↑

Properties

Label 32.0.14376676755...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $5^{16}\cdot 7^{16}\cdot 17^{28}$
Root discriminant $70.58$
Ramified primes $5, 7, 17$
Class number $546120$ (GRH)
Class group $[3, 182040]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1842570739, -4158477603, 8585381539, -12277480374, 15839257061, -17012899062, 16632009985, -14331413831, 11338690706, -8101027726, 5352278301, -3240171528, 1830768541, -961434547, 478360878, -225677648, 102440247, -44145382, 18026035, -6843359, 2403348, -780056, 275925, -129988, 63975, -21675, 3226, 296, 49, -188, 78, -14, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 14*x^31 + 78*x^30 - 188*x^29 + 49*x^28 + 296*x^27 + 3226*x^26 - 21675*x^25 + 63975*x^24 - 129988*x^23 + 275925*x^22 - 780056*x^21 + 2403348*x^20 - 6843359*x^19 + 18026035*x^18 - 44145382*x^17 + 102440247*x^16 - 225677648*x^15 + 478360878*x^14 - 961434547*x^13 + 1830768541*x^12 - 3240171528*x^11 + 5352278301*x^10 - 8101027726*x^9 + 11338690706*x^8 - 14331413831*x^7 + 16632009985*x^6 - 17012899062*x^5 + 15839257061*x^4 - 12277480374*x^3 + 8585381539*x^2 - 4158477603*x + 1842570739)
 
gp: K = bnfinit(x^32 - 14*x^31 + 78*x^30 - 188*x^29 + 49*x^28 + 296*x^27 + 3226*x^26 - 21675*x^25 + 63975*x^24 - 129988*x^23 + 275925*x^22 - 780056*x^21 + 2403348*x^20 - 6843359*x^19 + 18026035*x^18 - 44145382*x^17 + 102440247*x^16 - 225677648*x^15 + 478360878*x^14 - 961434547*x^13 + 1830768541*x^12 - 3240171528*x^11 + 5352278301*x^10 - 8101027726*x^9 + 11338690706*x^8 - 14331413831*x^7 + 16632009985*x^6 - 17012899062*x^5 + 15839257061*x^4 - 12277480374*x^3 + 8585381539*x^2 - 4158477603*x + 1842570739, 1)
 

Normalized defining polynomial

\( x^{32} - 14 x^{31} + 78 x^{30} - 188 x^{29} + 49 x^{28} + 296 x^{27} + 3226 x^{26} - 21675 x^{25} + 63975 x^{24} - 129988 x^{23} + 275925 x^{22} - 780056 x^{21} + 2403348 x^{20} - 6843359 x^{19} + 18026035 x^{18} - 44145382 x^{17} + 102440247 x^{16} - 225677648 x^{15} + 478360878 x^{14} - 961434547 x^{13} + 1830768541 x^{12} - 3240171528 x^{11} + 5352278301 x^{10} - 8101027726 x^{9} + 11338690706 x^{8} - 14331413831 x^{7} + 16632009985 x^{6} - 17012899062 x^{5} + 15839257061 x^{4} - 12277480374 x^{3} + 8585381539 x^{2} - 4158477603 x + 1842570739 \)

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:  \(143766767556164787106199922232179916833038973486480712890625=5^{16}\cdot 7^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.58$
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:  $5, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(595=5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{595}(1,·)$, $\chi_{595}(134,·)$, $\chi_{595}(519,·)$, $\chi_{595}(526,·)$, $\chi_{595}(274,·)$, $\chi_{595}(531,·)$, $\chi_{595}(281,·)$, $\chi_{595}(111,·)$, $\chi_{595}(36,·)$, $\chi_{595}(421,·)$, $\chi_{595}(169,·)$, $\chi_{595}(426,·)$, $\chi_{595}(174,·)$, $\chi_{595}(559,·)$, $\chi_{595}(314,·)$, $\chi_{595}(64,·)$, $\chi_{595}(321,·)$, $\chi_{595}(69,·)$, $\chi_{595}(76,·)$, $\chi_{595}(461,·)$, $\chi_{595}(594,·)$, $\chi_{595}(344,·)$, $\chi_{595}(356,·)$, $\chi_{595}(349,·)$, $\chi_{595}(484,·)$, $\chi_{595}(104,·)$, $\chi_{595}(489,·)$, $\chi_{595}(106,·)$, $\chi_{595}(491,·)$, $\chi_{595}(239,·)$, $\chi_{595}(246,·)$, $\chi_{595}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{24} - \frac{1}{4} a^{18} - \frac{1}{4} a^{12} - \frac{1}{2} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{25} - \frac{1}{4} a^{19} - \frac{1}{4} a^{13} - \frac{1}{2} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{26} - \frac{1}{4} a^{20} - \frac{1}{4} a^{14} - \frac{1}{2} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{27} - \frac{1}{4} a^{21} - \frac{1}{4} a^{15} + \frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{188} a^{28} - \frac{19}{188} a^{27} - \frac{2}{47} a^{26} + \frac{4}{47} a^{25} - \frac{3}{47} a^{23} + \frac{33}{188} a^{22} - \frac{39}{188} a^{21} - \frac{7}{47} a^{20} - \frac{23}{94} a^{19} + \frac{5}{94} a^{18} - \frac{11}{94} a^{17} - \frac{19}{188} a^{16} - \frac{25}{188} a^{15} - \frac{9}{94} a^{14} + \frac{13}{94} a^{13} - \frac{1}{47} a^{12} - \frac{33}{94} a^{11} - \frac{81}{188} a^{10} + \frac{5}{188} a^{9} + \frac{43}{94} a^{8} - \frac{9}{94} a^{7} + \frac{11}{47} a^{6} - \frac{9}{94} a^{5} - \frac{55}{188} a^{4} - \frac{37}{188} a^{3} - \frac{22}{47} a^{2} + \frac{1}{94} a - \frac{25}{94}$, $\frac{1}{188} a^{29} + \frac{7}{188} a^{27} + \frac{5}{188} a^{26} + \frac{11}{94} a^{25} - \frac{3}{47} a^{24} - \frac{7}{188} a^{23} + \frac{6}{47} a^{22} - \frac{17}{188} a^{21} + \frac{33}{188} a^{20} - \frac{9}{94} a^{19} - \frac{5}{47} a^{18} + \frac{33}{188} a^{17} - \frac{5}{94} a^{16} - \frac{23}{188} a^{15} + \frac{13}{188} a^{14} + \frac{5}{47} a^{13} + \frac{23}{94} a^{12} - \frac{19}{188} a^{11} - \frac{15}{94} a^{10} - \frac{7}{188} a^{9} + \frac{65}{188} a^{8} - \frac{4}{47} a^{7} + \frac{33}{94} a^{6} + \frac{73}{188} a^{5} + \frac{23}{94} a^{4} - \frac{39}{188} a^{3} + \frac{69}{188} a^{2} + \frac{41}{94} a + \frac{21}{47}$, $\frac{1}{366690052} a^{30} - \frac{127292}{91672513} a^{29} + \frac{513045}{366690052} a^{28} + \frac{4483979}{91672513} a^{27} - \frac{7982315}{91672513} a^{26} - \frac{24964425}{366690052} a^{25} - \frac{3315789}{91672513} a^{24} + \frac{25488843}{183345026} a^{23} - \frac{67064717}{366690052} a^{22} + \frac{33981857}{183345026} a^{21} - \frac{15830763}{91672513} a^{20} - \frac{63772195}{366690052} a^{19} - \frac{20872513}{183345026} a^{18} + \frac{14044627}{91672513} a^{17} - \frac{43861585}{366690052} a^{16} - \frac{5393790}{91672513} a^{15} + \frac{3134713}{91672513} a^{14} - \frac{980889}{7801916} a^{13} - \frac{17484574}{91672513} a^{12} + \frac{11687289}{91672513} a^{11} + \frac{164331591}{366690052} a^{10} + \frac{45079867}{91672513} a^{9} + \frac{84053661}{183345026} a^{8} + \frac{91539247}{366690052} a^{7} + \frac{29680161}{183345026} a^{6} - \frac{18819870}{91672513} a^{5} + \frac{53854053}{366690052} a^{4} - \frac{91063849}{183345026} a^{3} - \frac{395017}{1950479} a^{2} + \frac{51826079}{366690052} a + \frac{561421}{1534268}$, $\frac{1}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{31} + \frac{4289435506350556904628446356364189961152601570548449495218846086935768531833392032163641368490993568615848726539613603}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{30} - \frac{2221118958774948307853325476103719836939025727834269764249281716496935186477725215512541958423423241429130969006954753725127}{1324487763905962762423815355149358121491531882473421472548150934279512776178697648493848023405586828012812341104193867872401037} a^{29} + \frac{3660317396337084480504450073068943678665626379306128374616367582170163193262387812430811124089800113393933753477246119932957}{2648975527811925524847630710298716242983063764946842945096301868559025552357395296987696046811173656025624682208387735744802074} a^{28} - \frac{516297586144133773494198962350709161251858970670636152619076592206222395776858980944998472937293904028913904207898924375856531}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{27} - \frac{39795830784946064910878644958624953621398935797837338996516553544843471711955357328555402244139671527631752710110651103615957}{2648975527811925524847630710298716242983063764946842945096301868559025552357395296987696046811173656025624682208387735744802074} a^{26} - \frac{70856625096426076687877661697481983805049388069984051521248334329216968571160203515816168394997348024234552583092376894857831}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{25} + \frac{565789489958813507050550562376998980752211979190829757898285505322352739072423628163436372981151861270370483946296549247676825}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{24} + \frac{85214115880741582967321397739297437005556582612585125802741505141890547528464595836822076563531471292626884721855721512707070}{1324487763905962762423815355149358121491531882473421472548150934279512776178697648493848023405586828012812341104193867872401037} a^{23} + \frac{30214059210926396516210120323872576863679068255290758720364508373446743355649951144825453626258424509179655894755798615950931}{2648975527811925524847630710298716242983063764946842945096301868559025552357395296987696046811173656025624682208387735744802074} a^{22} + \frac{27839235576167350527145876747287739446493343111087235837983445817597865980410489130758684770173134878046856774678181520562381}{112722362885613852121175774906328350765236755955184806174310717811022363930101927531391321140901006639388284349293095138076684} a^{21} - \frac{631164986111536766337055967015236618995549427014017709881991149653418198234080620248858645903463784905384808203400974876966169}{2648975527811925524847630710298716242983063764946842945096301868559025552357395296987696046811173656025624682208387735744802074} a^{20} - \frac{669685542434726391939584133410090364648363477253765761227608223556138021209675192909643736216199954707907407139241251098406519}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{19} - \frac{636380267042039990195620409204804132836741826137160369469992484207727217002926716570219086974463820247825328320349873566918209}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{18} + \frac{567135940755293892087807221446913775351686284968153453544398773779549601602042695698713414749696983435411038649310047818490655}{2648975527811925524847630710298716242983063764946842945096301868559025552357395296987696046811173656025624682208387735744802074} a^{17} + \frac{660156235656222638941972514399870668026767130145055905140004587877758146323843471288952334724380301455658600559354217696392701}{2648975527811925524847630710298716242983063764946842945096301868559025552357395296987696046811173656025624682208387735744802074} a^{16} + \frac{524839682299017595554353992130497832455171596945722264847479876095939101881141397987882643957642484074534937322649229793472859}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{15} - \frac{76593168873333484668996492863599636602428202681193084520117994879087037866717258091029144791328688138638204098189297600300417}{2648975527811925524847630710298716242983063764946842945096301868559025552357395296987696046811173656025624682208387735744802074} a^{14} + \frac{593038839235550308066142708085158174453504214594729290308044175361963039235930490479596117264446016407874626733343390104034287}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{13} - \frac{865969664116819142696129485664964029065306732942354466769672508091755164561890697320055994012371007107751905205122882115056405}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{12} - \frac{383456645550856890806713206027225206053587344253354174216227328610093695304802401734607110637912089656931137811281926052591703}{1324487763905962762423815355149358121491531882473421472548150934279512776178697648493848023405586828012812341104193867872401037} a^{11} + \frac{256751075929918962903396584563742182427131219278267673052310124910462720588480943119293339453578956616219003573307334033919035}{1324487763905962762423815355149358121491531882473421472548150934279512776178697648493848023405586828012812341104193867872401037} a^{10} + \frac{2112315916704342894628621666030592972776521474891749999664769741004564346223691144205636670869490037520192058938661862446107407}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{9} - \frac{563563312725124710471953486533220497844083362395076869309619729335982616355514666831478947950461262828532337082857926884270569}{2648975527811925524847630710298716242983063764946842945096301868559025552357395296987696046811173656025624682208387735744802074} a^{8} - \frac{1003994210568448417099068157871914410716590433512373987520986335670340723438698916466248123534703622934410206580563695526716225}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{7} - \frac{2621654533413729514624065969230045074954499696261318364510974190737208712013283369173949008211461807002192113157821141277461371}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{6} + \frac{90844321784097255922264992913454489817180669376431862519229158796737522772435811826344615381348538571261877760008284736143217}{1324487763905962762423815355149358121491531882473421472548150934279512776178697648493848023405586828012812341104193867872401037} a^{5} - \frac{174084112661445761643369181824676240748150834214135502147061911022701268955193358256511763211956923254116275241248383069357969}{1324487763905962762423815355149358121491531882473421472548150934279512776178697648493848023405586828012812341104193867872401037} a^{4} + \frac{38956507708423207211542624225272738510947492395960207588754125250070956948790678093221795256242013939182443593114318520397665}{5297951055623851049695261420597432485966127529893685890192603737118051104714790593975392093622347312051249364416775471489604148} a^{3} + \frac{466432871782503843556613422244467181858233823090871791438457572350340242084020804586909980533333650992341629730037273231774459}{1324487763905962762423815355149358121491531882473421472548150934279512776178697648493848023405586828012812341104193867872401037} a^{2} + \frac{977071337855932268126500533048688885653556193047547392538830657536212794923587582493097160778592734537436674745302291676289623}{2648975527811925524847630710298716242983063764946842945096301868559025552357395296987696046811173656025624682208387735744802074} a - \frac{1684503737962347851221260113080685001622264351579451879782119711412738404255336217925920627157550515208930993631959970530329}{11083579614275839016098873264848185117083948807308966297474066395644458378064415468567765886239220318098848042712919396421766}$

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

Class group and class number

$C_{3}\times C_{182040}$, which has order $546120$ (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:  \( -1 \) (order $2$)
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:  \( 42597284566.379654 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-119}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-595}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}, \sqrt{85})\), \(\Q(\sqrt{17}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-7}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-119})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-7}, \sqrt{85})\), 4.4.4913.1, 4.0.6018425.1, 4.0.240737.1, 4.4.122825.1, 8.0.125333700625.1, 8.0.36221439480625.7, 8.0.36221439480625.3, 8.0.57954303169.1, 8.0.36221439480625.6, 8.8.15085980625.1, 8.0.36221439480625.8, 8.0.615764471170625.1, \(\Q(\zeta_{17})^+\), 8.8.256461670625.1, 8.0.985223153873.1, 16.0.1311992678048579469750390625.1, 16.0.379165883956039466757862890625.4, 16.0.379165883956039466757862890625.3, 16.0.379165883956039466757862890625.1, 16.0.970664662927461034900129.1, 16.0.379165883956039466757862890625.2, 16.16.65772588499765987890625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
5Data not computed
7Data not computed
17Data not computed