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Properties

Label 32.0.56950036795...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 5^{24}\cdot 17^{24}$
Root discriminant $79.18$
Ramified primes $2, 5, 17$
Class number $23400$ (GRH)
Class group $[2, 30, 390]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4879681, 5606442, 17135213, 23875530, 47580595, 21666686, 66179698, 16469642, 68825998, 4506654, 73270895, -4658378, 67449232, -20336812, 38788398, -12077360, 18328695, -3815316, 6110462, -1250168, 1817292, -421674, 419597, -68274, 75954, -2162, 5494, -250, 423, -58, 23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + 23*x^30 - 58*x^29 + 423*x^28 - 250*x^27 + 5494*x^26 - 2162*x^25 + 75954*x^24 - 68274*x^23 + 419597*x^22 - 421674*x^21 + 1817292*x^20 - 1250168*x^19 + 6110462*x^18 - 3815316*x^17 + 18328695*x^16 - 12077360*x^15 + 38788398*x^14 - 20336812*x^13 + 67449232*x^12 - 4658378*x^11 + 73270895*x^10 + 4506654*x^9 + 68825998*x^8 + 16469642*x^7 + 66179698*x^6 + 21666686*x^5 + 47580595*x^4 + 23875530*x^3 + 17135213*x^2 + 5606442*x + 4879681)
 
gp: K = bnfinit(x^32 - 2*x^31 + 23*x^30 - 58*x^29 + 423*x^28 - 250*x^27 + 5494*x^26 - 2162*x^25 + 75954*x^24 - 68274*x^23 + 419597*x^22 - 421674*x^21 + 1817292*x^20 - 1250168*x^19 + 6110462*x^18 - 3815316*x^17 + 18328695*x^16 - 12077360*x^15 + 38788398*x^14 - 20336812*x^13 + 67449232*x^12 - 4658378*x^11 + 73270895*x^10 + 4506654*x^9 + 68825998*x^8 + 16469642*x^7 + 66179698*x^6 + 21666686*x^5 + 47580595*x^4 + 23875530*x^3 + 17135213*x^2 + 5606442*x + 4879681, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} + 23 x^{30} - 58 x^{29} + 423 x^{28} - 250 x^{27} + 5494 x^{26} - 2162 x^{25} + 75954 x^{24} - 68274 x^{23} + 419597 x^{22} - 421674 x^{21} + 1817292 x^{20} - 1250168 x^{19} + 6110462 x^{18} - 3815316 x^{17} + 18328695 x^{16} - 12077360 x^{15} + 38788398 x^{14} - 20336812 x^{13} + 67449232 x^{12} - 4658378 x^{11} + 73270895 x^{10} + 4506654 x^{9} + 68825998 x^{8} + 16469642 x^{7} + 66179698 x^{6} + 21666686 x^{5} + 47580595 x^{4} + 23875530 x^{3} + 17135213 x^{2} + 5606442 x + 4879681 \)

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:  \(5695003679558247880375471569828315136000000000000000000000000=2^{48}\cdot 5^{24}\cdot 17^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.18$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(137,·)$, $\chi_{680}(13,·)$, $\chi_{680}(557,·)$, $\chi_{680}(273,·)$, $\chi_{680}(149,·)$, $\chi_{680}(89,·)$, $\chi_{680}(409,·)$, $\chi_{680}(157,·)$, $\chi_{680}(33,·)$, $\chi_{680}(421,·)$, $\chi_{680}(169,·)$, $\chi_{680}(429,·)$, $\chi_{680}(441,·)$, $\chi_{680}(577,·)$, $\chi_{680}(69,·)$, $\chi_{680}(81,·)$, $\chi_{680}(341,·)$, $\chi_{680}(217,·)$, $\chi_{680}(101,·)$, $\chi_{680}(477,·)$, $\chi_{680}(293,·)$, $\chi_{680}(353,·)$, $\chi_{680}(613,·)$, $\chi_{680}(361,·)$, $\chi_{680}(237,·)$, $\chi_{680}(497,·)$, $\chi_{680}(373,·)$, $\chi_{680}(489,·)$, $\chi_{680}(633,·)$, $\chi_{680}(509,·)$, $\chi_{680}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4} a^{24} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{4} a^{18} + \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{25} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{4} a^{19} + \frac{1}{4} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a$, $\frac{1}{12} a^{26} - \frac{1}{12} a^{25} + \frac{1}{12} a^{24} - \frac{1}{3} a^{23} - \frac{1}{3} a^{22} - \frac{1}{6} a^{21} + \frac{1}{4} a^{20} + \frac{5}{12} a^{19} - \frac{1}{4} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{12} a^{14} + \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{6} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{8} + \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{12}$, $\frac{1}{12} a^{27} + \frac{1}{3} a^{23} - \frac{5}{12} a^{21} - \frac{1}{3} a^{20} + \frac{1}{6} a^{19} - \frac{1}{6} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} - \frac{1}{4} a^{15} + \frac{1}{3} a^{14} - \frac{1}{6} a^{12} + \frac{1}{3} a^{11} + \frac{1}{4} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{5}{12} a^{3} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{15924} a^{28} - \frac{601}{15924} a^{27} - \frac{53}{7962} a^{26} - \frac{473}{3981} a^{25} + \frac{221}{5308} a^{24} + \frac{150}{1327} a^{23} + \frac{7661}{15924} a^{22} - \frac{863}{5308} a^{21} + \frac{42}{1327} a^{20} + \frac{179}{1327} a^{19} - \frac{2267}{5308} a^{18} + \frac{527}{1327} a^{17} + \frac{2833}{15924} a^{16} + \frac{5555}{15924} a^{15} + \frac{2669}{7962} a^{14} - \frac{139}{7962} a^{13} + \frac{2499}{5308} a^{12} - \frac{1229}{3981} a^{11} - \frac{1265}{5308} a^{10} - \frac{3623}{15924} a^{9} - \frac{629}{7962} a^{8} - \frac{1207}{3981} a^{7} + \frac{2437}{15924} a^{6} + \frac{213}{1327} a^{5} - \frac{3709}{15924} a^{4} + \frac{4877}{15924} a^{3} + \frac{218}{3981} a^{2} - \frac{360}{1327} a - \frac{571}{15924}$, $\frac{1}{3869090832521490133544296838467287517574217779532} a^{29} - \frac{92043539031028047263111113082922468857235809}{3869090832521490133544296838467287517574217779532} a^{28} + \frac{38640379059093456636041442513795751380941629169}{3869090832521490133544296838467287517574217779532} a^{27} - \frac{79029585359776446170081996176167029670170738683}{3869090832521490133544296838467287517574217779532} a^{26} + \frac{5212441888098017381675057548150697130331375969}{82321081543010428373282911456750798246259952756} a^{25} - \frac{2883152875653478824799481227460899493235922801}{67878786535464739184987663832759430132881013676} a^{24} + \frac{1109116474233534907404986269997403364965913347271}{3869090832521490133544296838467287517574217779532} a^{23} + \frac{4428552355531619227748051140742153552788495931}{82321081543010428373282911456750798246259952756} a^{22} - \frac{323600749470063253149307224275669370336647783917}{3869090832521490133544296838467287517574217779532} a^{21} - \frac{825376021596774532656857203129359238781757509393}{3869090832521490133544296838467287517574217779532} a^{20} - \frac{1852441927154373754647350699047546452470964779603}{3869090832521490133544296838467287517574217779532} a^{19} - \frac{11967470536789714953925711505235870653966388091}{3869090832521490133544296838467287517574217779532} a^{18} + \frac{355425207255607608977519173516841805195013880231}{1289696944173830044514765612822429172524739259844} a^{17} + \frac{471645495872928480133592293795300093033363967315}{3869090832521490133544296838467287517574217779532} a^{16} + \frac{180969627717629152732003252594504879218857376113}{3869090832521490133544296838467287517574217779532} a^{15} - \frac{1909476460600001023917735523617824360508391228921}{3869090832521490133544296838467287517574217779532} a^{14} + \frac{1190399764331520826100696301317240245504196639381}{3869090832521490133544296838467287517574217779532} a^{13} + \frac{132866075828363708919397959449781460336170810035}{1289696944173830044514765612822429172524739259844} a^{12} + \frac{1442222685272160485657279632498093222525189838545}{3869090832521490133544296838467287517574217779532} a^{11} + \frac{1188542501466511052989125065099047742275065502363}{3869090832521490133544296838467287517574217779532} a^{10} + \frac{23106770298384115019406352688368998770555345947}{3869090832521490133544296838467287517574217779532} a^{9} + \frac{1179929787056267029675149747446909698409299023491}{3869090832521490133544296838467287517574217779532} a^{8} + \frac{580557047714259391713487157744935194854845402445}{3869090832521490133544296838467287517574217779532} a^{7} + \frac{133969176284250145518118479071339485956646293317}{3869090832521490133544296838467287517574217779532} a^{6} - \frac{848824556894322277390742114006457237918383425207}{3869090832521490133544296838467287517574217779532} a^{5} - \frac{1475196411709799153613506937316511682428663574421}{3869090832521490133544296838467287517574217779532} a^{4} - \frac{390204643696289537149344435394918385584850914835}{3869090832521490133544296838467287517574217779532} a^{3} + \frac{1738876099680233407256516163937399238086798511221}{3869090832521490133544296838467287517574217779532} a^{2} + \frac{419847097304929802612085306139750289856804894817}{1289696944173830044514765612822429172524739259844} a + \frac{9892258076365419843126106292467569429141274729}{82321081543010428373282911456750798246259952756}$, $\frac{1}{181847269128510036276581951407962513325988235638004} a^{30} - \frac{1}{90923634564255018138290975703981256662994117819002} a^{29} + \frac{3036530408135948813786142048198528084000987205}{181847269128510036276581951407962513325988235638004} a^{28} + \frac{259125359009776069315717368260704344796252367167}{30307878188085006046096991901327085554331372606334} a^{27} - \frac{21644044186969714924302636313565107120276664179}{1289696944173830044514765612822429172524739259844} a^{26} - \frac{7633896465678345086955316389040471579702905801745}{181847269128510036276581951407962513325988235638004} a^{25} + \frac{12624044282197789720528956413613956559013606847437}{181847269128510036276581951407962513325988235638004} a^{24} + \frac{97413834717645507501697726314140803950117985015}{1934545416260745066772148419233643758787108889766} a^{23} - \frac{20595452271044646041125876931354927542871043500887}{60615756376170012092193983802654171108662745212668} a^{22} - \frac{4264470396545061879411109338762717840288484767860}{45461817282127509069145487851990628331497058909501} a^{21} + \frac{83071813373246684847181811299915459046985767725177}{181847269128510036276581951407962513325988235638004} a^{20} - \frac{31120030114387772594878100991143214436536285485929}{181847269128510036276581951407962513325988235638004} a^{19} + \frac{2587397790131417927885832290130845017023489249691}{60615756376170012092193983802654171108662745212668} a^{18} + \frac{8849358450498548510425598025612164480938759161595}{90923634564255018138290975703981256662994117819002} a^{17} + \frac{56638870619591747990477686249349151198964744677157}{181847269128510036276581951407962513325988235638004} a^{16} + \frac{1828254843774712958989116939269530888700864715092}{15153939094042503023048495950663542777165686303167} a^{15} - \frac{88055693532063425478265229625318666583734243871619}{181847269128510036276581951407962513325988235638004} a^{14} + \frac{12787909703850239509980147917386203187783336450841}{181847269128510036276581951407962513325988235638004} a^{13} - \frac{4111993231465432420624414295771870146523970399077}{9570908901500528225083260600419079648736222928316} a^{12} + \frac{4150142541530093952483100454833588648963098984477}{15153939094042503023048495950663542777165686303167} a^{11} - \frac{73406217006791926106480910191663625302426463311741}{181847269128510036276581951407962513325988235638004} a^{10} - \frac{3380197369325334201471333062729485178583172289365}{90923634564255018138290975703981256662994117819002} a^{9} - \frac{16494359108944952448363630694631132966925251566447}{60615756376170012092193983802654171108662745212668} a^{8} - \frac{19775824185600293542465036748858493536246506292197}{60615756376170012092193983802654171108662745212668} a^{7} - \frac{10133814935902454546136707730437950957535149124081}{181847269128510036276581951407962513325988235638004} a^{6} + \frac{19329564347130647878594628101287258588369194879593}{90923634564255018138290975703981256662994117819002} a^{5} + \frac{11223245907117255928121390190425433024767859944765}{181847269128510036276581951407962513325988235638004} a^{4} + \frac{17784777170474088579666355021224428350171383482255}{45461817282127509069145487851990628331497058909501} a^{3} - \frac{23584451438580087757743859302000015875649932966945}{181847269128510036276581951407962513325988235638004} a^{2} - \frac{99573595430850908764384699551307506910843064927}{203636359606394217554962991498278290398643041028} a + \frac{9236314546735603728338043215998004254217802577}{20580270385752607093320727864187699561564988189}$, $\frac{1}{8546821649039971704999351716174238126321447074986188} a^{31} - \frac{1}{4273410824519985852499675858087119063160723537493094} a^{30} + \frac{23}{8546821649039971704999351716174238126321447074986188} a^{29} - \frac{11114137497670285103039010138301463307344195023}{8546821649039971704999351716174238126321447074986188} a^{28} + \frac{7427458115531816182135474513465506172904092150535}{181847269128510036276581951407962513325988235638004} a^{27} - \frac{241668999013173668043850858590927685233934274475209}{8546821649039971704999351716174238126321447074986188} a^{26} - \frac{239605794666218101198023654350031690323142642810433}{2848940549679990568333117238724746042107149024995396} a^{25} + \frac{9973388773528968899783056844147814959929439396333}{90923634564255018138290975703981256662994117819002} a^{24} - \frac{435479403246520293172899355441828284860651850333337}{2848940549679990568333117238724746042107149024995396} a^{23} - \frac{7565370360684929744525594018963519421149949450989}{2848940549679990568333117238724746042107149024995396} a^{22} - \frac{3214906277444482222368840551458863255652667972607407}{8546821649039971704999351716174238126321447074986188} a^{21} + \frac{1021145687575683084138000368515471330376101841141943}{2848940549679990568333117238724746042107149024995396} a^{20} - \frac{23814134881453831632849817424077775462777006776443}{2848940549679990568333117238724746042107149024995396} a^{19} + \frac{1132844056208410408022885133911094751123910713644509}{4273410824519985852499675858087119063160723537493094} a^{18} - \frac{2822251647351643722238104642015692228463844591984321}{8546821649039971704999351716174238126321447074986188} a^{17} + \frac{51883953563731985208203718993052597107685260294249}{2848940549679990568333117238724746042107149024995396} a^{16} - \frac{795521952130843229717809641472292659081915659908447}{8546821649039971704999351716174238126321447074986188} a^{15} + \frac{909145386180041254997074388261998896435145720543239}{2848940549679990568333117238724746042107149024995396} a^{14} - \frac{115355022424146431327832686475748497954354502395101}{8546821649039971704999351716174238126321447074986188} a^{13} - \frac{3336052539307394975246400135035815056569972790876}{712235137419997642083279309681186510526787256248849} a^{12} + \frac{3966955003610174580972002863721213901976149330833521}{8546821649039971704999351716174238126321447074986188} a^{11} - \frac{2114769478801614688687904433154438419357029236484161}{8546821649039971704999351716174238126321447074986188} a^{10} + \frac{880876767741169401636311816678851326553013935505051}{8546821649039971704999351716174238126321447074986188} a^{9} + \frac{2224776442389272332706047731082914946476559388463373}{8546821649039971704999351716174238126321447074986188} a^{8} + \frac{22126058946432241814082951355350238929539158742213}{2848940549679990568333117238724746042107149024995396} a^{7} + \frac{2103377154732772629254712005917409969893120601497241}{4273410824519985852499675858087119063160723537493094} a^{6} - \frac{1025032925955114202127428172433688536799095872925097}{8546821649039971704999351716174238126321447074986188} a^{5} - \frac{2144672042805190355110687498549496188475602236459013}{8546821649039971704999351716174238126321447074986188} a^{4} + \frac{131920904595037935009334312984520146153773762118737}{2848940549679990568333117238724746042107149024995396} a^{3} + \frac{55022219544063040486556149479900343663094405518297}{181847269128510036276581951407962513325988235638004} a^{2} + \frac{322585279036757768224635601672323241839228083987}{1934545416260745066772148419233643758787108889766} a + \frac{652694732329466443406553943324344740526629093}{6860090128584202364440242621395899853854996063}$

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

Class group and class number

$C_{2}\times C_{30}\times C_{390}$, which has order $23400$ (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{2412836301769834464693654844934538307784}{34259093656463835018195544726443578245287911763} a^{31} + \frac{3534789586758950679357578635029562212886}{34259093656463835018195544726443578245287911763} a^{30} - \frac{17281625508224372513426609761189567904548}{11419697885487945006065181575481192748429303921} a^{29} + \frac{36072593397114889522517699620614147114410}{11419697885487945006065181575481192748429303921} a^{28} - \frac{921289752283894751998262560355301735902444}{34259093656463835018195544726443578245287911763} a^{27} - \frac{2829719752257010214986945976343721795388}{34259093656463835018195544726443578245287911763} a^{26} - \frac{12484717849282191734125413962272088608813136}{34259093656463835018195544726443578245287911763} a^{25} - \frac{702928987895863622669988221072203423473172}{11419697885487945006065181575481192748429303921} a^{24} - \frac{174579407953565890617839683028889978127156148}{34259093656463835018195544726443578245287911763} a^{23} + \frac{21615096801268401955928116567199540916636682}{11419697885487945006065181575481192748429303921} a^{22} - \frac{842785263906959815967825248497988585014955792}{34259093656463835018195544726443578245287911763} a^{21} + \frac{136523536334147719689826848026605478381714512}{11419697885487945006065181575481192748429303921} a^{20} - \frac{3391455008910940310958993838719536805231154732}{34259093656463835018195544726443578245287911763} a^{19} + \frac{245721626448025289854597139911770894362367364}{34259093656463835018195544726443578245287911763} a^{18} - \frac{11178423124612665168993584791011096706331496380}{34259093656463835018195544726443578245287911763} a^{17} + \frac{76096713863333294358339162871306666987268086}{34259093656463835018195544726443578245287911763} a^{16} - \frac{32689579695191580743386326760523133758665307188}{34259093656463835018195544726443578245287911763} a^{15} + \frac{1706314406619317550098934834292309218032862820}{34259093656463835018195544726443578245287911763} a^{14} - \frac{19389480336617826890768254303629828056904949196}{11419697885487945006065181575481192748429303921} a^{13} - \frac{12951261416726362187739358340054515987520350019}{34259093656463835018195544726443578245287911763} a^{12} - \frac{94407087177905495696043093619223946807172969676}{34259093656463835018195544726443578245287911763} a^{11} - \frac{32223676449386807110128753783499858318566616854}{11419697885487945006065181575481192748429303921} a^{10} - \frac{32248470534152895228240032147394112274867488944}{11419697885487945006065181575481192748429303921} a^{9} - \frac{108894387817601865491086232993855989919113916204}{34259093656463835018195544726443578245287911763} a^{8} - \frac{1579987049879936871264450910007268451758965632}{601036730815155000319220082920062776233121259} a^{7} - \frac{122056341199539259338596768742192757199635881292}{34259093656463835018195544726443578245287911763} a^{6} - \frac{104877102608122180757462434759535787287145891028}{34259093656463835018195544726443578245287911763} a^{5} - \frac{118048860539876063735112953313839853787246534034}{34259093656463835018195544726443578245287911763} a^{4} - \frac{1477728100988733003920268796328003803270796096}{728916886307741170599905206945608047772083229} a^{3} - \frac{42365466226384496591038812531731399632381086188}{11419697885487945006065181575481192748429303921} a^{2} - \frac{2895422874965665085531888526092068961078332}{5169623307147100500708547566990127998383569} a - \frac{1347817673779536235662564805657761170102778}{5169623307147100500708547566990127998383569} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 29176878483384.797 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{170}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{85})\), 4.0.614125.2, 4.0.39304000.2, \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{10}, \sqrt{34})\), 4.0.614125.1, 4.0.39304000.1, \(\Q(\sqrt{10}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{34})\), 4.4.122825.1, 4.4.7860800.1, 4.4.4913.1, 4.4.314432.1, \(\Q(\zeta_{5})\), 4.0.8000.2, 4.0.36125.1, 4.0.2312000.1, 8.0.1544804416000000.14, 8.8.213813760000.1, 8.0.1544804416000000.12, 8.0.377149515625.1, 8.0.1544804416000000.16, 8.0.1544804416000000.24, 8.0.1544804416000000.18, 8.8.61792176640000.3, 8.8.98867482624.1, 8.0.64000000.2, 8.0.5345344000000.5, 8.8.15085980625.1, 8.8.61792176640000.4, 8.0.1305015625.1, 8.0.5345344000000.2, 8.8.61792176640000.2, 8.8.61792176640000.1, 8.0.5345344000000.7, 8.0.5345344000000.3, 16.0.2386420683693101056000000000000.5, 16.16.3818273093908961689600000000.1, 16.0.28572702478336000000000000.1, 16.0.142241757136172119140625.1, 16.0.2386420683693101056000000000000.2, 16.0.2386420683693101056000000000000.4, 16.0.2386420683693101056000000000000.7

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
17Data not computed