi_{560}(181,·)$, $\chi_{560}(449,·)$, $\chi_{560}(69,·)$, $\chi_{560}(461,·)$, $\chi_{560}(141,·)$, $\chi_{560}(337,·)$, $\chi_{560}(477,·)$, $\chi_{560}(421,·)$, $\chi_{560}(97,·)$, $\chi_{560}(209,·)$, $\chi_{560}(489,·)$, $\chi_{560}(237,·)$, $\chi_{560}(113,·)$, $\chi_{560}(169,·)$, $\chi_{560}(377,·)$, $\chi_{560}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{164} a^{16} - \frac{2}{41} a^{15} - \frac{1}{41} a^{14} + \frac{1}{41} a^{13} - \frac{1}{41} a^{12} + \frac{9}{82} a^{11} - \frac{4}{41} a^{10} - \frac{17}{82} a^{9} + \frac{19}{164} a^{8} - \frac{27}{82} a^{7} - \frac{6}{41} a^{6} + \frac{17}{82} a^{5} + \frac{9}{41} a^{4} - \frac{19}{41} a^{3} + \frac{8}{41} a^{2} + \frac{19}{41} a + \frac{20}{41}$, $\frac{1}{164} a^{17} + \frac{7}{82} a^{15} - \frac{7}{41} a^{14} + \frac{7}{41} a^{13} - \frac{7}{82} a^{12} - \frac{9}{41} a^{11} + \frac{1}{82} a^{10} - \frac{7}{164} a^{9} + \frac{4}{41} a^{8} - \frac{23}{82} a^{7} + \frac{3}{82} a^{6} + \frac{31}{82} a^{5} - \frac{17}{82} a^{4} + \frac{20}{41} a^{3} + \frac{1}{41} a^{2} + \frac{8}{41} a - \frac{4}{41}$, $\frac{1}{164} a^{18} + \frac{1}{82} a^{15} + \frac{1}{82} a^{14} + \frac{3}{41} a^{13} + \frac{5}{41} a^{12} - \frac{1}{41} a^{11} - \frac{29}{164} a^{10} + \frac{4}{41} a^{8} - \frac{29}{82} a^{7} + \frac{35}{82} a^{6} + \frac{16}{41} a^{5} + \frac{17}{41} a^{4} - \frac{20}{41} a^{3} + \frac{19}{41} a^{2} + \frac{17}{41} a + \frac{7}{41}$, $\frac{1}{164} a^{19} + \frac{9}{82} a^{15} + \frac{5}{41} a^{14} + \frac{3}{41} a^{13} + \frac{1}{41} a^{12} + \frac{17}{164} a^{11} + \frac{8}{41} a^{10} + \frac{1}{82} a^{9} - \frac{7}{82} a^{8} - \frac{17}{41} a^{7} - \frac{13}{41} a^{6} - \frac{1}{2} a^{5} - \frac{35}{82} a^{4} + \frac{16}{41} a^{3} + \frac{1}{41} a^{2} + \frac{10}{41} a + \frac{1}{41}$, $\frac{1}{164} a^{20} + \frac{1}{82} a^{14} + \frac{7}{82} a^{13} + \frac{7}{164} a^{12} + \frac{9}{41} a^{11} - \frac{19}{82} a^{10} + \frac{6}{41} a^{9} - \frac{16}{41} a^{7} - \frac{15}{41} a^{6} - \frac{13}{82} a^{5} + \frac{18}{41} a^{4} + \frac{15}{41} a^{3} - \frac{11}{41} a^{2} - \frac{13}{41} a + \frac{9}{41}$, $\frac{1}{164} a^{21} + \frac{1}{82} a^{15} + \frac{7}{82} a^{14} + \frac{7}{164} a^{13} + \frac{9}{41} a^{12} - \frac{19}{82} a^{11} + \frac{6}{41} a^{10} + \frac{9}{82} a^{8} - \frac{15}{41} a^{7} - \frac{13}{82} a^{6} + \frac{18}{41} a^{5} - \frac{11}{82} a^{4} - \frac{11}{41} a^{3} - \frac{13}{41} a^{2} + \frac{9}{41} a$, $\frac{1}{164} a^{22} + \frac{15}{82} a^{15} + \frac{15}{164} a^{14} + \frac{7}{41} a^{13} - \frac{15}{82} a^{12} - \frac{3}{41} a^{11} + \frac{8}{41} a^{10} + \frac{1}{41} a^{9} - \frac{4}{41} a^{8} - \frac{1}{2} a^{7} - \frac{11}{41} a^{6} - \frac{2}{41} a^{5} - \frac{17}{82} a^{4} - \frac{16}{41} a^{3} - \frac{7}{41} a^{2} + \frac{3}{41} a + \frac{1}{41}$, $\frac{1}{164} a^{23} + \frac{9}{164} a^{15} - \frac{4}{41} a^{14} + \frac{7}{82} a^{13} + \frac{13}{82} a^{12} - \frac{4}{41} a^{11} - \frac{2}{41} a^{10} + \frac{5}{41} a^{9} + \frac{1}{41} a^{8} + \frac{9}{82} a^{7} + \frac{14}{41} a^{6} - \frac{35}{82} a^{5} - \frac{39}{82} a^{4} - \frac{11}{41} a^{3} + \frac{9}{41} a^{2} + \frac{5}{41} a + \frac{15}{41}$, $\frac{1}{13448} a^{24} - \frac{3}{3362} a^{23} + \frac{1}{6724} a^{22} - \frac{1}{1681} a^{21} - \frac{9}{13448} a^{20} + \frac{13}{6724} a^{19} - \frac{4}{1681} a^{18} + \frac{5}{3362} a^{17} + \frac{15}{13448} a^{16} - \frac{33}{1681} a^{15} + \frac{727}{6724} a^{14} - \frac{677}{3362} a^{13} - \frac{3055}{13448} a^{12} - \frac{935}{6724} a^{11} - \frac{248}{1681} a^{10} - \frac{143}{3362} a^{9} - \frac{368}{1681} a^{8} + \frac{111}{1681} a^{7} - \frac{411}{1681} a^{6} - \frac{829}{3362} a^{5} + \frac{163}{1681} a^{4} - \frac{692}{1681} a^{3} + \frac{688}{1681} a^{2} + \frac{326}{1681} a + \frac{550}{1681}$, $\frac{1}{13448} a^{25} + \frac{11}{6724} a^{23} + \frac{2}{1681} a^{22} - \frac{23}{13448} a^{21} + \frac{17}{6724} a^{19} - \frac{9}{3362} a^{18} + \frac{9}{13448} a^{17} - \frac{1}{6724} a^{16} - \frac{611}{6724} a^{15} + \frac{569}{3362} a^{14} + \frac{3235}{13448} a^{13} - \frac{469}{3362} a^{12} - \frac{1101}{6724} a^{11} + \frac{130}{1681} a^{10} + \frac{303}{6724} a^{9} + \frac{11}{164} a^{8} + \frac{593}{1681} a^{7} - \frac{201}{1681} a^{6} + \frac{669}{3362} a^{5} + \frac{649}{1681} a^{4} - \frac{31}{1681} a^{3} - \frac{233}{1681} a^{2} - \frac{663}{1681} a + \frac{450}{1681}$, $\frac{1}{13448} a^{26} + \frac{17}{6724} a^{23} + \frac{15}{13448} a^{22} + \frac{3}{3362} a^{21} - \frac{7}{6724} a^{20} - \frac{17}{6724} a^{19} - \frac{25}{13448} a^{18} - \frac{4}{1681} a^{17} + \frac{3}{6724} a^{16} - \frac{1657}{6724} a^{15} - \frac{463}{13448} a^{14} + \frac{58}{1681} a^{13} - \frac{1403}{6724} a^{12} + \frac{631}{6724} a^{11} + \frac{335}{1681} a^{10} + \frac{261}{1681} a^{9} - \frac{53}{6724} a^{8} + \frac{413}{3362} a^{7} - \frac{771}{1681} a^{6} + \frac{1537}{3362} a^{5} + \frac{729}{1681} a^{4} - \frac{794}{1681} a^{3} + \frac{68}{1681} a^{2} - \frac{39}{1681} a + \frac{815}{1681}$, $\frac{1}{13448} a^{27} + \frac{13}{13448} a^{23} + \frac{13}{6724} a^{22} + \frac{3}{3362} a^{21} + \frac{13}{6724} a^{20} - \frac{7}{13448} a^{19} - \frac{5}{6724} a^{18} - \frac{9}{6724} a^{17} + \frac{15}{6724} a^{16} - \frac{753}{13448} a^{15} - \frac{1239}{6724} a^{14} - \frac{459}{3362} a^{13} - \frac{611}{6724} a^{12} + \frac{1601}{6724} a^{11} - \frac{1185}{6724} a^{10} - \frac{251}{6724} a^{9} + \frac{239}{6724} a^{8} - \frac{281}{1681} a^{7} + \frac{413}{1681} a^{6} + \frac{431}{1681} a^{5} + \frac{101}{1681} a^{4} + \frac{759}{1681} a^{3} + \frac{267}{1681} a^{2} + \frac{473}{1681} a - \frac{660}{1681}$, $\frac{1}{1654104} a^{28} - \frac{7}{827052} a^{27} + \frac{19}{827052} a^{26} - \frac{11}{413526} a^{25} + \frac{7}{206763} a^{24} - \frac{743}{413526} a^{23} - \frac{337}{206763} a^{22} - \frac{253}{137842} a^{21} + \frac{149}{413526} a^{20} - \frac{289}{137842} a^{19} + \frac{539}{275684} a^{18} - \frac{1957}{827052} a^{17} + \frac{2477}{827052} a^{16} - \frac{13838}{68921} a^{15} - \frac{16981}{206763} a^{14} - \frac{20609}{413526} a^{13} - \frac{107831}{551368} a^{12} + \frac{23441}{275684} a^{11} - \frac{5159}{137842} a^{10} - \frac{98317}{827052} a^{9} + \frac{63839}{827052} a^{8} + \frac{70538}{206763} a^{7} - \frac{174245}{413526} a^{6} + \frac{29207}{206763} a^{5} + \frac{73642}{206763} a^{4} + \frac{22686}{68921} a^{3} - \frac{835}{1681} a^{2} - \frac{85130}{206763} a - \frac{35029}{206763}$, $\frac{1}{1654104} a^{29} - \frac{35}{1654104} a^{27} - \frac{1}{413526} a^{26} + \frac{55}{1654104} a^{25} + \frac{13}{827052} a^{24} - \frac{623}{551368} a^{23} - \frac{109}{413526} a^{22} + \frac{2495}{1654104} a^{21} - \frac{1241}{413526} a^{20} - \frac{469}{551368} a^{19} - \frac{53}{413526} a^{18} + \frac{1139}{551368} a^{17} + \frac{485}{413526} a^{16} + \frac{86245}{1654104} a^{15} - \frac{18005}{137842} a^{14} + \frac{36931}{206763} a^{13} + \frac{795}{68921} a^{12} + \frac{18187}{275684} a^{11} + \frac{18710}{206763} a^{10} - \frac{10030}{68921} a^{9} + \frac{6607}{275684} a^{8} - \frac{9926}{68921} a^{7} + \frac{2135}{68921} a^{6} - \frac{72928}{206763} a^{5} - \frac{1603}{10086} a^{4} - \frac{20236}{68921} a^{3} - \frac{2474}{206763} a^{2} - \frac{63884}{206763} a + \frac{7498}{206763}$, $\frac{1}{275869401107218635542460514725574640616} a^{30} - \frac{5}{91956467035739545180820171575191546872} a^{29} + \frac{2274588137655162792204623583303}{22989116758934886295205042893797886718} a^{28} - \frac{382130807126067349090376761993889}{275869401107218635542460514725574640616} a^{27} - \frac{4472056404392826386424429413207141}{137934700553609317771230257362787320308} a^{26} - \frac{60590084167791108994811619404321}{3364260989112422384664152618604568788} a^{25} - \frac{4090092021787040296081802037850711}{137934700553609317771230257362787320308} a^{24} - \frac{123130406785655108501288245490700455}{275869401107218635542460514725574640616} a^{23} + \frac{167027692798292078960824126976581469}{137934700553609317771230257362787320308} a^{22} + \frac{1334880689419981043141527515996323}{137934700553609317771230257362787320308} a^{21} + \frac{80888509521425739760785889810379732}{34483675138402329442807564340696830077} a^{20} + \frac{555017656187102806852166951638145707}{275869401107218635542460514725574640616} a^{19} + \frac{27193150972535029630781656681086835}{45978233517869772590410085787595773436} a^{18} + \frac{85749906420465622574069245478872640}{34483675138402329442807564340696830077} a^{17} + \frac{11401678671627672080973465351311691}{11494558379467443147602521446898943359} a^{16} + \frac{13015151484915578540428820478206809273}{91956467035739545180820171575191546872} a^{15} - \frac{5310381313474221180891813252097321}{1444342414173919557813929396468977176} a^{14} + \frac{15713944892792960590240386944556028565}{275869401107218635542460514725574640616} a^{13} - \frac{821359693063671868184391878148394282}{11494558379467443147602521446898943359} a^{12} - \frac{8983672775609579363762675889722073991}{137934700553609317771230257362787320308} a^{11} + \frac{1516775418324650431300566377951605801}{11494558379467443147602521446898943359} a^{10} - \frac{1349825068589122719224962220827859069}{137934700553609317771230257362787320308} a^{9} - \frac{960837829915907659386727214792924567}{137934700553609317771230257362787320308} a^{8} + \frac{5998713255673019544202988363776168750}{34483675138402329442807564340696830077} a^{7} + \frac{5966425520554814212076998349347688989}{22989116758934886295205042893797886718} a^{6} + \frac{1847617502053203116908929106496978508}{11494558379467443147602521446898943359} a^{5} + \frac{12664018253167306808794330053440091296}{34483675138402329442807564340696830077} a^{4} - \frac{11236070882281084267962387655654217648}{34483675138402329442807564340696830077} a^{3} - \frac{11974159304303126982072459306409439657}{34483675138402329442807564340696830077} a^{2} + \frac{1622794157382574365683376436991936982}{11494558379467443147602521446898943359} a + \frac{41479107863282928996275302898518331}{180542801771739944726741174558622147}$, $\frac{1}{11866158416886949186478658535501670133146267496} a^{31} + \frac{21506825}{11866158416886949186478658535501670133146267496} a^{30} - \frac{1322530974717338811601056988349818949669}{5933079208443474593239329267750835066573133748} a^{29} + \frac{1023862832483755151219123845887677945593}{3955386138962316395492886178500556711048755832} a^{28} - \frac{166532341140745211239437625757599495144289}{5933079208443474593239329267750835066573133748} a^{27} + \frac{134007874984081192827829574203990023186133}{11866158416886949186478658535501670133146267496} a^{26} + \frac{203966141534689779137597768206332168397853}{5933079208443474593239329267750835066573133748} a^{25} + \frac{409995014602099609787515941406867202633687}{11866158416886949186478658535501670133146267496} a^{24} + \frac{3174950639686584842930210430207966556180849}{1483269802110868648309832316937708766643283437} a^{23} - \frac{9807208563439860588267077044670769198354271}{3955386138962316395492886178500556711048755832} a^{22} - \frac{9324330191418073725603178118141434000443757}{5933079208443474593239329267750835066573133748} a^{21} + \frac{965893642464396627335553376609483427789969}{3955386138962316395492886178500556711048755832} a^{20} - \frac{6745510203181581211782845948931995074446997}{2966539604221737296619664633875417533286566874} a^{19} - \frac{2188179628296464349481781153456214315117193}{3955386138962316395492886178500556711048755832} a^{18} - \frac{12577332397609112464844624052603127023469381}{5933079208443474593239329267750835066573133748} a^{17} + \frac{678591358155028092741795829095597631030295}{3955386138962316395492886178500556711048755832} a^{16} - \frac{273019181803947225694478143194366072525834427}{3955386138962316395492886178500556711048755832} a^{15} - \frac{28898228968453110952384749131523935665711223}{494423267370289549436610772312569588881094479} a^{14} - \frac{300755243863924411020238554604980760269844529}{2966539604221737296619664633875417533286566874} a^{13} - \frac{1433082538393647487096497134652579188035902949}{5933079208443474593239329267750835066573133748} a^{12} + \frac{863105530453898971159670142824928449188641331}{5933079208443474593239329267750835066573133748} a^{11} - \frac{1391712760610725230143136407870466925212710299}{5933079208443474593239329267750835066573133748} a^{10} - \frac{351109280169775508363948675604322560301085}{7453617096034515820652423703204566666549163} a^{9} + \frac{131168674405591239529731389667280789657806283}{1483269802110868648309832316937708766643283437} a^{8} - \frac{45541792587384744943332791788106645085057456}{494423267370289549436610772312569588881094479} a^{7} - \frac{607073223500052251192075253224390864310493460}{1483269802110868648309832316937708766643283437} a^{6} + \frac{326707193845087724375679910059428087566965514}{1483269802110868648309832316937708766643283437} a^{5} + \frac{398914993885965338687841394719411942527615687}{1483269802110868648309832316937708766643283437} a^{4} - \frac{94228760674429794312680944009355390116651077}{494423267370289549436610772312569588881094479} a^{3} - \frac{429380116184866813195883142143577953215417514}{1483269802110868648309832316937708766643283437} a^{2} + \frac{349693483438662198830406149933750801934490745}{1483269802110868648309832316937708766643283437} a + \frac{3427935375743553288429843853408714846119757}{7765810482255856797433677052029888830593107}$

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

Class group and class number

$C_{2}\times C_{8}\times C_{40}$, which has order $640$ (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{9989108554088054390736347731}{12311201406070092625065178272294477} a^{30} - \frac{49945542770440271953681738655}{4103733802023364208355059424098159} a^{29} + \frac{5770338199713173015710351857211}{49244805624280370500260713089177908} a^{28} - \frac{20114477033193460696777677106547}{24622402812140185250130356544588954} a^{27} + \frac{226332322718695684873799648440999}{49244805624280370500260713089177908} a^{26} - \frac{531714508497849880752582599092475}{24622402812140185250130356544588954} a^{25} + \frac{2869437231621686061329973513342823}{32829870416186913666840475392785272} a^{24} - \frac{3808437095859389083589364993831676}{12311201406070092625065178272294477} a^{23} + \frac{23846355689597413046421249465726067}{24622402812140185250130356544588954} a^{22} - \frac{33224371809840443018941674361560323}{12311201406070092625065178272294477} a^{21} + \frac{220131048255308178867350311304543633}{32829870416186913666840475392785272} a^{20} - \frac{8908567118715279330458673521936957}{600546410052199640247081866941194} a^{19} + \frac{476626698281379309460471784867731285}{16414935208093456833420237696392636} a^{18} - \frac{1224767986943708709952985567717550323}{24622402812140185250130356544588954} a^{17} + \frac{7202165005636213902405409365429196967}{98489611248560741000521426178355816} a^{16} - \frac{364926629132769442151532471734711561}{4103733802023364208355059424098159} a^{15} + \frac{5302397780000184085545077575168876}{64456551864241322644320305090547} a^{14} - \frac{4202472243042333142223485443577913}{100091068342033273374513644490199} a^{13} - \frac{853656844494405986537101533358452563}{32829870416186913666840475392785272} a^{12} + \frac{1141139200533098540302876075835388955}{12311201406070092625065178272294477} a^{11} - \frac{479963512589257869275951154354431814}{4103733802023364208355059424098159} a^{10} + \frac{310171906318240906144107296591893634}{4103733802023364208355059424098159} a^{9} + \frac{104150683091439591826749148569328809}{8207467604046728416710118848196318} a^{8} - \frac{392904170663331119856822863746469396}{4103733802023364208355059424098159} a^{7} + \frac{1641020268603704356754624989266407854}{12311201406070092625065178272294477} a^{6} - \frac{1494683107540650099825064861048425092}{12311201406070092625065178272294477} a^{5} + \frac{327816035847392232039850517177757127}{4103733802023364208355059424098159} a^{4} - \frac{453979084081130324499557327800258432}{12311201406070092625065178272294477} a^{3} + \frac{148619981891638384826428463250417650}{12311201406070092625065178272294477} a^{2} - \frac{35685299116755840617648403955640992}{12311201406070092625065178272294477} a + \frac{27937786494084824863810300489786}{21485517288080440881440101696849} \) (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:  \( 11039633378602.484 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), 4.0.2508800.1, \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-7})\), 4.0.100352.5, 4.4.51200.1, \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-14})\), 4.0.256000.2, 4.4.12544000.1, 4.0.256000.4, 4.4.12544000.2, 4.4.6125.1, \(\Q(\zeta_{5})\), 4.4.392000.1, 4.0.8000.2, 8.0.6294077440000.6, 8.0.6146560000.2, 8.0.6294077440000.2, 8.0.6294077440000.7, 8.0.6294077440000.5, 8.8.2621440000.1, 8.0.10070523904.2, 8.0.157351936000000.59, 8.0.157351936000000.61, 8.0.37515625.1, 8.0.153664000000.2, 8.0.65536000000.1, 8.8.157351936000000.4, 8.8.153664000000.1, 8.0.64000000.2, 8.0.157351936000000.37, 8.0.157351936000000.14, 8.0.153664000000.1, 8.0.153664000000.5, 16.0.39615410820716953600000000.1, 16.0.24759631762948096000000000000.6, 16.0.23612624896000000000000.2, 16.0.24759631762948096000000000000.3, 16.0.24759631762948096000000000000.4, 16.0.4294967296000000000000.1, 16.16.24759631762948096000000000000.2

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$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}$