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Properties

Label 32.0.53818028735...0625.2
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 5^{16}\cdot 17^{30}$
Root discriminant $55.16$
Ramified primes $3, 5, 17$
Class number $340$ (GRH)
Class group $[2, 170]$ (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12752041, -12691334, -60418, 11294784, -11237834, 2492744, 8774568, -7260650, -1621426, 2204306, -317867, -308057, 198782, 534224, -227834, -427804, 213179, 218196, -131121, -87075, 59620, 27455, -20622, -6833, 5525, 1308, -1121, -187, 169, 18, -17, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 17*x^30 + 18*x^29 + 169*x^28 - 187*x^27 - 1121*x^26 + 1308*x^25 + 5525*x^24 - 6833*x^23 - 20622*x^22 + 27455*x^21 + 59620*x^20 - 87075*x^19 - 131121*x^18 + 218196*x^17 + 213179*x^16 - 427804*x^15 - 227834*x^14 + 534224*x^13 + 198782*x^12 - 308057*x^11 - 317867*x^10 + 2204306*x^9 - 1621426*x^8 - 7260650*x^7 + 8774568*x^6 + 2492744*x^5 - 11237834*x^4 + 11294784*x^3 - 60418*x^2 - 12691334*x + 12752041)
 
gp: K = bnfinit(x^32 - x^31 - 17*x^30 + 18*x^29 + 169*x^28 - 187*x^27 - 1121*x^26 + 1308*x^25 + 5525*x^24 - 6833*x^23 - 20622*x^22 + 27455*x^21 + 59620*x^20 - 87075*x^19 - 131121*x^18 + 218196*x^17 + 213179*x^16 - 427804*x^15 - 227834*x^14 + 534224*x^13 + 198782*x^12 - 308057*x^11 - 317867*x^10 + 2204306*x^9 - 1621426*x^8 - 7260650*x^7 + 8774568*x^6 + 2492744*x^5 - 11237834*x^4 + 11294784*x^3 - 60418*x^2 - 12691334*x + 12752041, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 17 x^{30} + 18 x^{29} + 169 x^{28} - 187 x^{27} - 1121 x^{26} + 1308 x^{25} + 5525 x^{24} - 6833 x^{23} - 20622 x^{22} + 27455 x^{21} + 59620 x^{20} - 87075 x^{19} - 131121 x^{18} + 218196 x^{17} + 213179 x^{16} - 427804 x^{15} - 227834 x^{14} + 534224 x^{13} + 198782 x^{12} - 308057 x^{11} - 317867 x^{10} + 2204306 x^{9} - 1621426 x^{8} - 7260650 x^{7} + 8774568 x^{6} + 2492744 x^{5} - 11237834 x^{4} + 11294784 x^{3} - 60418 x^{2} - 12691334 x + 12752041 \)

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:  \(53818028735688890166227692218037660486426411285400390625=3^{16}\cdot 5^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.16$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(86,·)$, $\chi_{255}(139,·)$, $\chi_{255}(14,·)$, $\chi_{255}(16,·)$, $\chi_{255}(151,·)$, $\chi_{255}(26,·)$, $\chi_{255}(29,·)$, $\chi_{255}(161,·)$, $\chi_{255}(164,·)$, $\chi_{255}(166,·)$, $\chi_{255}(44,·)$, $\chi_{255}(184,·)$, $\chi_{255}(191,·)$, $\chi_{255}(194,·)$, $\chi_{255}(196,·)$, $\chi_{255}(199,·)$, $\chi_{255}(74,·)$, $\chi_{255}(76,·)$, $\chi_{255}(206,·)$, $\chi_{255}(79,·)$, $\chi_{255}(209,·)$, $\chi_{255}(214,·)$, $\chi_{255}(224,·)$, $\chi_{255}(101,·)$, $\chi_{255}(106,·)$, $\chi_{255}(236,·)$, $\chi_{255}(109,·)$, $\chi_{255}(244,·)$, $\chi_{255}(121,·)$, $\chi_{255}(251,·)$, $\chi_{255}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1597} a^{18} - \frac{610}{1597} a^{17} - \frac{10}{1597} a^{16} - \frac{521}{1597} a^{15} + \frac{62}{1597} a^{14} + \frac{775}{1597} a^{13} - \frac{227}{1597} a^{12} + \frac{691}{1597} a^{11} + \frac{578}{1597} a^{10} + \frac{270}{1597} a^{9} + \frac{679}{1597} a^{8} + \frac{738}{1597} a^{7} - \frac{652}{1597} a^{6} - \frac{738}{1597} a^{5} - \frac{486}{1597} a^{4} - \frac{267}{1597} a^{3} + \frac{113}{1597} a^{2} + \frac{199}{1597} a - \frac{4}{1597}$, $\frac{1}{1597} a^{19} - \frac{9}{1597} a^{17} - \frac{233}{1597} a^{16} + \frac{55}{1597} a^{15} + \frac{267}{1597} a^{14} - \frac{189}{1597} a^{13} - \frac{437}{1597} a^{12} + \frac{480}{1597} a^{11} - \frac{87}{1597} a^{10} - \frac{709}{1597} a^{9} - \frac{292}{1597} a^{8} + \frac{771}{1597} a^{7} + \frac{792}{1597} a^{6} - \frac{312}{1597} a^{5} + \frac{315}{1597} a^{4} + \frac{137}{1597} a^{3} + \frac{458}{1597} a^{2} + \frac{14}{1597} a + \frac{754}{1597}$, $\frac{1}{1597} a^{20} + \frac{665}{1597} a^{17} - \frac{35}{1597} a^{16} + \frac{369}{1597} a^{15} + \frac{369}{1597} a^{14} + \frac{150}{1597} a^{13} + \frac{34}{1597} a^{12} - \frac{256}{1597} a^{11} - \frac{298}{1597} a^{10} + \frac{541}{1597} a^{9} + \frac{494}{1597} a^{8} - \frac{551}{1597} a^{7} + \frac{208}{1597} a^{6} + \frac{61}{1597} a^{5} + \frac{554}{1597} a^{4} - \frac{348}{1597} a^{3} - \frac{566}{1597} a^{2} - \frac{649}{1597} a - \frac{36}{1597}$, $\frac{1}{1597} a^{21} - \frac{23}{1597} a^{17} + \frac{631}{1597} a^{16} + \frac{285}{1597} a^{15} + \frac{442}{1597} a^{14} + \frac{490}{1597} a^{13} + \frac{581}{1597} a^{12} + \frac{123}{1597} a^{11} - \frac{549}{1597} a^{10} - \frac{192}{1597} a^{9} - \frac{135}{1597} a^{8} - \frac{283}{1597} a^{7} - \frac{743}{1597} a^{6} - \frac{552}{1597} a^{5} + \frac{248}{1597} a^{4} - \frac{278}{1597} a^{3} - \frac{735}{1597} a^{2} + \frac{180}{1597} a - \frac{534}{1597}$, $\frac{1}{1597} a^{22} - \frac{623}{1597} a^{17} + \frac{55}{1597} a^{16} - \frac{362}{1597} a^{15} + \frac{319}{1597} a^{14} - \frac{758}{1597} a^{13} - \frac{307}{1597} a^{12} - \frac{626}{1597} a^{11} + \frac{326}{1597} a^{10} - \frac{313}{1597} a^{9} - \frac{636}{1597} a^{8} + \frac{261}{1597} a^{7} + \frac{422}{1597} a^{6} - \frac{756}{1597} a^{5} - \frac{277}{1597} a^{4} - \frac{488}{1597} a^{3} - \frac{415}{1597} a^{2} - \frac{748}{1597} a - \frac{92}{1597}$, $\frac{1}{1597} a^{23} + \frac{111}{1597} a^{17} - \frac{204}{1597} a^{16} - \frac{73}{1597} a^{15} - \frac{460}{1597} a^{14} + \frac{224}{1597} a^{13} + \frac{86}{1597} a^{12} - \frac{371}{1597} a^{11} + \frac{456}{1597} a^{10} - \frac{111}{1597} a^{9} + \frac{73}{1597} a^{8} + \frac{260}{1597} a^{7} + \frac{283}{1597} a^{6} - \frac{115}{1597} a^{5} + \frac{164}{1597} a^{4} - \frac{668}{1597} a^{3} - \frac{617}{1597} a^{2} - \frac{681}{1597} a + \frac{702}{1597}$, $\frac{1}{1597} a^{24} + \frac{432}{1597} a^{17} - \frac{560}{1597} a^{16} - \frac{121}{1597} a^{15} - \frac{270}{1597} a^{14} + \frac{299}{1597} a^{13} - \frac{726}{1597} a^{12} + \frac{411}{1597} a^{11} - \frac{389}{1597} a^{10} + \frac{446}{1597} a^{9} - \frac{50}{1597} a^{8} - \frac{188}{1597} a^{7} + \frac{392}{1597} a^{6} + \frac{635}{1597} a^{5} + \frac{577}{1597} a^{4} + \frac{274}{1597} a^{3} - \frac{448}{1597} a^{2} - \frac{626}{1597} a + \frac{444}{1597}$, $\frac{1}{1597} a^{25} - \frac{545}{1597} a^{17} - \frac{592}{1597} a^{16} - \frac{375}{1597} a^{15} + \frac{664}{1597} a^{14} - \frac{156}{1597} a^{13} - \frac{539}{1597} a^{12} - \frac{262}{1597} a^{11} - \frac{118}{1597} a^{10} - \frac{109}{1597} a^{9} + \frac{332}{1597} a^{8} - \frac{621}{1597} a^{7} - \frac{370}{1597} a^{6} - \frac{7}{1597} a^{5} - \frac{578}{1597} a^{4} - \frac{88}{1597} a^{3} + \frac{65}{1597} a^{2} + \frac{714}{1597} a + \frac{131}{1597}$, $\frac{1}{1597} a^{26} + \frac{731}{1597} a^{17} + \frac{563}{1597} a^{16} - \frac{612}{1597} a^{15} + \frac{97}{1597} a^{14} + \frac{228}{1597} a^{13} + \frac{589}{1597} a^{12} - \frac{415}{1597} a^{11} + \frac{292}{1597} a^{10} + \frac{558}{1597} a^{9} + \frac{527}{1597} a^{8} - \frac{604}{1597} a^{7} + \frac{784}{1597} a^{6} - \frac{344}{1597} a^{5} + \frac{144}{1597} a^{4} - \frac{123}{1597} a^{3} + \frac{16}{1597} a^{2} - \frac{10}{1597} a - \frac{583}{1597}$, $\frac{1}{106999} a^{27} + \frac{9}{106999} a^{26} + \frac{28}{106999} a^{25} + \frac{19}{106999} a^{24} - \frac{22}{106999} a^{23} + \frac{13}{106999} a^{22} - \frac{14}{106999} a^{21} - \frac{29}{106999} a^{20} - \frac{25}{106999} a^{19} + \frac{11}{106999} a^{18} - \frac{37149}{106999} a^{17} + \frac{10006}{106999} a^{16} - \frac{820}{106999} a^{15} - \frac{13822}{106999} a^{14} + \frac{41129}{106999} a^{13} + \frac{44333}{106999} a^{12} + \frac{47395}{106999} a^{11} - \frac{38859}{106999} a^{10} - \frac{14244}{106999} a^{9} + \frac{471}{106999} a^{8} - \frac{49190}{106999} a^{7} + \frac{25880}{106999} a^{6} + \frac{3758}{106999} a^{5} - \frac{40090}{106999} a^{4} - \frac{26748}{106999} a^{3} - \frac{51709}{106999} a^{2} - \frac{31602}{106999} a - \frac{50770}{106999}$, $\frac{1}{106999} a^{28} + \frac{14}{106999} a^{26} - \frac{32}{106999} a^{25} + \frac{8}{106999} a^{24} + \frac{10}{106999} a^{23} + \frac{3}{106999} a^{22} + \frac{30}{106999} a^{21} - \frac{32}{106999} a^{20} - \frac{32}{106999} a^{19} + \frac{4}{106999} a^{18} + \frac{50619}{106999} a^{17} - \frac{44309}{106999} a^{16} - \frac{34180}{106999} a^{15} + \frac{35011}{106999} a^{14} - \frac{42217}{106999} a^{13} - \frac{24039}{106999} a^{12} - \frac{51488}{106999} a^{11} + \frac{48995}{106999} a^{10} - \frac{48816}{106999} a^{9} + \frac{42917}{106999} a^{8} - \frac{12537}{106999} a^{7} + \frac{37699}{106999} a^{6} - \frac{44901}{106999} a^{5} + \frac{34}{1597} a^{4} + \frac{45844}{106999} a^{3} + \frac{5984}{106999} a^{2} + \frac{47656}{106999} a - \frac{32639}{106999}$, $\frac{1}{106999} a^{29} - \frac{24}{106999} a^{26} + \frac{18}{106999} a^{25} + \frac{12}{106999} a^{24} - \frac{24}{106999} a^{23} - \frac{18}{106999} a^{22} + \frac{30}{106999} a^{21} - \frac{28}{106999} a^{20} + \frac{19}{106999} a^{19} + \frac{14}{106999} a^{18} + \frac{47915}{106999} a^{17} - \frac{6094}{106999} a^{16} + \frac{50511}{106999} a^{15} + \frac{19703}{106999} a^{14} + \frac{19972}{106999} a^{13} - \frac{53070}{106999} a^{12} - \frac{37263}{106999} a^{11} + \frac{11135}{106999} a^{10} - \frac{27074}{106999} a^{9} - \frac{37355}{106999} a^{8} - \frac{23773}{106999} a^{7} - \frac{53126}{106999} a^{6} + \frac{52310}{106999} a^{5} + \frac{4573}{106999} a^{4} + \frac{38220}{106999} a^{3} + \frac{10462}{106999} a^{2} + \frac{49865}{106999} a - \frac{49938}{106999}$, $\frac{1}{106999} a^{30} + \frac{33}{106999} a^{26} + \frac{14}{106999} a^{25} + \frac{30}{106999} a^{24} - \frac{10}{106999} a^{23} + \frac{7}{106999} a^{22} - \frac{29}{106999} a^{21} - \frac{7}{106999} a^{20} + \frac{17}{106999} a^{19} + \frac{6}{106999} a^{18} + \frac{22307}{106999} a^{17} + \frac{23660}{106999} a^{16} - \frac{21685}{106999} a^{15} - \frac{27475}{106999} a^{14} + \frac{14786}{106999} a^{13} - \frac{26444}{106999} a^{12} + \frac{10486}{106999} a^{11} - \frac{38038}{106999} a^{10} + \frac{7513}{106999} a^{9} - \frac{33641}{106999} a^{8} + \frac{46215}{106999} a^{7} - \frac{27390}{106999} a^{6} + \frac{14767}{106999} a^{5} + \frac{6623}{106999} a^{4} - \frac{3901}{106999} a^{3} + \frac{19606}{106999} a^{2} + \frac{27841}{106999} a + \frac{18876}{106999}$, $\frac{1}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{31} - \frac{411369574244351260668477206275118601481135568840809899454376300926}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{30} - \frac{3198505691828902844760542420274769103211173454749798089670173949552}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{29} - \frac{220624632501429578592401896582645408399875533967295768149010212691}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{28} + \frac{2077508979649774182243317588257673083467405718902368588936003388390}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{27} - \frac{211358296864247699164554649719854937251430304002860757860191874281173}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{26} + \frac{62459927946555393120806198544340229039874869038844060657591690047292}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{25} + \frac{175856171233463518900080063395297524500219412628744061699995231899287}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{24} + \frac{94310086061928554264730713147458612773128130341222478839858061729181}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{23} + \frac{288206661368081922306835038679040423695020710257547522747829105825598}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{22} + \frac{243356933364446609297662899298485221023183495092634326402887295355289}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{21} + \frac{50662010019224183016386940249014129915085453895495589413104029095414}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{20} + \frac{49980837761274526258487087578161345334937185339724624467454012116767}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{19} - \frac{271675006793200937527159990441177144966686819380795395346154482273341}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{18} - \frac{42567809663503745699910514131275162780839205869437783597282188409200282}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{17} + \frac{436731514061649061698689115178912890652997959277370124003893721793809032}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{16} + \frac{126560513598413243590986725785587020175955221979939201165831454574794204}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{15} + \frac{63253262137580722750443490736667357738553082418215653949010537538045356}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{14} + \frac{102472564215895023769145870139979532660398559361364326229431382195407983}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{13} + \frac{101620895732371580358077420692233293383444904374225098378938983242946094}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{12} - \frac{63084288882286551018342701142675728027413273490016364519802839872917437}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{11} + \frac{478678778057265440723472153979642672869007005902833669913793416392296479}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{10} - \frac{217101105885065176105306885778951449963944954292251200918710481492043989}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{9} - \frac{387883430591484634428663325579945774303076980962646875100508342938472534}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{8} + \frac{446177060149731407215102568607427886695968577511122461977968564637078964}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{7} + \frac{41396369256820561367552509335065411990389722968583670151419650357118766}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{6} + \frac{313449805818638837676949564497311943705620694614309655843050194109894779}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{5} + \frac{317700625071639423920028096475247932429694903585010498021368589896032447}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{4} + \frac{51296270564389951782426151558938548913378163327682396433057882261436116}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{3} - \frac{234183476359859014790093892836286501632684468981981645783221081347283530}{957632355060136550216271854499156265751831480135074558745658568584605053} a^{2} - \frac{252576740897730059744138397554143482892565376552343561689958303272366589}{957632355060136550216271854499156265751831480135074558745658568584605053} a + \frac{71121441713185787237000681049331271039514217195856153445227007252859}{268169239725605306697359802436056081140249644395148294244093690446543}$

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

Class group and class number

$C_{2}\times C_{170}$, which has order $340$ (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{53363709857688080351195169120994802054487992856769582328349465}{599644555454061709590652382278745313557815579295600850811307807504449} a^{31} - \frac{10452894276603848114363373631171153995191118728825908198373667}{599644555454061709590652382278745313557815579295600850811307807504449} a^{30} + \frac{887561832626004382746430913205458579790359352828911892310718056}{599644555454061709590652382278745313557815579295600850811307807504449} a^{29} + \frac{69951443004119040344209985849387347469276247538811551710808537}{599644555454061709590652382278745313557815579295600850811307807504449} a^{28} - \frac{8802203054252049763899067027321836266658184685388305841972502539}{599644555454061709590652382278745313557815579295600850811307807504449} a^{27} + \frac{64329187607312123495264083021017489862918296433679404076972209}{599644555454061709590652382278745313557815579295600850811307807504449} a^{26} + \frac{58072855091070282409175003641275144226200064590326535703123516102}{599644555454061709590652382278745313557815579295600850811307807504449} a^{25} - \frac{6610497722121391463399692319394315077043042978592923339753927739}{599644555454061709590652382278745313557815579295600850811307807504449} a^{24} - \frac{286426040401445834914638028807977907806824168318494041202508578359}{599644555454061709590652382278745313557815579295600850811307807504449} a^{23} + \frac{65400697437017333599912598064924735831295063826555525165390711128}{599644555454061709590652382278745313557815579295600850811307807504449} a^{22} + \frac{1066503011193931301720603955725638061313353376290740470674410040597}{599644555454061709590652382278745313557815579295600850811307807504449} a^{21} - \frac{404777067830329852622852170974381172994271723486562995245172778934}{599644555454061709590652382278745313557815579295600850811307807504449} a^{20} - \frac{3084906359643498348479620555326638834432199868854624766305910632023}{599644555454061709590652382278745313557815579295600850811307807504449} a^{19} + \frac{1776841539675655943816467587302294993074677623894809775335644709047}{599644555454061709590652382278745313557815579295600850811307807504449} a^{18} + \frac{6717940915135264206531768186958731261179557148014520794107595347979}{599644555454061709590652382278745313557815579295600850811307807504449} a^{17} - \frac{5981225035915653277758553298599965932374420598120126684318349236587}{599644555454061709590652382278745313557815579295600850811307807504449} a^{16} - \frac{10728761216887628421853754176203265260049208419930625743302906018500}{599644555454061709590652382278745313557815579295600850811307807504449} a^{15} + \frac{15459151211647339871891612574153576562402382112577412572135943783218}{599644555454061709590652382278745313557815579295600850811307807504449} a^{14} + \frac{10070930633451897124107205537683504863195373967997999917495279957188}{599644555454061709590652382278745313557815579295600850811307807504449} a^{13} - \frac{26045021732622976597342039598238341181957626079715667808752736568196}{599644555454061709590652382278745313557815579295600850811307807504449} a^{12} + \frac{640829477030701289942627428085707228462775340753540625521521376627}{599644555454061709590652382278745313557815579295600850811307807504449} a^{11} + \frac{33142410497342488783420383732141452353438071229780642629147786670619}{599644555454061709590652382278745313557815579295600850811307807504449} a^{10} - \frac{11680151232051770665583032442929048673694119948434461562846114189578}{599644555454061709590652382278745313557815579295600850811307807504449} a^{9} - \frac{157514961181276503352317073647887106031194869073093472864512350475738}{599644555454061709590652382278745313557815579295600850811307807504449} a^{8} - \frac{31283734420400925399957427400619031802162505149201381464973827400370}{599644555454061709590652382278745313557815579295600850811307807504449} a^{7} + \frac{436763450641480743581554158194597395397394262626166911710831094025096}{599644555454061709590652382278745313557815579295600850811307807504449} a^{6} - \frac{121341050303821931070021818855259542704570982346595941561060329341252}{599644555454061709590652382278745313557815579295600850811307807504449} a^{5} - \frac{363699399406408350832640496112842937265149092511777999613542627107738}{599644555454061709590652382278745313557815579295600850811307807504449} a^{4} + \frac{639176594991675844737387310805943190741742160033292485561691082480926}{599644555454061709590652382278745313557815579295600850811307807504449} a^{3} - \frac{447748942202169452550417159982957503220665239498273482348561178543193}{599644555454061709590652382278745313557815579295600850811307807504449} a^{2} - \frac{561441612424103393524532268794982625878895438486016558529444993507810}{599644555454061709590652382278745313557815579295600850811307807504449} a + \frac{332004615348343134683068093263486918404492682378104097308678142677}{167920626002257549591333627073297483494207667122822976984404314619} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3942730660019.478 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

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 $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-3}, \sqrt{17})\), 4.4.4913.1, 4.0.44217.1, 8.0.1955143089.1, \(\Q(\zeta_{17})^+\), 8.0.33237432513.1, 16.0.1104726920056229495169.1, 16.0.1118134004496021794140625.1, 16.16.7336077203498398991356640625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ R R $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
3Data not computed
5Data not computed
17Data not computed