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Properties

Label 32.0.11721315531...2529.1
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 7^{16}\cdot 17^{30}$
Root discriminant $65.26$
Ramified primes $3, 7, 17$
Class number Not computed
Class group Not computed
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![10201, -449753, 19379456, -24204327, 118464295, -98875264, 463378521, -360876953, 778203968, -414305095, 721594375, -306932144, 460124585, -153240517, 210412060, -57178411, 73305359, -16127049, 19686714, -3559665, 4170679, -611014, 693327, -82313, 90746, -8433, 9079, -646, 679, -33, 34, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + 34*x^30 - 33*x^29 + 679*x^28 - 646*x^27 + 9079*x^26 - 8433*x^25 + 90746*x^24 - 82313*x^23 + 693327*x^22 - 611014*x^21 + 4170679*x^20 - 3559665*x^19 + 19686714*x^18 - 16127049*x^17 + 73305359*x^16 - 57178411*x^15 + 210412060*x^14 - 153240517*x^13 + 460124585*x^12 - 306932144*x^11 + 721594375*x^10 - 414305095*x^9 + 778203968*x^8 - 360876953*x^7 + 463378521*x^6 - 98875264*x^5 + 118464295*x^4 - 24204327*x^3 + 19379456*x^2 - 449753*x + 10201)
 
gp: K = bnfinit(x^32 - x^31 + 34*x^30 - 33*x^29 + 679*x^28 - 646*x^27 + 9079*x^26 - 8433*x^25 + 90746*x^24 - 82313*x^23 + 693327*x^22 - 611014*x^21 + 4170679*x^20 - 3559665*x^19 + 19686714*x^18 - 16127049*x^17 + 73305359*x^16 - 57178411*x^15 + 210412060*x^14 - 153240517*x^13 + 460124585*x^12 - 306932144*x^11 + 721594375*x^10 - 414305095*x^9 + 778203968*x^8 - 360876953*x^7 + 463378521*x^6 - 98875264*x^5 + 118464295*x^4 - 24204327*x^3 + 19379456*x^2 - 449753*x + 10201, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} + 34 x^{30} - 33 x^{29} + 679 x^{28} - 646 x^{27} + 9079 x^{26} - 8433 x^{25} + 90746 x^{24} - 82313 x^{23} + 693327 x^{22} - 611014 x^{21} + 4170679 x^{20} - 3559665 x^{19} + 19686714 x^{18} - 16127049 x^{17} + 73305359 x^{16} - 57178411 x^{15} + 210412060 x^{14} - 153240517 x^{13} + 460124585 x^{12} - 306932144 x^{11} + 721594375 x^{10} - 414305095 x^{9} + 778203968 x^{8} - 360876953 x^{7} + 463378521 x^{6} - 98875264 x^{5} + 118464295 x^{4} - 24204327 x^{3} + 19379456 x^{2} - 449753 x + 10201 \)

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:  \(11721315531921426838644724774450299033325694662265349722529=3^{16}\cdot 7^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.26$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(357=3\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{357}(1,·)$, $\chi_{357}(134,·)$, $\chi_{357}(8,·)$, $\chi_{357}(265,·)$, $\chi_{357}(139,·)$, $\chi_{357}(146,·)$, $\chi_{357}(20,·)$, $\chi_{357}(281,·)$, $\chi_{357}(155,·)$, $\chi_{357}(286,·)$, $\chi_{357}(160,·)$, $\chi_{357}(167,·)$, $\chi_{357}(41,·)$, $\chi_{357}(43,·)$, $\chi_{357}(302,·)$, $\chi_{357}(125,·)$, $\chi_{357}(50,·)$, $\chi_{357}(181,·)$, $\chi_{357}(62,·)$, $\chi_{357}(64,·)$, $\chi_{357}(328,·)$, $\chi_{357}(335,·)$, $\chi_{357}(209,·)$, $\chi_{357}(344,·)$, $\chi_{357}(97,·)$, $\chi_{357}(106,·)$, $\chi_{357}(274,·)$, $\chi_{357}(239,·)$, $\chi_{357}(244,·)$, $\chi_{357}(169,·)$, $\chi_{357}(253,·)$, $\chi_{357}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{271} a^{18} - \frac{96}{271} a^{17} + \frac{20}{271} a^{16} + \frac{11}{271} a^{15} - \frac{23}{271} a^{14} + \frac{42}{271} a^{13} - \frac{81}{271} a^{12} + \frac{74}{271} a^{11} + \frac{34}{271} a^{10} + \frac{111}{271} a^{9} + \frac{108}{271} a^{8} - \frac{116}{271} a^{7} + \frac{47}{271} a^{6} + \frac{39}{271} a^{5} - \frac{122}{271} a^{4} + \frac{64}{271} a^{3} - \frac{69}{271} a^{2} - \frac{96}{271} a - \frac{120}{271}$, $\frac{1}{271} a^{19} + \frac{18}{271} a^{17} + \frac{34}{271} a^{16} - \frac{51}{271} a^{15} + \frac{2}{271} a^{14} - \frac{114}{271} a^{13} - \frac{114}{271} a^{12} + \frac{92}{271} a^{11} + \frac{123}{271} a^{10} - \frac{76}{271} a^{9} - \frac{46}{271} a^{8} + \frac{22}{271} a^{7} - \frac{56}{271} a^{6} + \frac{99}{271} a^{5} + \frac{5}{271} a^{4} + \frac{113}{271} a^{3} + \frac{55}{271} a^{2} - \frac{122}{271} a + \frac{133}{271}$, $\frac{1}{271} a^{20} - \frac{135}{271} a^{17} + \frac{131}{271} a^{16} + \frac{75}{271} a^{15} + \frac{29}{271} a^{14} - \frac{57}{271} a^{13} - \frac{76}{271} a^{12} - \frac{125}{271} a^{11} + \frac{125}{271} a^{10} + \frac{124}{271} a^{9} - \frac{25}{271} a^{8} + \frac{135}{271} a^{7} + \frac{66}{271} a^{6} + \frac{116}{271} a^{5} - \frac{130}{271} a^{4} - \frac{13}{271} a^{3} + \frac{36}{271} a^{2} - \frac{36}{271} a - \frac{8}{271}$, $\frac{1}{271} a^{21} - \frac{92}{271} a^{17} + \frac{65}{271} a^{16} - \frac{112}{271} a^{15} + \frac{90}{271} a^{14} - \frac{97}{271} a^{13} + \frac{51}{271} a^{12} + \frac{88}{271} a^{11} + \frac{107}{271} a^{10} + \frac{55}{271} a^{9} + \frac{81}{271} a^{8} + \frac{124}{271} a^{7} - \frac{43}{271} a^{6} - \frac{14}{271} a^{5} + \frac{48}{271} a^{4} + \frac{4}{271} a^{3} + \frac{134}{271} a^{2} + \frac{40}{271} a + \frac{60}{271}$, $\frac{1}{271} a^{22} - \frac{95}{271} a^{17} + \frac{102}{271} a^{16} + \frac{18}{271} a^{15} - \frac{45}{271} a^{14} + \frac{121}{271} a^{13} - \frac{47}{271} a^{12} - \frac{131}{271} a^{11} - \frac{69}{271} a^{10} - \frac{5}{271} a^{9} + \frac{33}{271} a^{8} + \frac{125}{271} a^{7} - \frac{26}{271} a^{6} + \frac{113}{271} a^{5} - \frac{109}{271} a^{4} + \frac{60}{271} a^{3} - \frac{75}{271} a^{2} - \frac{100}{271} a + \frac{71}{271}$, $\frac{1}{271} a^{23} - \frac{75}{271} a^{17} + \frac{21}{271} a^{16} - \frac{84}{271} a^{15} + \frac{104}{271} a^{14} - \frac{122}{271} a^{13} + \frac{33}{271} a^{12} - \frac{85}{271} a^{11} - \frac{27}{271} a^{10} + \frac{9}{271} a^{9} + \frac{87}{271} a^{8} + \frac{65}{271} a^{7} - \frac{29}{271} a^{6} + \frac{73}{271} a^{5} + \frac{123}{271} a^{4} + \frac{43}{271} a^{3} + \frac{120}{271} a^{2} - \frac{106}{271} a - \frac{18}{271}$, $\frac{1}{271} a^{24} - \frac{133}{271} a^{17} + \frac{61}{271} a^{16} + \frac{116}{271} a^{15} + \frac{50}{271} a^{14} - \frac{69}{271} a^{13} + \frac{73}{271} a^{12} + \frac{103}{271} a^{11} + \frac{120}{271} a^{10} + \frac{11}{271} a^{9} + \frac{35}{271} a^{8} - \frac{57}{271} a^{7} + \frac{75}{271} a^{6} + \frac{67}{271} a^{5} + \frac{107}{271} a^{4} + \frac{42}{271} a^{3} - \frac{132}{271} a^{2} + \frac{99}{271} a - \frac{57}{271}$, $\frac{1}{271} a^{25} + \frac{30}{271} a^{17} + \frac{66}{271} a^{16} - \frac{113}{271} a^{15} + \frac{124}{271} a^{14} - \frac{32}{271} a^{13} - \frac{101}{271} a^{12} - \frac{65}{271} a^{11} - \frac{74}{271} a^{10} - \frac{107}{271} a^{9} - \frac{56}{271} a^{8} + \frac{94}{271} a^{7} + \frac{85}{271} a^{6} - \frac{126}{271} a^{5} + \frac{76}{271} a^{4} - \frac{21}{271} a^{3} - \frac{135}{271} a^{2} - \frac{88}{271} a + \frac{29}{271}$, $\frac{1}{271} a^{26} - \frac{35}{271} a^{17} + \frac{100}{271} a^{16} + \frac{65}{271} a^{15} + \frac{116}{271} a^{14} - \frac{6}{271} a^{13} - \frac{74}{271} a^{12} - \frac{126}{271} a^{11} - \frac{43}{271} a^{10} - \frac{134}{271} a^{9} + \frac{106}{271} a^{8} + \frac{42}{271} a^{7} + \frac{90}{271} a^{6} - \frac{10}{271} a^{5} + \frac{116}{271} a^{4} + \frac{113}{271} a^{3} + \frac{85}{271} a^{2} - \frac{72}{271} a + \frac{77}{271}$, $\frac{1}{271} a^{27} - \frac{8}{271} a^{17} - \frac{48}{271} a^{16} - \frac{41}{271} a^{15} + \frac{2}{271} a^{14} + \frac{41}{271} a^{13} + \frac{20}{271} a^{12} + \frac{108}{271} a^{11} - \frac{28}{271} a^{10} - \frac{74}{271} a^{9} + \frac{28}{271} a^{8} + \frac{95}{271} a^{7} + \frac{9}{271} a^{6} + \frac{126}{271} a^{5} - \frac{92}{271} a^{4} - \frac{114}{271} a^{3} - \frac{48}{271} a^{2} - \frac{31}{271} a - \frac{135}{271}$, $\frac{1}{271} a^{28} - \frac{3}{271} a^{17} + \frac{119}{271} a^{16} + \frac{90}{271} a^{15} + \frac{128}{271} a^{14} + \frac{85}{271} a^{13} + \frac{2}{271} a^{12} + \frac{22}{271} a^{11} - \frac{73}{271} a^{10} + \frac{103}{271} a^{9} - \frac{125}{271} a^{8} - \frac{106}{271} a^{7} - \frac{40}{271} a^{6} - \frac{51}{271} a^{5} - \frac{6}{271} a^{4} - \frac{78}{271} a^{3} - \frac{41}{271} a^{2} - \frac{90}{271} a + \frac{124}{271}$, $\frac{1}{271} a^{29} + \frac{102}{271} a^{17} - \frac{121}{271} a^{16} - \frac{110}{271} a^{15} + \frac{16}{271} a^{14} + \frac{128}{271} a^{13} + \frac{50}{271} a^{12} - \frac{122}{271} a^{11} - \frac{66}{271} a^{10} - \frac{63}{271} a^{9} - \frac{53}{271} a^{8} - \frac{117}{271} a^{7} + \frac{90}{271} a^{6} + \frac{111}{271} a^{5} + \frac{98}{271} a^{4} - \frac{120}{271} a^{3} - \frac{26}{271} a^{2} + \frac{107}{271} a - \frac{89}{271}$, $\frac{1}{271} a^{30} - \frac{85}{271} a^{17} + \frac{18}{271} a^{16} - \frac{22}{271} a^{15} + \frac{35}{271} a^{14} + \frac{102}{271} a^{13} + \frac{10}{271} a^{12} - \frac{26}{271} a^{11} - \frac{8}{271} a^{10} + \frac{7}{271} a^{9} - \frac{22}{271} a^{8} - \frac{2}{271} a^{7} - \frac{76}{271} a^{6} - \frac{86}{271} a^{5} + \frac{129}{271} a^{4} - \frac{50}{271} a^{3} + \frac{99}{271} a^{2} - \frac{53}{271} a + \frac{45}{271}$, $\frac{1}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{31} - \frac{178917629368084556271583932580959394694716462862326984029866669882145015435915678113727320155550}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{30} + \frac{4579143795506344606573249277362640541862576950355024997346792202859015264900028372536048791273349}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{29} - \frac{7631600787687519059416266050792733894604688537570997510142594081224010141228296757157430592842450}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{28} + \frac{10507451545787111507363827758859496112929858611242066028165578527442687750931180424426724770405047}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{27} - \frac{7777213635457344353684732353998197658830018332282368871352067406925505624369867193596907150418227}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{26} + \frac{5456677498093818866254839634536935651606735506535422847216535863891197389192618807114025036447171}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{25} + \frac{4605210893306941758405415027627454967573957663217147957369151294317920234416963800765848936336285}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{24} - \frac{9612532801365618776055562126079505203622533335280889958537250009998745954620381173423058084161342}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{23} + \frac{5305631510120210873133444642616882899468318563372712614783096253748334696057425619463915006196567}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{22} - \frac{9319481153574629477978762983091312957936728636758305994469098602974249558252706076449250938550711}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{21} + \frac{3946343331005796265297201167370317048462563850014600766980824740878820221640901155798822771064501}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{20} + \frac{1720256882364614955658750232820988533244430768051772434844417515607041034437378100124310861457888}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{19} + \frac{4907901640648308725792579691236409638664040159431941581942591916658625168555834963198937581439464}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{18} + \frac{2019199960163877006758223155508325656950065721767980837149475760066338759648184708096695157896431521}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{17} - \frac{369420944156896900858312630525757710180328842451003997602654845909696807421151583309923343697525853}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{16} + \frac{67951989427835994314264218530556998674206098426127426968493630160675085062808438808822293664000640}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{15} - \frac{2150993814865520793055037509210527912303192842725377342635620762350624588674804705175248835132882720}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{14} - \frac{785605851940804960991528550390099811094015772491766674873525021161762416968439562207843516537499750}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{13} - \frac{576195794588541322796004324575552594354384983040611928923842666674756381981609622667748312254270692}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{12} + \frac{313330671223596843883055466530786884298452852294742959495575731973521245814739691727205333151424529}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{11} - \frac{1540707204613477100630962139818105237192708804349772520567976069653287041772767252476373324108829828}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{10} + \frac{839848212755731984946169779911652529943647779036169698325841404738611428471738543684090982939504918}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{9} - \frac{2696946764316054518470329776665427735351223320931565281780947894541199277422796887840633210609458418}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{8} + \frac{601641263655725173775263651597964391459875042164970858260454280153496822052761538812303968515947367}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{7} - \frac{2170063577549121239747768488869568260526397059356086102144447287079317945391931032002353952818438992}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{6} + \frac{93263976777318735913225498441030386343582469277043231828827566077950570855909866581063804046793699}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{5} + \frac{2268159036073745187441349040729267995235233911090911781233754288747988763739639942531346561116869827}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{4} + \frac{1211561792172319108219222541937109418401723700614281875136767660301307507907762936266506728073567472}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{3} + \frac{1853300743731841321389522386594656382584343094646265891344077974389754819147630414696904195420462661}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a^{2} + \frac{1088508522534383680319311794834397134318885572840219039559557617689151755994814224831690707762365900}{6198953222881717682715918432741914073308168239339780494203224227697501688575977213845579823535638697} a - \frac{801242871216897268791597072158802217183386497419388959114363303218778918466190202251661990897311}{61375774483977402799167509235068456171368002369700796972309150769282194936395813998471087361738997}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{51320067208624698541670161667699083878649875014787088530144897993855975970025785456292458722}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{31} + \frac{51551356898411763205837840737709456186972012094978285439364671357616107707000899684152209316}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{30} - \frac{1745562025149530297923761122989472688310603605426094331661376770303680351374643025070555638050}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{29} + \frac{1701622366624312329711992152349763889670773147880229271299708845937133624676311457279220958894}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{28} - \frac{34868778063393826183739017842637491753290287491080854515230623901881884255095487338537418071628}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{27} + \frac{33315867479178566790021182804274646260547022096487280629991657273117101993133876312160520923854}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{26} - \frac{466374690072010576090240906527313781302292062744434567779400797569671217678053933583734972859586}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{25} + \frac{434995187047614699164614123989801580378128782197197480823023495446664765428237475352986483310388}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{24} - \frac{4662836460362710862253897920343282460879823711424947127611002973025287367887041900198806353774322}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{23} + \frac{4246688477443978600851430251038485303663850032972926759329663182878650571169326855841663974139582}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{22} - \frac{35637713395060440748189788409443022675309475059719390309110742736200285342089388876100971079907020}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{21} + \frac{31530388176251179501334004344901387361823102286487895889516453969296952304107819593060315514627982}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{20} - \frac{214456541968137218870255227129811534675947639472952509904970277072036121601953247002376862136149186}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{19} + \frac{183733916271528131022940115778210550644446835965302691884966050147626342908529847517183355023364916}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{18} - \frac{1012755538272384105119137178420722193175082317925522102361835296840658299678914765057419678319101623}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{17} + \frac{832652328859330108676703949770991101881038487530487731052963554754606221243400875412549216552608574}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{16} - \frac{3773069777810562920888144380445409906041693363548515439229832249289459339837860179425129553356517366}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{15} + \frac{2953136108579148771406086718042420504982604834387212023791037180432963693692108262734241215037311160}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{14} - \frac{10837548146409311277476976651784763315920905694663776529661573201011721519325440019408987869169508362}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{13} + \frac{7918045875364304032021673223445677299196050530838391610688140567493827550185374173644280538401255218}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{12} - \frac{23718905935333963683672883472236602145632023769673583798134281933963377739158928768786939989059965984}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{11} + \frac{15866887754016362678502978072382755820787493532127963086530731982995526664580771848820925635629937198}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{10} - \frac{37243826045905752730318627606769974105597952896643456964435129475306870177870718188731842732067215406}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{9} + \frac{21433626078466820773876563367176776492561905880969409263538926329408654556353167869496424249173027456}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{8} - \frac{40224995061212421304020999086340382046436580611917784961083365223998390592539836619006195047926924690}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{7} + \frac{18673351501782991888518836882121851300078006386445798826931119701084983789887706189146118915235755666}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{6} - \frac{24034540468347117204757427471557504607817104674468519129268123201930324498926773733032878413541519872}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{5} + \frac{5114229700872282339738613793297547409290555777102260396360245475396336673368935325665883209310362734}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{4} - \frac{6153886526856497358763513906750644561711802587513800736903759463306061435999261572027576830397464686}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{3} + \frac{1169132351169548001573738555734968617716643994880429248864170512834973657558963660766212916961785344}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a^{2} - \frac{1009596979683134773773042831515918923976206579835706563148656780712765672147550290585098237680214418}{22874366136094899198213721154029203222539366196825758281192709327297054201387369792788117430020807} a + \frac{231986700114869600014233234545552634100922265351361089857116295916383808752803504555725120024874}{226478872634602962358551694594348546757815506899264933477155537894030239617696730621664529010107} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (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, 16.0.108265024508940221449655688273.1, 16.16.16501299269766837593302193.1

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 $16^{2}$ R $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ 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
7Data not computed
17Data not computed