Properties

Label 38.38.206...109.1
Degree $38$
Signature $[38, 0]$
Discriminant $2.060\times 10^{87}$
Root discriminant $198.49$
Ramified prime $229$
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 - x^37 - 111*x^36 + 252*x^35 + 5215*x^34 - 18518*x^33 - 127217*x^32 + 667591*x^31 + 1483161*x^30 - 13721975*x^29 + 465004*x^28 + 166721208*x^27 - 256518740*x^26 - 1121099509*x^25 + 3587854285*x^24 + 2545487107*x^23 - 24194172078*x^22 + 18477975516*x^21 + 81929300895*x^20 - 167013913064*x^19 - 73340427022*x^18 + 542830510766*x^17 - 389253844535*x^16 - 713755295161*x^15 + 1324741808499*x^14 - 116683382685*x^13 - 1503618080692*x^12 + 1234820609168*x^11 + 402888989840*x^10 - 1097103201368*x^9 + 414161581992*x^8 + 265775405099*x^7 - 274544542564*x^6 + 51663795877*x^5 + 34067812864*x^4 - 20708103251*x^3 + 4233902513*x^2 - 281738990*x - 6131569)
 
gp: K = bnfinit(x^38 - x^37 - 111*x^36 + 252*x^35 + 5215*x^34 - 18518*x^33 - 127217*x^32 + 667591*x^31 + 1483161*x^30 - 13721975*x^29 + 465004*x^28 + 166721208*x^27 - 256518740*x^26 - 1121099509*x^25 + 3587854285*x^24 + 2545487107*x^23 - 24194172078*x^22 + 18477975516*x^21 + 81929300895*x^20 - 167013913064*x^19 - 73340427022*x^18 + 542830510766*x^17 - 389253844535*x^16 - 713755295161*x^15 + 1324741808499*x^14 - 116683382685*x^13 - 1503618080692*x^12 + 1234820609168*x^11 + 402888989840*x^10 - 1097103201368*x^9 + 414161581992*x^8 + 265775405099*x^7 - 274544542564*x^6 + 51663795877*x^5 + 34067812864*x^4 - 20708103251*x^3 + 4233902513*x^2 - 281738990*x - 6131569, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6131569, -281738990, 4233902513, -20708103251, 34067812864, 51663795877, -274544542564, 265775405099, 414161581992, -1097103201368, 402888989840, 1234820609168, -1503618080692, -116683382685, 1324741808499, -713755295161, -389253844535, 542830510766, -73340427022, -167013913064, 81929300895, 18477975516, -24194172078, 2545487107, 3587854285, -1121099509, -256518740, 166721208, 465004, -13721975, 1483161, 667591, -127217, -18518, 5215, 252, -111, -1, 1]);
 

\( x^{38} - x^{37} - 111 x^{36} + 252 x^{35} + 5215 x^{34} - 18518 x^{33} - 127217 x^{32} + 667591 x^{31} + 1483161 x^{30} - 13721975 x^{29} + 465004 x^{28} + 166721208 x^{27} - 256518740 x^{26} - 1121099509 x^{25} + 3587854285 x^{24} + 2545487107 x^{23} - 24194172078 x^{22} + 18477975516 x^{21} + 81929300895 x^{20} - 167013913064 x^{19} - 73340427022 x^{18} + 542830510766 x^{17} - 389253844535 x^{16} - 713755295161 x^{15} + 1324741808499 x^{14} - 116683382685 x^{13} - 1503618080692 x^{12} + 1234820609168 x^{11} + 402888989840 x^{10} - 1097103201368 x^{9} + 414161581992 x^{8} + 265775405099 x^{7} - 274544542564 x^{6} + 51663795877 x^{5} + 34067812864 x^{4} - 20708103251 x^{3} + 4233902513 x^{2} - 281738990 x - 6131569 \)

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:  \(206\!\cdots\!109\)\(\medspace = 229^{37}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $198.49$
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:  $229$
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:  \(229\)
Dirichlet character group:    $\lbrace$$\chi_{229}(1,·)$, $\chi_{229}(4,·)$, $\chi_{229}(44,·)$, $\chi_{229}(11,·)$, $\chi_{229}(15,·)$, $\chi_{229}(16,·)$, $\chi_{229}(17,·)$, $\chi_{229}(26,·)$, $\chi_{229}(27,·)$, $\chi_{229}(161,·)$, $\chi_{229}(165,·)$, $\chi_{229}(168,·)$, $\chi_{229}(169,·)$, $\chi_{229}(42,·)$, $\chi_{229}(43,·)$, $\chi_{229}(172,·)$, $\chi_{229}(176,·)$, $\chi_{229}(53,·)$, $\chi_{229}(57,·)$, $\chi_{229}(186,·)$, $\chi_{229}(187,·)$, $\chi_{229}(60,·)$, $\chi_{229}(61,·)$, $\chi_{229}(64,·)$, $\chi_{229}(68,·)$, $\chi_{229}(202,·)$, $\chi_{229}(203,·)$, $\chi_{229}(212,·)$, $\chi_{229}(213,·)$, $\chi_{229}(214,·)$, $\chi_{229}(185,·)$, $\chi_{229}(218,·)$, $\chi_{229}(225,·)$, $\chi_{229}(228,·)$, $\chi_{229}(104,·)$, $\chi_{229}(108,·)$, $\chi_{229}(121,·)$, $\chi_{229}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{7787737} a^{36} - \frac{2637475}{7787737} a^{35} + \frac{615065}{7787737} a^{34} - \frac{2506933}{7787737} a^{33} + \frac{1207515}{7787737} a^{32} + \frac{119873}{7787737} a^{31} + \frac{1574909}{7787737} a^{30} + \frac{3751364}{7787737} a^{29} - \frac{755682}{7787737} a^{28} + \frac{3174239}{7787737} a^{27} - \frac{96975}{7787737} a^{26} + \frac{913786}{7787737} a^{25} - \frac{3040176}{7787737} a^{24} - \frac{1682192}{7787737} a^{23} - \frac{1864227}{7787737} a^{22} - \frac{3799130}{7787737} a^{21} + \frac{1276549}{7787737} a^{20} + \frac{62153}{7787737} a^{19} - \frac{2536689}{7787737} a^{18} + \frac{2163242}{7787737} a^{17} - \frac{945908}{7787737} a^{16} + \frac{2582888}{7787737} a^{15} - \frac{2012910}{7787737} a^{14} - \frac{1100709}{7787737} a^{13} + \frac{1058155}{7787737} a^{12} + \frac{1861100}{7787737} a^{11} - \frac{37384}{7787737} a^{10} - \frac{3344653}{7787737} a^{9} + \frac{205591}{7787737} a^{8} - \frac{727001}{7787737} a^{7} - \frac{3708005}{7787737} a^{6} + \frac{2492503}{7787737} a^{5} + \frac{1487749}{7787737} a^{4} - \frac{1386641}{7787737} a^{3} - \frac{567980}{7787737} a^{2} - \frac{2451025}{7787737} a - \frac{3398}{17041}$, $\frac{1}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{37} - \frac{6371031559139371259160190253501375647129865956719470876485274273845477532170581730719425026601859955010255}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{36} + \frac{122058667936970520889953645721130955863217645511646798688864411569713422246624395772179586197195950818705394309153}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{35} + \frac{160704271902037249936081467262591432781485795795943875209807657809800252483301946469740040307822148180402222033133}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{34} + \frac{83709616017500617236944242212136172888657778710271220948784321202620027299818306775674498330067594150644550639092}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{33} + \frac{157967844533440587163859660915169870321642186995411913217792972366957178740270076939139136911036017375464585349328}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{32} + \frac{214658959246101704958430538984258940005450894773915432921102673110503922472763500308812538176884277829950867603491}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{31} + \frac{124988966080596855034989647743735762563944441017416761778279207828723667843358422933165060373095764024980964122693}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{30} - \frac{144987292661759355754303112936716089964440430544390459289363637320935881909579238523302272514732999205422827247217}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{29} - \frac{110601742359626578142310411447479855450449481455738438704995433407733805441008515739511040290457707589563925575228}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{28} + \frac{189755185574584299516974564297968228147281686390762287810534640521878338726892352860573958544154996404856106016384}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{27} + \frac{124139834363251923202429598815646036197799390721521400804814369412260393598864273098656053285894729446080626089386}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{26} + \frac{206460526179633438691040738936250990125428736805479838530200652542615490509758733518359240736792211387848131471189}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{25} + \frac{31863580595570699638711090337932371214046389233572310540837282684917504541569321087768539892927567471405016603558}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{24} - \frac{5575460530690221783354284796753081373888011163556188016990239655546852794690119458212584714814448358081422387839}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{23} + \frac{208599712695303141762740231620332713143953070280909790905762678595019002371850393577439919321179198667186705436360}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{22} - \frac{174178573782393632760014019503706131588780616368710684680912319666926051006224496350440855370333216177313751495990}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{21} - \frac{155447741496896774313010163880719398243030586637866778807032968990050051999949687385053750430390822523100636592377}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{20} - \frac{223478176570809442913876000100829452019910349543681291798038752370654359763340237555157461085129267984488784921885}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{19} - \frac{134763558431866940375521282600678689727412670474301370464367701110934995081915841768623797316141530480108687575835}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{18} + \frac{49421846949018226926601028340873334595639315679654941597738090401444470192015069720919776275326653596333574355911}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{17} + \frac{125321137815520327538274400851755392253544709038296101216832476951810367948799163687690703365397360116796854438461}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{16} - \frac{39088333200966073292503678245786562881183794779756337184103617538944266507143854923884505504827506007621910521076}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{15} + \frac{45113167656025557657185558223632191456240691902140357287051463907376519132079737552429157733478659851027981746841}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{14} + \frac{133475500854050452469007218700740514793912994283101282001291508616923793255321792993718215916789158837355573097935}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{13} + \frac{230124575064127550208672804648609932781306167577848321146430344952673621283229990781640090111370531937884125907935}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{12} - \frac{165855764131521657020605927503575549688524958820807278348715113984817329554194438583370924252708446684190822594590}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{11} + \frac{142903408331597338544613282032739292042067039701665291766945642886475124580231666383715205747694081660827444856236}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{10} + \frac{49285233861862339173821007616946917799266591182482003497554268933495309116516649410171557811945334822729079280076}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{9} + \frac{181545081650255060084177747698417089077035299867366520888766465459855730537918439490205625115041124927128044132206}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{8} - \frac{130431460877364864909473671157383226914806384345589296145904176837460591154657995899001749678605067020933323656996}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{7} - \frac{47376325724498189570812373885605475878908035516923904187932092754781941076913565387610188683988819763071921356600}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{6} + \frac{230507322510700855649253896602236096868502638445891068379787622638529109153029732998509765650900698319889210112402}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{5} - \frac{187329197710155627263403938395057161250062511545859114866717437066566070478948897745245126178738772288067092810016}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{4} + \frac{88977683957311568228257155312047387092190263284307710136792354964573889730599839447546616101820530518651096585205}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{3} - \frac{125584688155816982660918951102437847159889810575591239382348163115913869718511134581213065265547898714759680139621}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a^{2} + \frac{77683777021885330308689880407193806016043991196558320750784984498574285567770210849089223792106881876764750305583}{476766562830589883421748674448158620276914142566604219189996874478400738371517626957878040437247199030001405328831} a + \frac{181947159564466116010976432906852082485342456379063977763890953196123737384208610139615973960540230849530749844}{1043252872714638694577130578661178600168302281327361529956229484635450193373123910192293305114326474901534803783}$

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 $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
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

\(\Q(\sqrt{229}) \), 19.19.2999429662895796650415561622892044448017561.1

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 $38$ $19^{2}$ $19^{2}$ $38$ $19^{2}$ $38$ $19^{2}$ $19^{2}$ $38$ $38$ $38$ $19^{2}$ $38$ $19^{2}$ $38$ $19^{2}$ $38$

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
229Data not computed