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Properties

Label 32.0.55549814836...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 5^{16}\cdot 11^{16}$
Root discriminant $59.33$
Ramified primes $2, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22688836, 29361904, 226590728, -1453633536, 4388439484, -9105819856, 14623867272, -19475722016, 22339693138, -22729087880, 20941585712, -17739551640, 13969080118, -10275000064, 7076120604, -4553339576, 2732603853, -1523844232, 787966480, -376678008, 166198466, -67485584, 25163388, -8576968, 2661064, -745264, 186928, -41328, 7940, -1280, 168, -16, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^31 + 168*x^30 - 1280*x^29 + 7940*x^28 - 41328*x^27 + 186928*x^26 - 745264*x^25 + 2661064*x^24 - 8576968*x^23 + 25163388*x^22 - 67485584*x^21 + 166198466*x^20 - 376678008*x^19 + 787966480*x^18 - 1523844232*x^17 + 2732603853*x^16 - 4553339576*x^15 + 7076120604*x^14 - 10275000064*x^13 + 13969080118*x^12 - 17739551640*x^11 + 20941585712*x^10 - 22729087880*x^9 + 22339693138*x^8 - 19475722016*x^7 + 14623867272*x^6 - 9105819856*x^5 + 4388439484*x^4 - 1453633536*x^3 + 226590728*x^2 + 29361904*x + 22688836)
 
gp: K = bnfinit(x^32 - 16*x^31 + 168*x^30 - 1280*x^29 + 7940*x^28 - 41328*x^27 + 186928*x^26 - 745264*x^25 + 2661064*x^24 - 8576968*x^23 + 25163388*x^22 - 67485584*x^21 + 166198466*x^20 - 376678008*x^19 + 787966480*x^18 - 1523844232*x^17 + 2732603853*x^16 - 4553339576*x^15 + 7076120604*x^14 - 10275000064*x^13 + 13969080118*x^12 - 17739551640*x^11 + 20941585712*x^10 - 22729087880*x^9 + 22339693138*x^8 - 19475722016*x^7 + 14623867272*x^6 - 9105819856*x^5 + 4388439484*x^4 - 1453633536*x^3 + 226590728*x^2 + 29361904*x + 22688836, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{31} + 168 x^{30} - 1280 x^{29} + 7940 x^{28} - 41328 x^{27} + 186928 x^{26} - 745264 x^{25} + 2661064 x^{24} - 8576968 x^{23} + 25163388 x^{22} - 67485584 x^{21} + 166198466 x^{20} - 376678008 x^{19} + 787966480 x^{18} - 1523844232 x^{17} + 2732603853 x^{16} - 4553339576 x^{15} + 7076120604 x^{14} - 10275000064 x^{13} + 13969080118 x^{12} - 17739551640 x^{11} + 20941585712 x^{10} - 22729087880 x^{9} + 22339693138 x^{8} - 19475722016 x^{7} + 14623867272 x^{6} - 9105819856 x^{5} + 4388439484 x^{4} - 1453633536 x^{3} + 226590728 x^{2} + 29361904 x + 22688836 \)

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:  \(555498148363897775465439454409629483663360000000000000000=2^{96}\cdot 5^{16}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.33$
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$$\chi_{880}(1,·)$, $\chi_{880}(131,·)$, $\chi_{880}(769,·)$, $\chi_{880}(109,·)$, $\chi_{880}(529,·)$, $\chi_{880}(659,·)$, $\chi_{880}(661,·)$, $\chi_{880}(791,·)$, $\chi_{880}(111,·)$, $\chi_{880}(419,·)$, $\chi_{880}(549,·)$, $\chi_{880}(551,·)$, $\chi_{880}(681,·)$, $\chi_{880}(309,·)$, $\chi_{880}(219,·)$, $\chi_{880}(439,·)$, $\chi_{880}(441,·)$, $\chi_{880}(571,·)$, $\chi_{880}(199,·)$, $\chi_{880}(329,·)$, $\chi_{880}(331,·)$, $\chi_{880}(461,·)$, $\chi_{880}(89,·)$, $\chi_{880}(859,·)$, $\chi_{880}(221,·)$, $\chi_{880}(351,·)$, $\chi_{880}(749,·)$, $\chi_{880}(879,·)$, $\chi_{880}(241,·)$, $\chi_{880}(771,·)$, $\chi_{880}(639,·)$, $\chi_{880}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{3}{7} a^{14} + \frac{1}{7} a^{13} - \frac{3}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} - \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{42} a^{16} - \frac{1}{21} a^{15} - \frac{1}{3} a^{14} - \frac{4}{21} a^{13} + \frac{5}{21} a^{12} - \frac{1}{21} a^{11} - \frac{4}{21} a^{10} + \frac{8}{21} a^{9} - \frac{3}{14} a^{8} + \frac{8}{21} a^{7} + \frac{2}{21} a^{6} + \frac{8}{21} a^{5} - \frac{10}{21} a^{4} + \frac{2}{21} a^{3} - \frac{1}{21} a^{2} + \frac{5}{21} a - \frac{8}{21}$, $\frac{1}{42} a^{17} + \frac{3}{7} a^{14} + \frac{2}{7} a^{13} + \frac{1}{7} a^{12} - \frac{3}{7} a^{11} + \frac{3}{7} a^{10} + \frac{11}{42} a^{9} + \frac{8}{21} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{3}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{21} a - \frac{4}{21}$, $\frac{1}{42} a^{18} - \frac{2}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{4} - \frac{4}{21} a^{2} - \frac{1}{21} a + \frac{3}{7}$, $\frac{1}{42} a^{19} - \frac{2}{7} a^{14} - \frac{1}{7} a^{13} - \frac{3}{7} a^{12} - \frac{1}{6} a^{11} - \frac{1}{3} a^{10} - \frac{2}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{4}{21} a^{3} - \frac{1}{21} a^{2} + \frac{3}{7} a$, $\frac{1}{42} a^{20} - \frac{2}{7} a^{14} - \frac{1}{7} a^{13} - \frac{1}{42} a^{12} + \frac{5}{21} a^{11} - \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{10}{21} a^{4} - \frac{4}{21} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{42} a^{21} - \frac{2}{7} a^{14} + \frac{11}{42} a^{13} + \frac{8}{21} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{21} a^{5} - \frac{10}{21} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{42} a^{22} + \frac{5}{42} a^{14} - \frac{1}{3} a^{13} - \frac{2}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{4}{21} a^{6} - \frac{1}{21} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{42} a^{23} - \frac{1}{42} a^{15} + \frac{5}{21} a^{14} - \frac{3}{7} a^{13} - \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{8}{21} a^{7} - \frac{10}{21} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{42} a^{24} + \frac{1}{21} a^{15} - \frac{4}{21} a^{14} - \frac{1}{3} a^{13} + \frac{5}{21} a^{12} + \frac{8}{21} a^{11} - \frac{1}{21} a^{10} - \frac{4}{21} a^{9} + \frac{1}{42} a^{8} + \frac{10}{21} a^{7} + \frac{8}{21} a^{6} - \frac{1}{3} a^{5} - \frac{1}{21} a^{4} + \frac{5}{21} a^{3} - \frac{1}{3} a^{2} + \frac{2}{21} a - \frac{5}{21}$, $\frac{1}{42} a^{25} + \frac{1}{21} a^{15} - \frac{5}{21} a^{14} - \frac{5}{21} a^{13} + \frac{10}{21} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{6} a^{9} + \frac{1}{21} a^{8} + \frac{1}{21} a^{7} - \frac{2}{21} a^{6} - \frac{2}{21} a^{5} + \frac{1}{21} a^{4} - \frac{2}{21} a^{3} - \frac{2}{21} a^{2} - \frac{3}{7} a - \frac{8}{21}$, $\frac{1}{1722} a^{26} - \frac{13}{1722} a^{25} - \frac{1}{574} a^{24} - \frac{11}{1722} a^{23} - \frac{19}{1722} a^{22} - \frac{5}{1722} a^{21} + \frac{1}{123} a^{20} + \frac{10}{861} a^{19} - \frac{1}{574} a^{18} + \frac{10}{861} a^{17} - \frac{4}{861} a^{16} - \frac{47}{1722} a^{15} + \frac{67}{1722} a^{14} - \frac{167}{574} a^{13} - \frac{430}{861} a^{12} - \frac{35}{123} a^{11} - \frac{316}{861} a^{10} + \frac{263}{574} a^{9} + \frac{29}{1722} a^{8} + \frac{61}{287} a^{7} + \frac{64}{861} a^{6} - \frac{137}{861} a^{5} - \frac{44}{123} a^{4} + \frac{2}{41} a^{3} - \frac{263}{861} a^{2} - \frac{11}{41} a - \frac{367}{861}$, $\frac{1}{1722} a^{27} - \frac{4}{861} a^{25} - \frac{3}{574} a^{24} + \frac{1}{861} a^{23} - \frac{1}{287} a^{22} - \frac{5}{861} a^{21} - \frac{1}{574} a^{20} + \frac{11}{1722} a^{19} - \frac{19}{1722} a^{18} + \frac{1}{287} a^{17} + \frac{13}{1722} a^{16} + \frac{8}{123} a^{15} - \frac{20}{861} a^{14} - \frac{61}{123} a^{13} - \frac{115}{246} a^{12} - \frac{16}{861} a^{11} + \frac{39}{574} a^{10} - \frac{187}{861} a^{9} + \frac{187}{861} a^{8} + \frac{352}{861} a^{7} - \frac{53}{123} a^{6} + \frac{166}{861} a^{5} + \frac{49}{123} a^{4} + \frac{108}{287} a^{3} - \frac{124}{861} a^{2} - \frac{131}{861} a - \frac{97}{861}$, $\frac{1}{773178} a^{28} - \frac{1}{55227} a^{27} - \frac{223}{773178} a^{26} + \frac{1859}{386589} a^{25} - \frac{7345}{773178} a^{24} + \frac{2252}{386589} a^{23} - \frac{694}{128863} a^{22} - \frac{805}{110454} a^{21} - \frac{8779}{773178} a^{20} - \frac{1888}{386589} a^{19} - \frac{3826}{386589} a^{18} + \frac{180}{18409} a^{17} + \frac{866}{386589} a^{16} - \frac{8909}{128863} a^{15} - \frac{17531}{128863} a^{14} + \frac{793}{36818} a^{13} - \frac{79994}{386589} a^{12} + \frac{8161}{128863} a^{11} + \frac{249775}{773178} a^{10} + \frac{76309}{386589} a^{9} + \frac{24245}{110454} a^{8} - \frac{142388}{386589} a^{7} + \frac{7472}{18409} a^{6} - \frac{16294}{55227} a^{5} + \frac{17606}{55227} a^{4} - \frac{3559}{18409} a^{3} - \frac{109745}{386589} a^{2} - \frac{2382}{128863} a + \frac{3439}{128863}$, $\frac{1}{773178} a^{29} + \frac{5}{128863} a^{27} + \frac{7}{36818} a^{26} - \frac{8275}{773178} a^{25} - \frac{8975}{773178} a^{24} - \frac{2969}{257726} a^{23} - \frac{2867}{773178} a^{22} + \frac{2131}{773178} a^{21} - \frac{2726}{386589} a^{20} + \frac{1297}{110454} a^{19} + \frac{617}{128863} a^{18} - \frac{1528}{128863} a^{17} - \frac{228}{128863} a^{16} - \frac{2991}{257726} a^{15} + \frac{244861}{773178} a^{14} - \frac{143921}{386589} a^{13} + \frac{30659}{386589} a^{12} + \frac{257309}{773178} a^{11} + \frac{101323}{257726} a^{10} + \frac{182597}{773178} a^{9} - \frac{36577}{773178} a^{8} - \frac{59155}{128863} a^{7} - \frac{9182}{18409} a^{6} - \frac{48893}{386589} a^{5} - \frac{100451}{386589} a^{4} - \frac{10315}{55227} a^{3} - \frac{3032}{18409} a^{2} - \frac{44827}{386589} a + \frac{33329}{128863}$, $\frac{1}{2747932532538411982056238076743514297170126923337626} a^{30} - \frac{5}{915977510846137327352079358914504765723375641112542} a^{29} - \frac{51249941280785204838794573749633374948872639}{2747932532538411982056238076743514297170126923337626} a^{28} + \frac{102499882561570409677589147499266749897745423}{392561790362630283150891153820502042452875274762518} a^{27} + \frac{278321693589110767136265911000636604882659344810}{1373966266269205991028119038371757148585063461668813} a^{26} - \frac{3639168867612921514152943220958750730516134863827}{1373966266269205991028119038371757148585063461668813} a^{25} + \frac{5881219483797807094262235472895006180497157401137}{915977510846137327352079358914504765723375641112542} a^{24} + \frac{21758792242783965799186521357163667076921996031097}{2747932532538411982056238076743514297170126923337626} a^{23} + \frac{26183217036491435462627989898019878030166454984753}{2747932532538411982056238076743514297170126923337626} a^{22} + \frac{16770335270251753433054333774229625595169661895693}{2747932532538411982056238076743514297170126923337626} a^{21} + \frac{26994423703534371338943310294640374067375910923563}{2747932532538411982056238076743514297170126923337626} a^{20} + \frac{2689776442128594183259940284906503320949409526699}{392561790362630283150891153820502042452875274762518} a^{19} - \frac{954072742617673852973772925104781584390630251358}{457988755423068663676039679457252382861687820556271} a^{18} + \frac{9059041099072419831399989355949976312319981023513}{1373966266269205991028119038371757148585063461668813} a^{17} - \frac{12589517363576980669731230217874342171100434415678}{1373966266269205991028119038371757148585063461668813} a^{16} - \frac{18884156131584868293745021713569657754747232693561}{2747932532538411982056238076743514297170126923337626} a^{15} + \frac{488999662572483694853675572144660458488016436712257}{1373966266269205991028119038371757148585063461668813} a^{14} + \frac{8056774935723966833290833596924222418044068455687}{457988755423068663676039679457252382861687820556271} a^{13} - \frac{40052921578343695520136181521047388521959143753611}{80821545074659175942830531668926891093239027156989} a^{12} - \frac{12715090653422388621661561891651435108435545528183}{65426965060438380525148525636750340408812545793753} a^{11} + \frac{449781249387346285402488625616533202493997765579948}{1373966266269205991028119038371757148585063461668813} a^{10} + \frac{294275703986390533274838627041559264366356821469837}{1373966266269205991028119038371757148585063461668813} a^{9} - \frac{550849640916780758016473312282874046118972541222329}{2747932532538411982056238076743514297170126923337626} a^{8} - \frac{49752603661162216472716492384220184735907348974846}{457988755423068663676039679457252382861687820556271} a^{7} - \frac{17847100202848860263699463821505312021880141452164}{1373966266269205991028119038371757148585063461668813} a^{6} - \frac{638323735127090455776823638655036981345570071534592}{1373966266269205991028119038371757148585063461668813} a^{5} - \frac{676446197194298518722841127906219818976686502228533}{1373966266269205991028119038371757148585063461668813} a^{4} + \frac{26749941530439369935458452078847044773070562020190}{65426965060438380525148525636750340408812545793753} a^{3} - \frac{208817199500648101167984644127971603052055011231934}{1373966266269205991028119038371757148585063461668813} a^{2} + \frac{461320891291789290740704417177960676016690587165963}{1373966266269205991028119038371757148585063461668813} a + \frac{127595965523557286792710790931351066264788971468009}{1373966266269205991028119038371757148585063461668813}$, $\frac{1}{153328719016900833397740717601708868995811275038973584891222} a^{31} + \frac{4649818}{25554786502816805566290119600284811499301879173162264148537} a^{30} + \frac{32485055001831196340465524485037679196896054877859234}{76664359508450416698870358800854434497905637519486792445611} a^{29} + \frac{16017409223197724107624029550014391721752624896370785}{51109573005633611132580239200569622998603758346324528297074} a^{28} - \frac{1773026684923315439565484284501773539189743815907691072}{25554786502816805566290119600284811499301879173162264148537} a^{27} + \frac{16424986854790569017705760881164521483426694135349738219}{76664359508450416698870358800854434497905637519486792445611} a^{26} + \frac{589399489393293634626391859052412898475644356265619332917}{51109573005633611132580239200569622998603758346324528297074} a^{25} + \frac{19480551311436149158877163455999267351929519441593927455}{76664359508450416698870358800854434497905637519486792445611} a^{24} + \frac{851331384331687423545732775722724390761096248632258015978}{76664359508450416698870358800854434497905637519486792445611} a^{23} - \frac{786339358174222178702367546129650084240864848589506108996}{76664359508450416698870358800854434497905637519486792445611} a^{22} + \frac{234510217759198965364189667537975138004670196059594574111}{21904102716700119056820102514529838427973039291281940698746} a^{21} - \frac{22821757739844659747565230624023428417826455865216944424}{4509668206379436276404138752991437323406213971734517202683} a^{20} + \frac{681265413968574702284075674432565574504210601141106895991}{76664359508450416698870358800854434497905637519486792445611} a^{19} - \frac{1482672334141020179891344526712332279437007901023351828903}{153328719016900833397740717601708868995811275038973584891222} a^{18} - \frac{228331549393203081894129100343652198677140332495622286369}{51109573005633611132580239200569622998603758346324528297074} a^{17} + \frac{34023396183238253647541616720272529830896275274887268985}{3650683786116686509470017085754973071328839881880323449791} a^{16} + \frac{456690417689307080419958321150814654852324577022441460397}{7301367572233373018940034171509946142657679763760646899582} a^{15} + \frac{3151503878031794976433077482814810560387877105583907843410}{76664359508450416698870358800854434497905637519486792445611} a^{14} + \frac{9889681795137084988935442353104563404468901056232654392731}{153328719016900833397740717601708868995811275038973584891222} a^{13} - \frac{75620593277169985252942854123824935445548202541938041460567}{153328719016900833397740717601708868995811275038973584891222} a^{12} - \frac{29283609925972814149668944348085205632953161155970980112145}{76664359508450416698870358800854434497905637519486792445611} a^{11} - \frac{6257119590180398058122944452321756009178429516769303548045}{21904102716700119056820102514529838427973039291281940698746} a^{10} + \frac{19245162602590501262906858970644416222155448451451994817674}{76664359508450416698870358800854434497905637519486792445611} a^{9} - \frac{4436056481417218329235200495914122868860776509121932982131}{25554786502816805566290119600284811499301879173162264148537} a^{8} - \frac{9053875574906261423974440775418768518973345525636286910892}{76664359508450416698870358800854434497905637519486792445611} a^{7} - \frac{22630850139524203036124127199218213439075497042949232161549}{76664359508450416698870358800854434497905637519486792445611} a^{6} + \frac{35035157622061459846878961239157201625811424285379951079256}{76664359508450416698870358800854434497905637519486792445611} a^{5} + \frac{12164930496145137923060245763574447750239762693756980131667}{76664359508450416698870358800854434497905637519486792445611} a^{4} - \frac{11823355151326700654166706922542005941682497440934181619214}{25554786502816805566290119600284811499301879173162264148537} a^{3} + \frac{22474580985679722580142389960066741858249749183687985220449}{76664359508450416698870358800854434497905637519486792445611} a^{2} + \frac{20457428901897318734783091027781151300008087231124302835200}{76664359508450416698870358800854434497905637519486792445611} a - \frac{28745776590783560898058938769967880976980641111516156297587}{76664359508450416698870358800854434497905637519486792445611}$

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{94368846925509499649011763501309708277597842339770}{4509668206379436276404138752991437323406213971734517202683} a^{31} - \frac{1462717127345397244559682334270300478302766556266435}{4509668206379436276404138752991437323406213971734517202683} a^{30} + \frac{4979292127859693808495631572101064453181595719161939}{1503222735459812092134712917663812441135404657244839067561} a^{29} - \frac{110552215829355380438983004151799519036448838454227809}{4509668206379436276404138752991437323406213971734517202683} a^{28} + \frac{666016385216229547915419382410695833634778483221317697}{4509668206379436276404138752991437323406213971734517202683} a^{27} - \frac{1120874566333243437369167404445647847234269983340275105}{1503222735459812092134712917663812441135404657244839067561} a^{26} + \frac{4912439810682664363082767464578222337907183475626447459}{1503222735459812092134712917663812441135404657244839067561} a^{25} - \frac{18949033262379116215919156010707453742107159366330276660}{1503222735459812092134712917663812441135404657244839067561} a^{24} + \frac{65359282470151550158350715201341927153709321163655844701}{1503222735459812092134712917663812441135404657244839067561} a^{23} - \frac{203118403481020085971988830527856554605381709987675700254}{1503222735459812092134712917663812441135404657244839067561} a^{22} + \frac{245720669319062199940846172979289342474527919176764403955}{644238315197062325200591250427348189058030567390645314669} a^{21} - \frac{2951336854253732243972288480266794352741050763250710868977}{3006445470919624184269425835327624882270809314489678135122} a^{20} + \frac{10432227849074037385800809096485057355479412663692719672001}{4509668206379436276404138752991437323406213971734517202683} a^{19} - \frac{22548644443182145735479355442401572033348123991286189367238}{4509668206379436276404138752991437323406213971734517202683} a^{18} + \frac{29885181876160516476659626002501713111509642458659450227751}{3006445470919624184269425835327624882270809314489678135122} a^{17} - \frac{558540282088022166251349037197604822685472180930026015757}{30678015009383920247647202401302294717049074637649776889} a^{16} + \frac{6621227114084128373975752465505047441032293468189595312677}{214746105065687441733530416809116063019343522463548438223} a^{15} - \frac{72780203935798272409419862961049774988145089879671216117943}{1503222735459812092134712917663812441135404657244839067561} a^{14} + \frac{319357431429125045904725846974805833113539885557776026514115}{4509668206379436276404138752991437323406213971734517202683} a^{13} - \frac{290615930518740119405844936409463942656539276833306346260787}{3006445470919624184269425835327624882270809314489678135122} a^{12} + \frac{555369539815552560414274127635182333526816460709690623503664}{4509668206379436276404138752991437323406213971734517202683} a^{11} - \frac{93735568438266153388648426905726093793064762976299948801255}{644238315197062325200591250427348189058030567390645314669} a^{10} + \frac{473087592526763621554629173626438838455928187888668299579565}{3006445470919624184269425835327624882270809314489678135122} a^{9} - \frac{229510784019154867989535514852391952585286540436410616802008}{1503222735459812092134712917663812441135404657244839067561} a^{8} + \frac{582024509411338131443839121300394175383795041097677813469256}{4509668206379436276404138752991437323406213971734517202683} a^{7} - \frac{413371193591586261208619989355299511903013445206715848776588}{4509668206379436276404138752991437323406213971734517202683} a^{6} + \frac{172895747479584792059798496113211784996779188999701908228}{3347934822850361006981543246467288287606691886959552489} a^{5} - \frac{94726510633299832197718924691529218039595215016125627151242}{4509668206379436276404138752991437323406213971734517202683} a^{4} + \frac{9410882068735205803749166827533234014682497165781872898508}{1503222735459812092134712917663812441135404657244839067561} a^{3} - \frac{9002355127595947948540427074771599218412223748176604686906}{4509668206379436276404138752991437323406213971734517202683} a^{2} + \frac{3028895223031183741300153074220647097147872946792839344332}{1503222735459812092134712917663812441135404657244839067561} a - \frac{1172239655428345734493004209702424518590967990890077630579}{1503222735459812092134712917663812441135404657244839067561} \) (order $16$)
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

$C_2^3\times C_4$ (as 32T34):

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 $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{110}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(i, \sqrt{110})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{22})\), \(\Q(\sqrt{-5}, \sqrt{22})\), \(\Q(\sqrt{5}, \sqrt{-22})\), \(\Q(\sqrt{-5}, \sqrt{-22})\), \(\Q(\sqrt{5}, \sqrt{22})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{55})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{-55})\), \(\Q(\sqrt{-2}, \sqrt{55})\), \(\Q(\sqrt{-10}, \sqrt{11})\), \(\Q(\sqrt{10}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{55})\), \(\Q(\sqrt{-2}, \sqrt{-55})\), \(\Q(\sqrt{-10}, \sqrt{-11})\), \(\Q(\sqrt{10}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-5}, \sqrt{11})\), \(\Q(\sqrt{-5}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(\sqrt{-10}, \sqrt{22})\), \(\Q(\sqrt{10}, \sqrt{22})\), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{11})\), \(\Q(\sqrt{10}, \sqrt{-22})\), \(\Q(\sqrt{-10}, \sqrt{-22})\), 4.0.2048.2, \(\Q(\zeta_{16})^+\), 4.4.6195200.2, 4.0.6195200.5, 4.4.51200.1, 4.0.51200.2, 4.0.247808.2, 4.4.247808.1, 8.0.599695360000.8, 8.0.599695360000.9, 8.0.599695360000.6, 8.0.40960000.1, 8.0.2342560000.1, 8.0.959512576.1, 8.0.599695360000.3, 8.0.599695360000.2, 8.0.599695360000.5, 8.0.37480960000.9, 8.0.599695360000.7, 8.0.599695360000.1, 8.0.599695360000.4, 8.8.599695360000.1, 8.0.37480960000.2, \(\Q(\zeta_{16})\), 8.0.153522012160000.43, 8.0.10485760000.3, 8.0.245635219456.2, 8.0.38380503040000.50, 8.0.38380503040000.59, 8.0.38380503040000.23, 8.0.38380503040000.20, 8.0.153522012160000.31, 8.8.153522012160000.3, 8.8.153522012160000.4, 8.0.153522012160000.46, 8.0.10485760000.1, 8.0.10485760000.2, 8.0.153522012160000.26, 8.0.153522012160000.32, 8.0.2621440000.1, 8.8.2621440000.1, 8.8.38380503040000.3, 8.0.38380503040000.63, 8.0.245635219456.1, 8.8.245635219456.1, 8.8.153522012160000.2, 8.0.153522012160000.39, 8.0.61408804864.1, 8.0.61408804864.2, 8.0.38380503040000.15, 8.0.38380503040000.48, 16.0.359634524805529600000000.1, 16.0.23569008217655187865600000000.6, 16.0.23569008217655187865600000000.9, 16.0.109951162777600000000.1, 16.0.23569008217655187865600000000.5, 16.0.60336661037197280935936.1, 16.0.23569008217655187865600000000.2, 16.0.23569008217655187865600000000.3, 16.0.23569008217655187865600000000.8, 16.0.1473063013603449241600000000.2, 16.0.1473063013603449241600000000.1, 16.0.23569008217655187865600000000.1, 16.0.23569008217655187865600000000.7, 16.0.23569008217655187865600000000.4, 16.16.23569008217655187865600000000.1

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$