↑

Properties

Label 39.39.2702763658...8889.1
Degree $39$
Signature $[39, 0]$
Discriminant $13^{74}$
Root discriminant $129.91$
Ramified prime $13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{39}$ (as 39T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 78, -1833, 7137, 161980, -427921, -3966105, 9129731, 41256709, -93363413, -206676184, 480801555, 581665721, -1417531106, -1007230666, 2610137920, 1129454313, -3171548939, -844358905, 2636840661, 427950341, -1537691324, -148392244, 639669732, 35251762, -191726951, -5687020, 41530255, 608296, -6470152, -41041, 714415, 1573, -54301, -26, 2691, 0, -78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^39 - 78*x^37 + 2691*x^35 - 26*x^34 - 54301*x^33 + 1573*x^32 + 714415*x^31 - 41041*x^30 - 6470152*x^29 + 608296*x^28 + 41530255*x^27 - 5687020*x^26 - 191726951*x^25 + 35251762*x^24 + 639669732*x^23 - 148392244*x^22 - 1537691324*x^21 + 427950341*x^20 + 2636840661*x^19 - 844358905*x^18 - 3171548939*x^17 + 1129454313*x^16 + 2610137920*x^15 - 1007230666*x^14 - 1417531106*x^13 + 581665721*x^12 + 480801555*x^11 - 206676184*x^10 - 93363413*x^9 + 41256709*x^8 + 9129731*x^7 - 3966105*x^6 - 427921*x^5 + 161980*x^4 + 7137*x^3 - 1833*x^2 + 78*x - 1)
 
gp: K = bnfinit(x^39 - 78*x^37 + 2691*x^35 - 26*x^34 - 54301*x^33 + 1573*x^32 + 714415*x^31 - 41041*x^30 - 6470152*x^29 + 608296*x^28 + 41530255*x^27 - 5687020*x^26 - 191726951*x^25 + 35251762*x^24 + 639669732*x^23 - 148392244*x^22 - 1537691324*x^21 + 427950341*x^20 + 2636840661*x^19 - 844358905*x^18 - 3171548939*x^17 + 1129454313*x^16 + 2610137920*x^15 - 1007230666*x^14 - 1417531106*x^13 + 581665721*x^12 + 480801555*x^11 - 206676184*x^10 - 93363413*x^9 + 41256709*x^8 + 9129731*x^7 - 3966105*x^6 - 427921*x^5 + 161980*x^4 + 7137*x^3 - 1833*x^2 + 78*x - 1, 1)
 

Normalized defining polynomial

\( x^{39} - 78 x^{37} + 2691 x^{35} - 26 x^{34} - 54301 x^{33} + 1573 x^{32} + 714415 x^{31} - 41041 x^{30} - 6470152 x^{29} + 608296 x^{28} + 41530255 x^{27} - 5687020 x^{26} - 191726951 x^{25} + 35251762 x^{24} + 639669732 x^{23} - 148392244 x^{22} - 1537691324 x^{21} + 427950341 x^{20} + 2636840661 x^{19} - 844358905 x^{18} - 3171548939 x^{17} + 1129454313 x^{16} + 2610137920 x^{15} - 1007230666 x^{14} - 1417531106 x^{13} + 581665721 x^{12} + 480801555 x^{11} - 206676184 x^{10} - 93363413 x^{9} + 41256709 x^{8} + 9129731 x^{7} - 3966105 x^{6} - 427921 x^{5} + 161980 x^{4} + 7137 x^{3} - 1833 x^{2} + 78 x - 1 \)

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

Invariants

Degree:  $39$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[39, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(27027636582498189040621249864144468324898507852136260989871841246090732111847218889=13^{74}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.91$
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:  $13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(169=13^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{169}(1,·)$, $\chi_{169}(3,·)$, $\chi_{169}(133,·)$, $\chi_{169}(9,·)$, $\chi_{169}(139,·)$, $\chi_{169}(14,·)$, $\chi_{169}(16,·)$, $\chi_{169}(146,·)$, $\chi_{169}(131,·)$, $\chi_{169}(22,·)$, $\chi_{169}(152,·)$, $\chi_{169}(27,·)$, $\chi_{169}(157,·)$, $\chi_{169}(159,·)$, $\chi_{169}(35,·)$, $\chi_{169}(165,·)$, $\chi_{169}(40,·)$, $\chi_{169}(42,·)$, $\chi_{169}(29,·)$, $\chi_{169}(48,·)$, $\chi_{169}(53,·)$, $\chi_{169}(55,·)$, $\chi_{169}(61,·)$, $\chi_{169}(66,·)$, $\chi_{169}(68,·)$, $\chi_{169}(74,·)$, $\chi_{169}(79,·)$, $\chi_{169}(81,·)$, $\chi_{169}(87,·)$, $\chi_{169}(92,·)$, $\chi_{169}(94,·)$, $\chi_{169}(144,·)$, $\chi_{169}(100,·)$, $\chi_{169}(105,·)$, $\chi_{169}(107,·)$, $\chi_{169}(113,·)$, $\chi_{169}(118,·)$, $\chi_{169}(120,·)$, $\chi_{169}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $\frac{1}{239} a^{35} - \frac{80}{239} a^{34} - \frac{54}{239} a^{33} + \frac{64}{239} a^{32} - \frac{36}{239} a^{31} - \frac{50}{239} a^{30} + \frac{61}{239} a^{29} + \frac{29}{239} a^{28} + \frac{106}{239} a^{27} + \frac{117}{239} a^{26} - \frac{20}{239} a^{25} + \frac{59}{239} a^{24} + \frac{63}{239} a^{23} + \frac{47}{239} a^{22} + \frac{31}{239} a^{21} - \frac{55}{239} a^{20} + \frac{83}{239} a^{19} - \frac{42}{239} a^{18} + \frac{102}{239} a^{17} + \frac{114}{239} a^{16} - \frac{1}{239} a^{15} + \frac{107}{239} a^{14} + \frac{117}{239} a^{13} + \frac{19}{239} a^{12} + \frac{44}{239} a^{11} - \frac{102}{239} a^{10} - \frac{34}{239} a^{9} - \frac{61}{239} a^{8} + \frac{20}{239} a^{7} + \frac{47}{239} a^{6} - \frac{9}{239} a^{5} + \frac{8}{239} a^{4} + \frac{49}{239} a^{3} + \frac{74}{239} a^{2} - \frac{73}{239} a + \frac{56}{239}$, $\frac{1}{669917} a^{36} + \frac{866}{669917} a^{35} - \frac{135962}{669917} a^{34} - \frac{151400}{669917} a^{33} + \frac{313131}{669917} a^{32} - \frac{232237}{669917} a^{31} + \frac{165471}{669917} a^{30} + \frac{328283}{669917} a^{29} - \frac{300846}{669917} a^{28} + \frac{259089}{669917} a^{27} + \frac{256452}{669917} a^{26} - \frac{63554}{669917} a^{25} + \frac{36518}{669917} a^{24} - \frac{330642}{669917} a^{23} + \frac{37084}{669917} a^{22} - \frac{86405}{669917} a^{21} - \frac{278997}{669917} a^{20} - \frac{17602}{669917} a^{19} - \frac{189961}{669917} a^{18} + \frac{205112}{669917} a^{17} + \frac{35904}{669917} a^{16} + \frac{28558}{669917} a^{15} - \frac{136227}{669917} a^{14} - \frac{222226}{669917} a^{13} + \frac{332}{669917} a^{12} + \frac{154808}{669917} a^{11} - \frac{270040}{669917} a^{10} + \frac{176183}{669917} a^{9} + \frac{203302}{669917} a^{8} - \frac{19990}{669917} a^{7} - \frac{62858}{669917} a^{6} - \frac{144258}{669917} a^{5} - \frac{130047}{669917} a^{4} + \frac{226156}{669917} a^{3} + \frac{72082}{669917} a^{2} - \frac{169382}{669917} a + \frac{119657}{669917}$, $\frac{1}{972049567} a^{37} - \frac{63}{972049567} a^{36} + \frac{665643}{972049567} a^{35} - \frac{126936784}{972049567} a^{34} - \frac{115255696}{972049567} a^{33} - \frac{293516289}{972049567} a^{32} - \frac{448181525}{972049567} a^{31} + \frac{395351837}{972049567} a^{30} - \frac{82028015}{972049567} a^{29} + \frac{101213}{4067153} a^{28} + \frac{413491459}{972049567} a^{27} + \frac{204176921}{972049567} a^{26} + \frac{386701233}{972049567} a^{25} - \frac{230238821}{972049567} a^{24} + \frac{275745433}{972049567} a^{23} - \frac{10633457}{972049567} a^{22} - \frac{199148310}{972049567} a^{21} + \frac{308167051}{972049567} a^{20} - \frac{274587287}{972049567} a^{19} - \frac{297417736}{972049567} a^{18} - \frac{336860577}{972049567} a^{17} + \frac{317250558}{972049567} a^{16} - \frac{282171271}{972049567} a^{15} + \frac{104581377}{972049567} a^{14} - \frac{194492834}{972049567} a^{13} + \frac{203871144}{972049567} a^{12} - \frac{165865982}{972049567} a^{11} - \frac{424597383}{972049567} a^{10} - \frac{224108004}{972049567} a^{9} + \frac{293957232}{972049567} a^{8} - \frac{309785114}{972049567} a^{7} - \frac{125519462}{972049567} a^{6} - \frac{41349516}{972049567} a^{5} + \frac{170064289}{972049567} a^{4} - \frac{102520580}{972049567} a^{3} + \frac{103302855}{972049567} a^{2} - \frac{225728201}{972049567} a - \frac{344788600}{972049567}$, $\frac{1}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{38} - \frac{971052844547018770541642618398936396046536970980225780077505919816576893483611970584361755}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{37} + \frac{393842214476860451957559160825572645066927113614885757854501266463717848982542671373959967306}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{36} - \frac{1626223028682619863780783905994956213007899379201303582270020771743571687594985708794561060832440}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{35} - \frac{475809498300979103032812888916736838894212271328450969295162473310631800598584591045311494107359717}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{34} + \frac{699984700718888682765985611945933269528499716600161976796330050588393348928823321636277873955155806}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{33} + \frac{231441985911380631237887401834245160238117427113171797128045867320412914189758220476466397688398908}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{32} - \frac{274257046520194639324095181152772865995890232670492313678442988252117912561564802172150539540994438}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{31} - \frac{404883784125629500072460622454490245509568415956751029586503900873937659071386115820158848922512765}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{30} - \frac{197386248155985361751897170869667831700045839109484346933681628900732082478756972099676292732628292}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{29} - \frac{278626626717207383670908127398027104260502311970671115173927696028433084479609502370210772486702151}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{28} + \frac{667981236204155202362023955356560302477358077722192347266233259051348301177812022898620169240099817}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{27} + \frac{1028507258094052034126750769722843694189483428208865565834632286799046345867697368864931644928275687}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{26} + \frac{395229060593361683183430550492878236026712879078401199404513490825236965644596515039139996677372819}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{25} + \frac{143113392131111296050771068394548124024833188268645826138609203408664900857872555151712741824726269}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{24} + \frac{715933642673330117289211747190131753483125223671989487514644489232478447191243585521619330539262205}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{23} + \frac{1118459419838989785439543582954618713248412194082368039760946874609277478949605557330670020908534498}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{22} - \frac{740058702720205171399522556157438807956754892222840835669371721774126939763892278219948013519851138}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{21} - \frac{750653854111231863020736652036859668457457605312449025110429137881801077751846649974062142534325774}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{20} + \frac{488811205594439448869542684431748394696012835531133616307661115671141274545092598953878285518492474}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{19} - \frac{874962402158993342651778807354997582515593394229995954091176422133062983242199785362607923846116234}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{18} - \frac{372262157171515309780928212107941157593757506051132975564781040607597388612098442651731827696045988}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{17} + \frac{14469398165464928566795850445022708638941837952458950168968884622634336361236650864597763193179119}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{16} - \frac{1213493993259068598287804260858224611229522243387457619035634071802124294645653500537172842313191158}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{15} + \frac{1006072659622951735673537107038991792906439226519849903141972339089612497180178995936236196307747261}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{14} + \frac{691122824758668047256503435538713987123045713294308166167211251232046807627241459514255528196427463}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{13} - \frac{1033429001624645438404755023798018143346830157435827572660906875222147806498956366136286607653322951}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{12} + \frac{51690442201253059254069564627515827546580934844195180148575809097281300610025984631942047336795201}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{11} + \frac{422486783668111886633446448398778259787423648873419938152346602034843964492188849036749551060734537}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{10} + \frac{1051355588079749177225533592884535020758039632990474723528210215091396453776906973422668208738841515}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{9} - \frac{74192352762506471860655012330120905355262509907424235571631384658980227550817942714653273023436950}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{8} + \frac{193202719157214332043040657366137238354900719988407600260907966007087756739918273711934548885791698}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{7} - \frac{829616175591123835748760264130511745292120567642767969574627467111198476560283004562116815987009242}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{6} - \frac{583142424840379530520551864902375381043883449537803643364197310640718109036592149331166641729525471}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{5} - \frac{1219807779865181837640862570620404420754395286560499508148600679767982280313192386877987800246359264}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{4} + \frac{1151257576414222529434317718968142319931827345448530914973322309781034691074775093560735843831122755}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{3} + \frac{401838266537914076950471362225902156300808929263183132950589933733982623150799268332831857416392545}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a^{2} + \frac{797604595624038181531194517511636678780909491780639542893108694056975906587938640199297168888266157}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779} a + \frac{1098775328186896977127036766520899034735129816587275813068104251765302256718715345286581837807777056}{2463831950877657452333716197413652317132260985717413099466878409150422997441710293154961360842050779}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $38$
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:  \( 80306583605632290000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{39}$ (as 39T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 39
The 39 conjugacy class representatives for $C_{39}$
Character table for $C_{39}$ is not computed

Intermediate fields

3.3.169.1, 13.13.542800770374370512771595361.1

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 $39$ $39$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{3}$ $39$ $39$ R $39$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{13}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{13}$ $39$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{3}$ $39$ $39$ $39$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{3}$ ${\href{/LocalNumberField/53.13.0.1}{13} }^{3}$ $39$

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
13Data not computed