Properties

Label 38.38.687...664.1
Degree $38$
Signature $[38, 0]$
Discriminant $6.877\times 10^{95}$
Root discriminant $332.69$
Ramified primes $2, 191$
Class number not computed
Class group not computed
Galois group $C_{38}$ (as 38T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^38 - 191*x^36 + 15662*x^34 - 730575*x^32 + 21705622*x^30 - 436357836*x^28 + 6157331558*x^26 - 62411853712*x^24 + 460944322269*x^22 - 2499589479956*x^20 + 9974553348782*x^18 - 29202012585177*x^16 + 62167403988174*x^14 - 94680464008195*x^12 + 100459885220950*x^10 - 71207905317537*x^8 + 31469275400290*x^6 - 7629511512714*x^4 + 744125807285*x^2 - 1101076991)
 
gp: K = bnfinit(x^38 - 191*x^36 + 15662*x^34 - 730575*x^32 + 21705622*x^30 - 436357836*x^28 + 6157331558*x^26 - 62411853712*x^24 + 460944322269*x^22 - 2499589479956*x^20 + 9974553348782*x^18 - 29202012585177*x^16 + 62167403988174*x^14 - 94680464008195*x^12 + 100459885220950*x^10 - 71207905317537*x^8 + 31469275400290*x^6 - 7629511512714*x^4 + 744125807285*x^2 - 1101076991, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1101076991, 0, 744125807285, 0, -7629511512714, 0, 31469275400290, 0, -71207905317537, 0, 100459885220950, 0, -94680464008195, 0, 62167403988174, 0, -29202012585177, 0, 9974553348782, 0, -2499589479956, 0, 460944322269, 0, -62411853712, 0, 6157331558, 0, -436357836, 0, 21705622, 0, -730575, 0, 15662, 0, -191, 0, 1]);
 

\(x^{38} - 191 x^{36} + 15662 x^{34} - 730575 x^{32} + 21705622 x^{30} - 436357836 x^{28} + 6157331558 x^{26} - 62411853712 x^{24} + 460944322269 x^{22} - 2499589479956 x^{20} + 9974553348782 x^{18} - 29202012585177 x^{16} + 62167403988174 x^{14} - 94680464008195 x^{12} + 100459885220950 x^{10} - 71207905317537 x^{8} + 31469275400290 x^{6} - 7629511512714 x^{4} + 744125807285 x^{2} - 1101076991\)  Toggle raw display

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

Invariants

Degree:  $38$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[38, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(687\!\cdots\!664\)\(\medspace = 2^{38}\cdot 191^{37}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $332.69$
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:  $2, 191$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $38$
This field is Galois and abelian over $\Q$.
Conductor:  \(764=2^{2}\cdot 191\)
Dirichlet character group:    $\lbrace$$\chi_{764}(1,·)$, $\chi_{764}(643,·)$, $\chi_{764}(5,·)$, $\chi_{764}(567,·)$, $\chi_{764}(139,·)$, $\chi_{764}(275,·)$, $\chi_{764}(25,·)$, $\chi_{764}(153,·)$, $\chi_{764}(155,·)$, $\chi_{764}(31,·)$, $\chi_{764}(197,·)$, $\chi_{764}(419,·)$, $\chi_{764}(423,·)$, $\chi_{764}(605,·)$, $\chi_{764}(177,·)$, $\chi_{764}(709,·)$, $\chi_{764}(695,·)$, $\chi_{764}(543,·)$, $\chi_{764}(159,·)$, $\chi_{764}(11,·)$, $\chi_{764}(69,·)$, $\chi_{764}(55,·)$, $\chi_{764}(611,·)$, $\chi_{764}(341,·)$, $\chi_{764}(625,·)$, $\chi_{764}(345,·)$, $\chi_{764}(733,·)$, $\chi_{764}(609,·)$, $\chi_{764}(587,·)$, $\chi_{764}(739,·)$, $\chi_{764}(489,·)$, $\chi_{764}(753,·)$, $\chi_{764}(221,·)$, $\chi_{764}(759,·)$, $\chi_{764}(121,·)$, $\chi_{764}(763,·)$, $\chi_{764}(125,·)$, $\chi_{764}(639,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{6}$, $\frac{1}{7} a^{13} - \frac{1}{7} a$, $\frac{1}{49} a^{14} - \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{49} a^{15} - \frac{2}{49} a^{9} + \frac{1}{49} a^{3}$, $\frac{1}{49} a^{16} - \frac{2}{49} a^{10} + \frac{1}{49} a^{4}$, $\frac{1}{49} a^{17} - \frac{2}{49} a^{11} + \frac{1}{49} a^{5}$, $\frac{1}{49} a^{18} - \frac{2}{49} a^{12} + \frac{1}{49} a^{6}$, $\frac{1}{49} a^{19} - \frac{2}{49} a^{13} + \frac{1}{49} a^{7}$, $\frac{1}{49} a^{20} - \frac{3}{49} a^{8} + \frac{2}{49} a^{2}$, $\frac{1}{343} a^{21} - \frac{3}{343} a^{15} + \frac{3}{343} a^{9} - \frac{1}{343} a^{3}$, $\frac{1}{343} a^{22} - \frac{3}{343} a^{16} + \frac{3}{343} a^{10} - \frac{1}{343} a^{4}$, $\frac{1}{343} a^{23} - \frac{3}{343} a^{17} + \frac{3}{343} a^{11} - \frac{1}{343} a^{5}$, $\frac{1}{2401} a^{24} + \frac{3}{2401} a^{22} + \frac{1}{343} a^{20} + \frac{4}{2401} a^{18} - \frac{2}{2401} a^{16} - \frac{2}{343} a^{14} + \frac{87}{2401} a^{12} + \frac{93}{2401} a^{10} + \frac{15}{343} a^{8} - \frac{778}{2401} a^{6} + \frac{935}{2401} a^{4} - \frac{9}{49} a^{2}$, $\frac{1}{2401} a^{25} + \frac{3}{2401} a^{23} + \frac{4}{2401} a^{19} - \frac{2}{2401} a^{17} + \frac{1}{343} a^{15} + \frac{87}{2401} a^{13} + \frac{93}{2401} a^{11} + \frac{12}{343} a^{9} - \frac{92}{2401} a^{7} + \frac{935}{2401} a^{5} - \frac{62}{343} a^{3} - \frac{2}{7} a$, $\frac{1}{2401} a^{26} - \frac{2}{2401} a^{22} - \frac{17}{2401} a^{20} - \frac{2}{343} a^{18} - \frac{8}{2401} a^{16} - \frac{18}{2401} a^{14} - \frac{24}{343} a^{12} + \frac{169}{2401} a^{10} - \frac{113}{2401} a^{8} + \frac{124}{343} a^{6} - \frac{1188}{2401} a^{4} + \frac{10}{49} a^{2}$, $\frac{1}{2401} a^{27} - \frac{2}{2401} a^{23} - \frac{3}{2401} a^{21} - \frac{2}{343} a^{19} - \frac{8}{2401} a^{17} - \frac{11}{2401} a^{15} - \frac{24}{343} a^{13} + \frac{169}{2401} a^{11} - \frac{169}{2401} a^{9} - \frac{23}{343} a^{7} - \frac{1188}{2401} a^{5} + \frac{75}{343} a^{3} + \frac{3}{7} a$, $\frac{1}{1831963} a^{28} + \frac{22}{261709} a^{26} + \frac{36}{261709} a^{24} + \frac{2096}{1831963} a^{22} + \frac{1418}{261709} a^{20} - \frac{2250}{261709} a^{18} - \frac{14967}{1831963} a^{16} - \frac{1915}{261709} a^{14} - \frac{13738}{261709} a^{12} - \frac{22320}{1831963} a^{10} + \frac{9582}{261709} a^{8} + \frac{27957}{261709} a^{6} + \frac{20784}{1831963} a^{4} - \frac{13747}{37387} a^{2} - \frac{202}{763}$, $\frac{1}{1831963} a^{29} + \frac{22}{261709} a^{27} + \frac{36}{261709} a^{25} + \frac{2096}{1831963} a^{23} - \frac{108}{261709} a^{21} - \frac{2250}{261709} a^{19} - \frac{14967}{1831963} a^{17} + \frac{2663}{261709} a^{15} - \frac{13738}{261709} a^{13} - \frac{22320}{1831963} a^{11} + \frac{5004}{261709} a^{9} - \frac{9430}{261709} a^{7} + \frac{20784}{1831963} a^{5} - \frac{13529}{37387} a^{3} - \frac{93}{763} a$, $\frac{1}{1831963} a^{30} + \frac{27}{261709} a^{26} - \frac{88}{1831963} a^{24} + \frac{15}{37387} a^{22} - \frac{1096}{261709} a^{20} - \frac{17333}{1831963} a^{18} + \frac{2}{763} a^{16} - \frac{2446}{261709} a^{14} - \frac{95071}{1831963} a^{12} + \frac{1446}{37387} a^{10} + \frac{17725}{261709} a^{8} + \frac{242145}{1831963} a^{6} + \frac{4566}{37387} a^{4} - \frac{2327}{5341} a^{2} - \frac{25}{109}$, $\frac{1}{12823741} a^{31} + \frac{3}{12823741} a^{29} + \frac{93}{1831963} a^{27} - \frac{95}{12823741} a^{25} - \frac{16630}{12823741} a^{23} + \frac{2395}{1831963} a^{21} + \frac{44526}{12823741} a^{19} - \frac{49255}{12823741} a^{17} + \frac{14699}{1831963} a^{15} + \frac{634273}{12823741} a^{13} + \frac{803518}{12823741} a^{11} - \frac{45089}{1831963} a^{9} + \frac{226472}{12823741} a^{7} + \frac{6061996}{12823741} a^{5} + \frac{32504}{261709} a^{3} - \frac{781}{5341} a$, $\frac{1}{12823741} a^{32} + \frac{3}{12823741} a^{30} + \frac{1130}{12823741} a^{26} + \frac{912}{12823741} a^{24} + \frac{506}{1831963} a^{22} - \frac{66851}{12823741} a^{20} + \frac{88141}{12823741} a^{18} - \frac{16365}{1831963} a^{16} + \frac{19519}{12823741} a^{14} + \frac{394466}{12823741} a^{12} + \frac{72813}{1831963} a^{10} - \frac{507744}{12823741} a^{8} - \frac{759637}{12823741} a^{6} + \frac{435594}{1831963} a^{4} - \frac{8073}{37387} a^{2} - \frac{289}{763}$, $\frac{1}{12823741} a^{33} - \frac{2}{12823741} a^{29} + \frac{255}{12823741} a^{27} - \frac{340}{1831963} a^{25} + \frac{14694}{12823741} a^{23} - \frac{10277}{12823741} a^{21} + \frac{12094}{1831963} a^{19} + \frac{51284}{12823741} a^{17} - \frac{9125}{12823741} a^{15} + \frac{70618}{1831963} a^{13} - \frac{834014}{12823741} a^{11} - \frac{474676}{12823741} a^{9} + \frac{97703}{1831963} a^{7} + \frac{3867729}{12823741} a^{5} + \frac{68682}{261709} a^{3} - \frac{1094}{5341} a$, $\frac{1}{686313794579} a^{34} - \frac{19890}{686313794579} a^{32} - \frac{45819}{686313794579} a^{30} + \frac{5876}{686313794579} a^{28} - \frac{113957209}{686313794579} a^{26} + \frac{135161525}{686313794579} a^{24} + \frac{64903026}{686313794579} a^{22} + \frac{1815319684}{686313794579} a^{20} + \frac{904338957}{686313794579} a^{18} + \frac{6515432227}{686313794579} a^{16} + \frac{3123289335}{686313794579} a^{14} + \frac{25923280105}{686313794579} a^{12} - \frac{10106665799}{686313794579} a^{10} + \frac{14261364695}{686313794579} a^{8} - \frac{138914368254}{686313794579} a^{6} - \frac{22900353157}{98044827797} a^{4} - \frac{567990517}{2000914853} a^{2} - \frac{13811668}{40834997}$, $\frac{1}{686313794579} a^{35} - \frac{19890}{686313794579} a^{33} + \frac{1100}{98044827797} a^{31} + \frac{166433}{686313794579} a^{29} - \frac{79116340}{686313794579} a^{27} + \frac{18582460}{98044827797} a^{25} - \frac{117873992}{98044827797} a^{23} + \frac{711650866}{686313794579} a^{21} + \frac{469617993}{98044827797} a^{19} + \frac{3879353882}{686313794579} a^{17} + \frac{626360390}{686313794579} a^{15} - \frac{5453698715}{98044827797} a^{13} + \frac{32896814043}{686313794579} a^{11} + \frac{19379600741}{686313794579} a^{9} - \frac{4106997927}{98044827797} a^{7} + \frac{164129491825}{686313794579} a^{5} - \frac{2481362025}{14006403971} a^{3} - \frac{138480015}{285844979} a$, $\frac{1}{2062542955254745697899364932789220287969378086401589779656925811407} a^{36} + \frac{1168342676101525121910429055608846037432410856244325741}{2062542955254745697899364932789220287969378086401589779656925811407} a^{34} + \frac{2432759269999120291030790858729256467582956755589365706146}{2062542955254745697899364932789220287969378086401589779656925811407} a^{32} - \frac{1096043756435039866967406469486714456662858600353261977246}{4200698483207221380650437744988228692401992029331140080767669677} a^{30} - \frac{449736736798541680905083588803796827466584543324054239321142}{2062542955254745697899364932789220287969378086401589779656925811407} a^{28} + \frac{39815730701276646269484506251933576977906155164004698359575422}{2062542955254745697899364932789220287969378086401589779656925811407} a^{26} - \frac{27257255573328474613450733483299264397315103126101617826612589}{294648993607820813985623561827031469709911155200227111379560830201} a^{24} + \frac{2149022474936384792693744152850431368474268540940431723208988704}{2062542955254745697899364932789220287969378086401589779656925811407} a^{22} - \frac{18708388471700767671480554940339382792534161476666090078926839954}{2062542955254745697899364932789220287969378086401589779656925811407} a^{20} - \frac{12921786021085845514751119852968978292230311341749314687229940725}{2062542955254745697899364932789220287969378086401589779656925811407} a^{18} + \frac{15827169373077791237576285880871221717135539671040340339847714233}{2062542955254745697899364932789220287969378086401589779656925811407} a^{16} - \frac{12730583139712485492090760560124111093783359632563541143596267510}{2062542955254745697899364932789220287969378086401589779656925811407} a^{14} + \frac{1230697497043802811608138654204130615049885995532262455798452699}{2062542955254745697899364932789220287969378086401589779656925811407} a^{12} - \frac{126554232896656229790687425423925616143316236118869447391118565095}{2062542955254745697899364932789220287969378086401589779656925811407} a^{10} - \frac{119194212370317605371189452077327285912581954178139492506487437627}{2062542955254745697899364932789220287969378086401589779656925811407} a^{8} + \frac{285714677978779848052842990476188151822672334182833248210397278575}{2062542955254745697899364932789220287969378086401589779656925811407} a^{6} - \frac{12728335344569793871353572061857509554725424868008298756676948937}{42092713372545830569374794546718781387130165028603873054222975743} a^{4} + \frac{112339459981333290150396049781604823479631051976180713098898980}{859034966786649603456628460137117987492452347522528021514754607} a^{2} - \frac{6980166186967240546840086351825007698655450829274522772310521}{17531325852788767417482213472186081377396986684133224928872543}$, $\frac{1}{14437800686783219885295554529524542015785646604811128457598480679849} a^{37} - \frac{4842152731527444309500352941279880054149224709392305725}{14437800686783219885295554529524542015785646604811128457598480679849} a^{35} - \frac{38856338932708085218045576068897847418537088762551474149541}{14437800686783219885295554529524542015785646604811128457598480679849} a^{33} + \frac{380588812114336805696161327180443205959430494883066945170576}{14437800686783219885295554529524542015785646604811128457598480679849} a^{31} - \frac{1610919370836901357782911089737722042914469672338309163816347}{14437800686783219885295554529524542015785646604811128457598480679849} a^{29} - \frac{2853726353981103059244495271872135785304977367784028851819772507}{14437800686783219885295554529524542015785646604811128457598480679849} a^{27} - \frac{96233107978838675264979097801741591929253633288867875945567550}{2062542955254745697899364932789220287969378086401589779656925811407} a^{25} + \frac{9106317883911119660741230602101301739499129153899319157653226802}{14437800686783219885295554529524542015785646604811128457598480679849} a^{23} - \frac{16345250343866654660585802727904227031991914074007354137539041961}{14437800686783219885295554529524542015785646604811128457598480679849} a^{21} + \frac{101411560245919007105860458181118538075719153175605544775800590425}{14437800686783219885295554529524542015785646604811128457598480679849} a^{19} + \frac{43119247471453836607257190378265303571622542807248262294479434187}{14437800686783219885295554529524542015785646604811128457598480679849} a^{17} + \frac{873230471341924410479352299621103205582517933886418406714739994}{132456887034708439314638114949766440511794922979918609702738354861} a^{15} - \frac{911752588592610572536092954759116579308784682983133486226392886321}{14437800686783219885295554529524542015785646604811128457598480679849} a^{13} + \frac{45596173565812936048130375262679084934545526701537620526068727789}{14437800686783219885295554529524542015785646604811128457598480679849} a^{11} - \frac{211980420451848105896940731255805669710997567553510166409555853866}{14437800686783219885295554529524542015785646604811128457598480679849} a^{9} + \frac{969554905357273218248402659089942683283004529111767326087495280979}{14437800686783219885295554529524542015785646604811128457598480679849} a^{7} + \frac{131356325228488111886117251933185617201165984535272561206599826231}{294648993607820813985623561827031469709911155200227111379560830201} a^{5} + \frac{972563529441152991481690404375771283653525129520990135756300819}{6013244767506547224196399220959825912447166432657696150603282249} a^{3} - \frac{40454182563391060836609165632860511926885033905122790502284690}{122719280969521371922375494305302569641778906788932574502107801} a$  Toggle raw display

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $37$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_{38}$ (as 38T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 38
The 38 conjugacy class representatives for $C_{38}$
Character table for $C_{38}$ is not computed

Intermediate fields

\(\Q(\sqrt{191}) \), 19.19.114445997944945591651333831028437092270721.1

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 $38$ $19^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{38}$ $19^{2}$ $19^{2}$ $19^{2}$ $19^{2}$ $38$ $38$ $19^{2}$ $38$ $38$ $38$ $19^{2}$ $38$ $38$

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
191Data not computed