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Properties

Label 32.0.664...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $6.649\times 10^{48}$
Root discriminant $33.55$
Ramified primes $2, 3, 5, 11$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group $C_2^3\times D_4$ (as 32T273)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 22*x^30 - 40*x^29 + 178*x^28 - 712*x^27 + 1264*x^26 + 844*x^25 - 9359*x^24 + 23304*x^23 - 31738*x^22 + 20712*x^21 + 4242*x^20 - 53384*x^19 + 309318*x^18 - 1090772*x^17 + 2225756*x^16 - 2949980*x^15 + 2866688*x^14 - 2095500*x^13 + 1173576*x^12 - 535544*x^11 + 258862*x^10 - 119572*x^9 + 42617*x^8 - 13676*x^7 + 3860*x^6 - 916*x^5 + 264*x^4 - 80*x^3 + 20*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^32 - 8*x^31 + 22*x^30 - 40*x^29 + 178*x^28 - 712*x^27 + 1264*x^26 + 844*x^25 - 9359*x^24 + 23304*x^23 - 31738*x^22 + 20712*x^21 + 4242*x^20 - 53384*x^19 + 309318*x^18 - 1090772*x^17 + 2225756*x^16 - 2949980*x^15 + 2866688*x^14 - 2095500*x^13 + 1173576*x^12 - 535544*x^11 + 258862*x^10 - 119572*x^9 + 42617*x^8 - 13676*x^7 + 3860*x^6 - 916*x^5 + 264*x^4 - 80*x^3 + 20*x^2 - 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 20, -80, 264, -916, 3860, -13676, 42617, -119572, 258862, -535544, 1173576, -2095500, 2866688, -2949980, 2225756, -1090772, 309318, -53384, 4242, 20712, -31738, 23304, -9359, 844, 1264, -712, 178, -40, 22, -8, 1]);
 

\( x^{32} - 8 x^{31} + 22 x^{30} - 40 x^{29} + 178 x^{28} - 712 x^{27} + 1264 x^{26} + 844 x^{25} - 9359 x^{24} + 23304 x^{23} - 31738 x^{22} + 20712 x^{21} + 4242 x^{20} - 53384 x^{19} + 309318 x^{18} - 1090772 x^{17} + 2225756 x^{16} - 2949980 x^{15} + 2866688 x^{14} - 2095500 x^{13} + 1173576 x^{12} - 535544 x^{11} + 258862 x^{10} - 119572 x^{9} + 42617 x^{8} - 13676 x^{7} + 3860 x^{6} - 916 x^{5} + 264 x^{4} - 80 x^{3} + 20 x^{2} - 4 x + 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(6649076259173893054484111016591360000000000000000\)\(\medspace = 2^{72}\cdot 3^{16}\cdot 5^{16}\cdot 11^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $33.55$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 5, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $16$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{5} a^{22} - \frac{1}{5} a^{21} + \frac{2}{5} a^{20} + \frac{2}{5} a^{19} - \frac{1}{5} a^{18} + \frac{1}{5} a^{17} - \frac{1}{5} a^{16} + \frac{2}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{23} + \frac{1}{5} a^{21} - \frac{1}{5} a^{20} + \frac{1}{5} a^{19} - \frac{1}{5} a^{16} + \frac{2}{5} a^{15} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{24} + \frac{2}{5} a^{20} - \frac{1}{5} a^{19} - \frac{2}{5} a^{18} - \frac{1}{5} a^{17} - \frac{1}{5} a^{16} + \frac{1}{5} a^{14} - \frac{2}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{10}$, $\frac{1}{10} a^{25} + \frac{2}{5} a^{21} - \frac{1}{5} a^{20} - \frac{2}{5} a^{19} - \frac{1}{5} a^{18} - \frac{1}{5} a^{17} + \frac{1}{5} a^{15} - \frac{2}{5} a^{14} - \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a$, $\frac{1}{4130} a^{26} + \frac{76}{2065} a^{25} + \frac{171}{4130} a^{24} - \frac{157}{2065} a^{23} - \frac{21}{295} a^{22} - \frac{185}{413} a^{21} + \frac{802}{2065} a^{20} - \frac{356}{2065} a^{19} - \frac{271}{2065} a^{18} + \frac{783}{2065} a^{17} - \frac{879}{2065} a^{16} - \frac{72}{295} a^{15} + \frac{729}{2065} a^{14} + \frac{136}{295} a^{13} + \frac{642}{2065} a^{12} + \frac{34}{413} a^{11} - \frac{393}{2065} a^{10} + \frac{272}{2065} a^{9} + \frac{43}{295} a^{8} + \frac{206}{2065} a^{7} - \frac{418}{2065} a^{6} - \frac{111}{295} a^{5} + \frac{37}{413} a^{4} + \frac{334}{2065} a^{3} - \frac{1927}{4130} a^{2} + \frac{593}{2065} a + \frac{667}{4130}$, $\frac{1}{4130} a^{27} + \frac{39}{826} a^{25} + \frac{9}{295} a^{24} + \frac{176}{2065} a^{23} - \frac{57}{2065} a^{22} - \frac{257}{2065} a^{21} - \frac{85}{413} a^{20} - \frac{135}{413} a^{19} - \frac{151}{2065} a^{18} - \frac{951}{2065} a^{17} + \frac{16}{35} a^{16} - \frac{307}{2065} a^{15} + \frac{828}{2065} a^{14} - \frac{338}{2065} a^{13} - \frac{772}{2065} a^{12} + \frac{199}{2065} a^{11} - \frac{58}{413} a^{10} + \frac{257}{2065} a^{9} + \frac{142}{413} a^{8} - \frac{151}{413} a^{7} - \frac{17}{2065} a^{6} - \frac{242}{2065} a^{5} - \frac{528}{2065} a^{4} + \frac{1439}{4130} a^{3} - \frac{396}{2065} a^{2} + \frac{1289}{4130} a + \frac{104}{413}$, $\frac{1}{20650} a^{28} - \frac{1}{20650} a^{27} + \frac{1}{10325} a^{26} + \frac{372}{10325} a^{25} - \frac{563}{20650} a^{24} + \frac{149}{2065} a^{23} + \frac{87}{10325} a^{22} - \frac{71}{175} a^{21} + \frac{154}{1475} a^{20} + \frac{674}{10325} a^{19} + \frac{719}{2065} a^{18} - \frac{131}{10325} a^{17} - \frac{4651}{10325} a^{16} - \frac{886}{2065} a^{15} - \frac{619}{2065} a^{14} + \frac{299}{1475} a^{13} - \frac{44}{413} a^{12} - \frac{37}{1475} a^{11} + \frac{2882}{10325} a^{10} - \frac{1657}{10325} a^{9} + \frac{561}{2065} a^{8} - \frac{2263}{10325} a^{7} - \frac{2151}{10325} a^{6} + \frac{3473}{10325} a^{5} + \frac{539}{2950} a^{4} + \frac{179}{20650} a^{3} - \frac{4636}{10325} a^{2} - \frac{4096}{10325} a - \frac{8349}{20650}$, $\frac{1}{12038950} a^{29} - \frac{7}{1719850} a^{28} - \frac{6}{1203895} a^{27} + \frac{19}{547225} a^{26} - \frac{37146}{1203895} a^{25} - \frac{596681}{12038950} a^{24} - \frac{194448}{6019475} a^{23} + \frac{33601}{1203895} a^{22} + \frac{73373}{171985} a^{21} - \frac{24127}{1203895} a^{20} - \frac{2538542}{6019475} a^{19} - \frac{3137}{14575} a^{18} + \frac{314836}{859925} a^{17} + \frac{1413663}{6019475} a^{16} + \frac{464197}{1203895} a^{15} - \frac{2255697}{6019475} a^{14} - \frac{2891439}{6019475} a^{13} - \frac{408262}{859925} a^{12} - \frac{2317176}{6019475} a^{11} - \frac{136459}{859925} a^{10} - \frac{2441059}{6019475} a^{9} + \frac{1923252}{6019475} a^{8} + \frac{2611883}{6019475} a^{7} + \frac{212991}{6019475} a^{6} - \frac{1379}{6490} a^{5} - \frac{274469}{2407790} a^{4} + \frac{37916}{113575} a^{3} - \frac{2444573}{6019475} a^{2} + \frac{331036}{6019475} a - \frac{860323}{12038950}$, $\frac{1}{344218798616325118397450} a^{30} - \frac{3917290964605222}{172109399308162559198725} a^{29} + \frac{729763623864407779}{49174114088046445485350} a^{28} - \frac{1550059117592932313}{68843759723265023679490} a^{27} + \frac{30341025692676898891}{344218798616325118397450} a^{26} + \frac{139965456512815986583}{5551916106714921264475} a^{25} - \frac{338697368081426936298}{6884375972326502367949} a^{24} + \frac{60598835233770501571}{983482281760928909707} a^{23} - \frac{3126960349749035085624}{172109399308162559198725} a^{22} + \frac{377104012061634710419}{793130872387845894925} a^{21} + \frac{60052599823788859367247}{172109399308162559198725} a^{20} - \frac{8407118673259446076422}{24587057044023222742675} a^{19} - \frac{1076915750828879305761}{3247347156757784135825} a^{18} - \frac{12192266285197850396579}{34421879861632511839745} a^{17} - \frac{62275806870520938645028}{172109399308162559198725} a^{16} - \frac{15651922279846769234877}{172109399308162559198725} a^{15} - \frac{64651745992095315122009}{172109399308162559198725} a^{14} + \frac{12161209391466930238058}{34421879861632511839745} a^{13} + \frac{79714821338251643523059}{172109399308162559198725} a^{12} + \frac{621215311322823205249}{3129261805602955621795} a^{11} - \frac{18997300653109690056603}{172109399308162559198725} a^{10} - \frac{21329048131812293525969}{172109399308162559198725} a^{9} - \frac{23882958964807325970012}{172109399308162559198725} a^{8} + \frac{179602243801542285659}{847829553242180094575} a^{7} - \frac{276467327157251806351}{31292618056029556217950} a^{6} + \frac{77042554782340344635824}{172109399308162559198725} a^{5} + \frac{393863406935379865887}{1601017667982907527430} a^{4} + \frac{1919611530449826835929}{5834216925700425735550} a^{3} - \frac{5904207849132205186027}{49174114088046445485350} a^{2} - \frac{83715555791169020873587}{172109399308162559198725} a + \frac{11635411890120489447654}{172109399308162559198725}$, $\frac{1}{193814307807186708025455162505127797600180444550} a^{31} + \frac{1942394754339472322678}{13843879129084764858961083036080556971441460325} a^{30} - \frac{391727328050509611591493439501360165579}{96907153903593354012727581252563898800090222275} a^{29} - \frac{2434845587695232457682345610521692163212679}{193814307807186708025455162505127797600180444550} a^{28} - \frac{23222758584822109192183421497290391950595999}{193814307807186708025455162505127797600180444550} a^{27} + \frac{22603525734539370797078795788475384221497571}{193814307807186708025455162505127797600180444550} a^{26} + \frac{2534071115638185520008251827053740679380644373}{96907153903593354012727581252563898800090222275} a^{25} - \frac{4742642115961029882585433104143742898215755759}{96907153903593354012727581252563898800090222275} a^{24} - \frac{307177951510283876302654296599696626414630329}{13843879129084764858961083036080556971441460325} a^{23} + \frac{4045228867683344078780904038126228537126978999}{96907153903593354012727581252563898800090222275} a^{22} + \frac{3700608047832338196193138479498714528052825281}{19381430780718670802545516250512779760018044455} a^{21} - \frac{10514801169236165993007941182948883274888177}{58342657377238623728312812313403912582835775} a^{20} - \frac{17614645889004898001865995903032320677392626263}{96907153903593354012727581252563898800090222275} a^{19} - \frac{30758945669578251902285894424847224585633919766}{96907153903593354012727581252563898800090222275} a^{18} - \frac{38266452847350594567373721023754165826986428582}{96907153903593354012727581252563898800090222275} a^{17} + \frac{6556901159214751767352248966699223338049567282}{13843879129084764858961083036080556971441460325} a^{16} + \frac{47392264785010650115361136724337269091924279634}{96907153903593354012727581252563898800090222275} a^{15} + \frac{9098861708152721892979410790559356858464720027}{19381430780718670802545516250512779760018044455} a^{14} + \frac{32034890567293847935920331322778651505520564627}{96907153903593354012727581252563898800090222275} a^{13} + \frac{15992294018711036009223420621641300569959349612}{96907153903593354012727581252563898800090222275} a^{12} + \frac{16855805072709355364196840939436506310661752559}{96907153903593354012727581252563898800090222275} a^{11} - \frac{28386757704754661537642585267774374919885988046}{96907153903593354012727581252563898800090222275} a^{10} - \frac{1121538548784472331665368301072140195324496}{12108853417917450207763036517876283743607425} a^{9} - \frac{11744460888097923093683489830499658198139291929}{96907153903593354012727581252563898800090222275} a^{8} + \frac{7296646517047610666223564992496688238156235251}{17619482527926064365950469318647981600016404050} a^{7} + \frac{35088509966468154223984328057747649929754919719}{96907153903593354012727581252563898800090222275} a^{6} - \frac{23586028286287042050365513978791756317285042501}{96907153903593354012727581252563898800090222275} a^{5} + \frac{25290778378146779618322193109348439929730711971}{193814307807186708025455162505127797600180444550} a^{4} - \frac{14534482224336662733712969820872705121486271347}{38762861561437341605091032501025559520036088910} a^{3} + \frac{82859454875692580391327294230521738183177108999}{193814307807186708025455162505127797600180444550} a^{2} + \frac{12416978368082645458855182577719475653279567}{365687373221106996274443702839863769056944235} a - \frac{41715382046717370505174446345877109842027550093}{96907153903593354012727581252563898800090222275}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{5813233793250990397555049726219043664958222326}{13843879129084764858961083036080556971441460325} a^{31} + \frac{313058792723162497869348165189543391911301676767}{96907153903593354012727581252563898800090222275} a^{30} - \frac{37054210403733855916135027276944665787622580007}{4507309483888062977336166569886692967446056850} a^{29} + \frac{1363141573795627555646626794476227343974898529287}{96907153903593354012727581252563898800090222275} a^{28} - \frac{13540841942993408602557041640135108047184002083461}{193814307807186708025455162505127797600180444550} a^{27} + \frac{53595764744693003614729576396491644755877091488007}{193814307807186708025455162505127797600180444550} a^{26} - \frac{42768372550827290812157084203186147561824293553582}{96907153903593354012727581252563898800090222275} a^{25} - \frac{49247822236294657827509483759437767783244213564157}{96907153903593354012727581252563898800090222275} a^{24} + \frac{368843992106941123511534181055872872832359940628812}{96907153903593354012727581252563898800090222275} a^{23} - \frac{166566622846549580070499961434818690426282847612934}{19381430780718670802545516250512779760018044455} a^{22} + \frac{1012396760840747374585713091184219639881990414719708}{96907153903593354012727581252563898800090222275} a^{21} - \frac{43133024291071871246068493738469773567943068001007}{8809741263963032182975234659323990800008202025} a^{20} - \frac{395447275940503425117720491653817533475422107485363}{96907153903593354012727581252563898800090222275} a^{19} + \frac{2099354223268852302892452740124769298329985158495238}{96907153903593354012727581252563898800090222275} a^{18} - \frac{11929910309621938335029349421532955824181382382849829}{96907153903593354012727581252563898800090222275} a^{17} + \frac{5798948450096472874577707663504427609376473559138033}{13843879129084764858961083036080556971441460325} a^{16} - \frac{15468404626972161061208414672302436173494312698823118}{19381430780718670802545516250512779760018044455} a^{15} + \frac{93590828169507739053280346457161095951761270523335492}{96907153903593354012727581252563898800090222275} a^{14} - \frac{82446463767698257007722316487106048698558356976218752}{96907153903593354012727581252563898800090222275} a^{13} + \frac{10559940074280426442942612294571259929802601080873907}{19381430780718670802545516250512779760018044455} a^{12} - \frac{24670440691204120787232873477468689131102953593769149}{96907153903593354012727581252563898800090222275} a^{11} + \frac{9255120962340904556387960040955490913520952281597767}{96907153903593354012727581252563898800090222275} a^{10} - \frac{988904651754516272201153935897453050299687714441863}{19381430780718670802545516250512779760018044455} a^{9} + \frac{2098497702604096510018743795660950122343799718049527}{96907153903593354012727581252563898800090222275} a^{8} - \frac{43218920416338525607714988932101176226223579612404}{8809741263963032182975234659323990800008202025} a^{7} + \frac{3048847681603319526363160213496805227585448841102}{2253654741944031488668083284943346483723028425} a^{6} - \frac{44053440887251912559993267187161402082137346322999}{193814307807186708025455162505127797600180444550} a^{5} - \frac{217049248544852947607796794169263992303004737449}{96907153903593354012727581252563898800090222275} a^{4} - \frac{728295092022594877282397190043773960422084001043}{38762861561437341605091032501025559520036088910} a^{3} + \frac{1268108014959344308335996092678611078172330091927}{193814307807186708025455162505127797600180444550} a^{2} - \frac{222631420193665279006894984012361680594655562}{1828436866105534981372218514199318845284721175} a - \frac{246257323728857193349491022855375283225169988}{1167556071127630771237681701838119262651689425} \) (order $24$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 184122377962.3511 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 184122377962.3511 \cdot 24}{24\sqrt{6649076259173893054484111016591360000000000000000}}\approx 0.421311983406428$ (assuming GRH)

Galois group

$C_2^3\times D_4$ (as 32T273):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 64
The 40 conjugacy class representatives for $C_2^3\times D_4$
Character table for $C_2^3\times D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{30}) \), 4.4.17600.1, 4.4.4400.1, 4.4.158400.1, 4.4.39600.1, 4.0.4400.1, 4.0.17600.1, 4.0.39600.1, 4.0.158400.1, \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(i, \sqrt{30})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{6})\), 8.0.3317760000.2, 8.0.3317760000.1, 8.8.3317760000.1, 8.0.40960000.1, 8.0.3317760000.5, 8.0.3317760000.6, 8.0.3317760000.7, 8.0.12960000.1, \(\Q(\zeta_{24})\), 8.0.3317760000.8, 8.0.207360000.1, 8.0.3317760000.3, 8.0.207360000.2, 8.0.3317760000.9, 8.0.3317760000.4, 8.8.401448960000.5, 8.8.25090560000.1, 8.0.25090560000.24, 8.0.401448960000.60, 8.8.4956160000.1, 8.8.401448960000.1, 8.0.4956160000.5, 8.0.401448960000.14, 8.0.4956160000.2, 8.0.4956160000.6, 8.0.401448960000.11, 8.0.401448960000.6, 8.0.25090560000.13, 8.0.25090560000.10, 8.0.1568160000.8, 8.0.1568160000.5, 8.0.4956160000.13, 8.0.309760000.3, 8.0.401448960000.31, 8.0.25090560000.4, 8.0.401448960000.55, 8.0.401448960000.19, 8.0.401448960000.47, 8.0.401448960000.43, 8.8.401448960000.2, 8.8.401448960000.3, 8.0.401448960000.22, 8.0.401448960000.53, 16.0.11007531417600000000.1, 16.16.2578580279761305600000000.1, 16.0.2578580279761305600000000.1, 16.0.393016351129600000000.2, Deg 16, Deg 16, Deg 16, 16.0.161161267485081600000000.3, 16.0.629536201113600000000.4, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 32 siblings: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$