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Properties

Label 38.38.6139778515...0496.1
Degree $38$
Signature $[38, 0]$
Discriminant $2^{38}\cdot 19^{73}$
Root discriminant $572.25$
Ramified primes $2, 19$
Class number Not computed
Class group Not computed
Galois group $C_{38}$ (as 38T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8330392699, 0, 134224048088799, 0, -1646505887125241, 0, 7975348801613157, 0, -19840358033605744, 0, 27980982110305398, 0, -23833233673330582, 0, 12897097470403170, 0, -4643480159473720, 0, 1152429903953389, 0, -201840423651116, 0, 25265325956832, 0, -2266887392060, 0, 144858382999, 0, -6488139646, 0, 198348562, 0, -3980025, 0, 49457, 0, -342, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^38 - 342*x^36 + 49457*x^34 - 3980025*x^32 + 198348562*x^30 - 6488139646*x^28 + 144858382999*x^26 - 2266887392060*x^24 + 25265325956832*x^22 - 201840423651116*x^20 + 1152429903953389*x^18 - 4643480159473720*x^16 + 12897097470403170*x^14 - 23833233673330582*x^12 + 27980982110305398*x^10 - 19840358033605744*x^8 + 7975348801613157*x^6 - 1646505887125241*x^4 + 134224048088799*x^2 - 8330392699)
 
gp: K = bnfinit(x^38 - 342*x^36 + 49457*x^34 - 3980025*x^32 + 198348562*x^30 - 6488139646*x^28 + 144858382999*x^26 - 2266887392060*x^24 + 25265325956832*x^22 - 201840423651116*x^20 + 1152429903953389*x^18 - 4643480159473720*x^16 + 12897097470403170*x^14 - 23833233673330582*x^12 + 27980982110305398*x^10 - 19840358033605744*x^8 + 7975348801613157*x^6 - 1646505887125241*x^4 + 134224048088799*x^2 - 8330392699, 1)
 

Normalized defining polynomial

\( x^{38} - 342 x^{36} + 49457 x^{34} - 3980025 x^{32} + 198348562 x^{30} - 6488139646 x^{28} + 144858382999 x^{26} - 2266887392060 x^{24} + 25265325956832 x^{22} - 201840423651116 x^{20} + 1152429903953389 x^{18} - 4643480159473720 x^{16} + 12897097470403170 x^{14} - 23833233673330582 x^{12} + 27980982110305398 x^{10} - 19840358033605744 x^{8} + 7975348801613157 x^{6} - 1646505887125241 x^{4} + 134224048088799 x^{2} - 8330392699 \)

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

Invariants

Degree:  $38$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[38, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(613977851508884585934104054696506599029285341338287807495000891446051265879187437942845703843204086890496=2^{38}\cdot 19^{73}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $572.25$
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:  $2, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1444=2^{2}\cdot 19^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1444}(1,·)$, $\chi_{1444}(1291,·)$, $\chi_{1444}(1293,·)$, $\chi_{1444}(911,·)$, $\chi_{1444}(913,·)$, $\chi_{1444}(531,·)$, $\chi_{1444}(533,·)$, $\chi_{1444}(151,·)$, $\chi_{1444}(153,·)$, $\chi_{1444}(1443,·)$, $\chi_{1444}(1063,·)$, $\chi_{1444}(1065,·)$, $\chi_{1444}(683,·)$, $\chi_{1444}(685,·)$, $\chi_{1444}(303,·)$, $\chi_{1444}(305,·)$, $\chi_{1444}(1215,·)$, $\chi_{1444}(1217,·)$, $\chi_{1444}(835,·)$, $\chi_{1444}(837,·)$, $\chi_{1444}(455,·)$, $\chi_{1444}(457,·)$, $\chi_{1444}(75,·)$, $\chi_{1444}(77,·)$, $\chi_{1444}(1367,·)$, $\chi_{1444}(1369,·)$, $\chi_{1444}(987,·)$, $\chi_{1444}(989,·)$, $\chi_{1444}(607,·)$, $\chi_{1444}(609,·)$, $\chi_{1444}(227,·)$, $\chi_{1444}(229,·)$, $\chi_{1444}(1139,·)$, $\chi_{1444}(1141,·)$, $\chi_{1444}(759,·)$, $\chi_{1444}(761,·)$, $\chi_{1444}(379,·)$, $\chi_{1444}(381,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{127} a^{34} - \frac{15}{127} a^{32} + \frac{8}{127} a^{30} - \frac{10}{127} a^{28} + \frac{4}{127} a^{26} - \frac{6}{127} a^{24} - \frac{56}{127} a^{22} + \frac{20}{127} a^{20} + \frac{24}{127} a^{18} - \frac{31}{127} a^{16} - \frac{22}{127} a^{14} + \frac{54}{127} a^{12} + \frac{7}{127} a^{10} - \frac{62}{127} a^{8} + \frac{57}{127} a^{6} - \frac{23}{127} a^{4} - \frac{37}{127} a^{2} + \frac{5}{127}$, $\frac{1}{127} a^{35} - \frac{15}{127} a^{33} + \frac{8}{127} a^{31} - \frac{10}{127} a^{29} + \frac{4}{127} a^{27} - \frac{6}{127} a^{25} - \frac{56}{127} a^{23} + \frac{20}{127} a^{21} + \frac{24}{127} a^{19} - \frac{31}{127} a^{17} - \frac{22}{127} a^{15} + \frac{54}{127} a^{13} + \frac{7}{127} a^{11} - \frac{62}{127} a^{9} + \frac{57}{127} a^{7} - \frac{23}{127} a^{5} - \frac{37}{127} a^{3} + \frac{5}{127} a$, $\frac{1}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{36} + \frac{1135681081077959949837745328247425808771833811697177591799512953125125427614065168022415928810652457666264143733553882740215619894784372205267027}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{34} + \frac{216666501878480505656804668437399000456209901974094604483909138157635592278018720700120429062946500556067134089982274626854625116149484181770347441}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{32} + \frac{10837253139923341187240567913342870606539965465711115253574028850216334200734605953798006306138349105210939000599675277311501324311274144166029165}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{30} - \frac{115848558104784516521203757271645638284359455501889088583559241378105415604314667315152718259527078137640394379467342986270116992314636777328781214}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{28} - \frac{14512159105409114997677197973568134905285000371699993108853664305071872565006416609315863371762423373889038039179760119993468515021136924840213903}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{26} + \frac{7977977434648735284869151971490146881847549238318315379740466424426365831895659118896910349425591020868139565958952392566042727101016332480301237}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{24} - \frac{130546107694769596180530664461835397152236644361845069776519077755783996945852146293322434263568088954178305078482757747111230405568911729386815354}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{22} - \frac{207578061398533087176979256077424909238453899029494085686315161035081256439941061250418515927108653843340657888619077122341802615824297011108761342}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{20} + \frac{255813343712777330906793165852071016726353686290215399424984935182598313512515137172799088802618608964340485741781704203717721291419300475243850797}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{18} - \frac{44917000066688416577949082279951103684444851183393092906346035946152895998508125165207022945412260568885408453188060279522624384836088515062438345}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{16} - \frac{42306322446458951852384416173818672852900993763786424384876864732912209759757271020243017280810151899595464499179371744766004009712769993905433370}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{14} - \frac{169165546432974749109807029486495398207208811558721498513270024623677281499498851580693133876371250136689187835065933578235891313128820829261397595}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{12} - \frac{1947348751607607544708220100536735790397808212572222659528468529837878668945227755533959019788164198916600047368208072096891051611344949038650439}{4250408085721976445186842810289240123668750662728421158858313889165230926979628885939045662642596163450713817067888099572977276637546159054556789} a^{10} - \frac{232402957391102238921916670269247318679131970796166373999794949520355576117696857620654713898853309868781950765408089668735453103381637148047033826}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{8} + \frac{241479918752493278394796091951187812635402264623615058048499679798519736335745370018499431013029836035477532543667405854489731786391916032491660732}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{6} + \frac{44499810401184072310229244688146489782115420116840522378244350252652976849930244539965723964592226763383318669969021499585171491583912078242758966}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{4} - \frac{86872225162315798405560511063770939443623118427264952672144587412216467100340503707834044898127154897266313664991622095789459297175501160902545457}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203} a^{2} + \frac{156278228393948426524989147006366386473725419280293101960503290819569192446018802528260478885331351234366202723926833893281911847678659314031739365}{539801826886691008538729036906733495705931334166509487175005863923984327726412868514258799155609712758240654767621788645768114132968362199928712203}$, $\frac{1}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{37} + \frac{38195302739378758296398807238587359177096165289089289711092608120991890235266559234216286740435179777225780624315776016645514023484884569636452572981}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{35} + \frac{976535403792216469965074598191875855407168237130692434794238609479411179519498773820898909138067285245185030914583878746539735068760502219013464780741}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{33} - \frac{302792485181779400590368628079902412779678925997969740662971565749170000134755006857229754046015054931287092102732816426401162663611114643061514175984}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{31} + \frac{3582247097514777178827415148353188029313201424110567859389651419335975146308782340010918825035474512170700584617314523843907021996877439506856660578078}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{29} + \frac{3783469243713969035662289703009752370988338442424681480879983821520568576854643727886488424555284642087625212369673140522367437421290142160852667287291}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{27} - \frac{841292296061859036966327675659818964491368646127248995342081760871606391934953968250340167279150203425432720527949404148485118231248960429972163172689}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{25} - \frac{5126807013190199600699188041565741992201576204751537038459479257947179884113622008839418577932919999192830606613499460826788956196824881663132657992383}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{23} - \frac{427127067007485291194633844807149281410775107964797572123733078816569206024127825622740042963074752503159937871821435574431325422644235604768901762080}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{21} - \frac{3457098380520000136783639325051190762657537804138273938375628123847932046617476274376707906680950024876624940452168552586148739709391464759205175787216}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{19} + \frac{1704955281690275498852781802151525253624550858494370013729199970836016250690755863922651691998180327553723553742525308426225436171447155062311738787537}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{17} + \frac{2492962299420321551372922089631205058782950431468325261144258543489017479084382912085466026629469045385337931801780431335092314994040148538752451689167}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{15} + \frac{851540357894793831266765315824702308302908055237950495576992242027972791559459476199543093883434975770607697617679555123153669733148283168999991652592}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{13} - \frac{13079214720650635325223427380557530943684688922037995375845475599870000578431733811610609394549656747656072649722492080539374741665060646376563287033}{88999294906932464785767301604646398949499970126870410645334234525230770380026449242677677130073321066494496615584508916958571195513579024443364604871} a^{11} + \frac{1913437330293385796630440695491875912040949816410746348821096583997119895139774575716487752069002502552873001335343986757271969761972300646303165191222}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{9} + \frac{388405747939226270158638028898805752866560387787633948969625157293924068911823373749515905526185481867008466162451480211257066543433657345530835859368}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{7} - \frac{5455940834529585216074438302069371467376125647783268238992190079710026043669426933644484730579252261832783999394719775338157820256974892604394924759416}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{5} - \frac{4032389517436401261676930718833121826902256014339602220465330514844018483659696549789677307554652111074301154572531481253585310123464290949992164474711}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a^{3} - \frac{5539948882536116087546916348225487598766962282150162969567572710949928212729684680368116088533691143379166093664256390904412916798552473891007869190116}{11302910453180423027792447303790092666586496206112542151957447784704307838263359053820064995519311775444801070179232632453738541830224536104307304818617} a$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $37$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{19}) \), 19.19.10842505080063916320800450434338728415281531281.1

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

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
2Data not computed
19Data not computed