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Properties

Label 39.39.1112962024...6929.1
Degree $39$
Signature $[39, 0]$
Discriminant $7^{26}\cdot 53^{36}$
Root discriminant $142.90$
Ramified primes $7, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{39}$ (as 39T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17513, -1233420, 11997840, 23663051, -294156298, -133518320, 2936606920, -163687701, -15459668136, 5015492400, 47511398377, -24031853409, -89832821948, 58560262932, 107935481554, -85491405725, -83924117638, 80500790810, 41988060380, -51075711901, -12656284391, 22412813436, 1579984745, -6887826619, 368114118, 1482194078, -219014481, -220059875, 50319569, 21722262, -6805710, -1308560, 574075, 36492, -29490, 433, 838, -55, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^39 - 10*x^38 - 55*x^37 + 838*x^36 + 433*x^35 - 29490*x^34 + 36492*x^33 + 574075*x^32 - 1308560*x^31 - 6805710*x^30 + 21722262*x^29 + 50319569*x^28 - 220059875*x^27 - 219014481*x^26 + 1482194078*x^25 + 368114118*x^24 - 6887826619*x^23 + 1579984745*x^22 + 22412813436*x^21 - 12656284391*x^20 - 51075711901*x^19 + 41988060380*x^18 + 80500790810*x^17 - 83924117638*x^16 - 85491405725*x^15 + 107935481554*x^14 + 58560262932*x^13 - 89832821948*x^12 - 24031853409*x^11 + 47511398377*x^10 + 5015492400*x^9 - 15459668136*x^8 - 163687701*x^7 + 2936606920*x^6 - 133518320*x^5 - 294156298*x^4 + 23663051*x^3 + 11997840*x^2 - 1233420*x - 17513)
 
gp: K = bnfinit(x^39 - 10*x^38 - 55*x^37 + 838*x^36 + 433*x^35 - 29490*x^34 + 36492*x^33 + 574075*x^32 - 1308560*x^31 - 6805710*x^30 + 21722262*x^29 + 50319569*x^28 - 220059875*x^27 - 219014481*x^26 + 1482194078*x^25 + 368114118*x^24 - 6887826619*x^23 + 1579984745*x^22 + 22412813436*x^21 - 12656284391*x^20 - 51075711901*x^19 + 41988060380*x^18 + 80500790810*x^17 - 83924117638*x^16 - 85491405725*x^15 + 107935481554*x^14 + 58560262932*x^13 - 89832821948*x^12 - 24031853409*x^11 + 47511398377*x^10 + 5015492400*x^9 - 15459668136*x^8 - 163687701*x^7 + 2936606920*x^6 - 133518320*x^5 - 294156298*x^4 + 23663051*x^3 + 11997840*x^2 - 1233420*x - 17513, 1)
 

Normalized defining polynomial

\( x^{39} - 10 x^{38} - 55 x^{37} + 838 x^{36} + 433 x^{35} - 29490 x^{34} + 36492 x^{33} + 574075 x^{32} - 1308560 x^{31} - 6805710 x^{30} + 21722262 x^{29} + 50319569 x^{28} - 220059875 x^{27} - 219014481 x^{26} + 1482194078 x^{25} + 368114118 x^{24} - 6887826619 x^{23} + 1579984745 x^{22} + 22412813436 x^{21} - 12656284391 x^{20} - 51075711901 x^{19} + 41988060380 x^{18} + 80500790810 x^{17} - 83924117638 x^{16} - 85491405725 x^{15} + 107935481554 x^{14} + 58560262932 x^{13} - 89832821948 x^{12} - 24031853409 x^{11} + 47511398377 x^{10} + 5015492400 x^{9} - 15459668136 x^{8} - 163687701 x^{7} + 2936606920 x^{6} - 133518320 x^{5} - 294156298 x^{4} + 23663051 x^{3} + 11997840 x^{2} - 1233420 x - 17513 \)

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:  \(1112962024555065990379787974028986797706599025588599389261176471461970163669515376929=7^{26}\cdot 53^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $142.90$
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:  $7, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(371=7\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{371}(256,·)$, $\chi_{371}(1,·)$, $\chi_{371}(130,·)$, $\chi_{371}(261,·)$, $\chi_{371}(134,·)$, $\chi_{371}(44,·)$, $\chi_{371}(142,·)$, $\chi_{371}(15,·)$, $\chi_{371}(16,·)$, $\chi_{371}(275,·)$, $\chi_{371}(148,·)$, $\chi_{371}(281,·)$, $\chi_{371}(155,·)$, $\chi_{371}(289,·)$, $\chi_{371}(36,·)$, $\chi_{371}(169,·)$, $\chi_{371}(172,·)$, $\chi_{371}(46,·)$, $\chi_{371}(309,·)$, $\chi_{371}(183,·)$, $\chi_{371}(312,·)$, $\chi_{371}(319,·)$, $\chi_{371}(331,·)$, $\chi_{371}(333,·)$, $\chi_{371}(205,·)$, $\chi_{371}(81,·)$, $\chi_{371}(100,·)$, $\chi_{371}(95,·)$, $\chi_{371}(225,·)$, $\chi_{371}(354,·)$, $\chi_{371}(99,·)$, $\chi_{371}(228,·)$, $\chi_{371}(102,·)$, $\chi_{371}(107,·)$, $\chi_{371}(365,·)$, $\chi_{371}(240,·)$, $\chi_{371}(116,·)$, $\chi_{371}(121,·)$, $\chi_{371}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{83} a^{31} - \frac{11}{83} a^{30} - \frac{37}{83} a^{29} + \frac{37}{83} a^{28} + \frac{11}{83} a^{27} - \frac{9}{83} a^{26} + \frac{11}{83} a^{25} - \frac{27}{83} a^{24} + \frac{16}{83} a^{23} - \frac{41}{83} a^{22} - \frac{3}{83} a^{21} - \frac{3}{83} a^{20} - \frac{8}{83} a^{19} + \frac{6}{83} a^{18} - \frac{13}{83} a^{17} + \frac{13}{83} a^{16} - \frac{39}{83} a^{15} + \frac{10}{83} a^{13} - \frac{27}{83} a^{12} - \frac{1}{83} a^{11} + \frac{22}{83} a^{10} - \frac{33}{83} a^{9} - \frac{18}{83} a^{8} - \frac{41}{83} a^{7} + \frac{27}{83} a^{6} + \frac{30}{83} a^{5} - \frac{28}{83} a^{4} + \frac{12}{83} a^{3} - \frac{27}{83} a^{2} + \frac{4}{83} a$, $\frac{1}{83} a^{32} + \frac{8}{83} a^{30} - \frac{38}{83} a^{29} + \frac{3}{83} a^{28} + \frac{29}{83} a^{27} - \frac{5}{83} a^{26} + \frac{11}{83} a^{25} - \frac{32}{83} a^{24} - \frac{31}{83} a^{23} - \frac{39}{83} a^{22} - \frac{36}{83} a^{21} - \frac{41}{83} a^{20} + \frac{1}{83} a^{19} - \frac{30}{83} a^{18} + \frac{36}{83} a^{17} + \frac{21}{83} a^{16} - \frac{14}{83} a^{15} + \frac{10}{83} a^{14} + \frac{34}{83} a^{12} + \frac{11}{83} a^{11} - \frac{40}{83} a^{10} + \frac{34}{83} a^{9} + \frac{10}{83} a^{8} - \frac{9}{83} a^{7} - \frac{5}{83} a^{6} - \frac{30}{83} a^{5} + \frac{36}{83} a^{4} + \frac{22}{83} a^{3} + \frac{39}{83} a^{2} - \frac{39}{83} a$, $\frac{1}{1909} a^{33} - \frac{2}{1909} a^{32} - \frac{8}{1909} a^{31} - \frac{44}{1909} a^{30} + \frac{754}{1909} a^{29} - \frac{901}{1909} a^{28} - \frac{73}{1909} a^{27} - \frac{250}{1909} a^{26} - \frac{396}{1909} a^{25} + \frac{133}{1909} a^{24} + \frac{763}{1909} a^{23} - \frac{879}{1909} a^{22} + \frac{328}{1909} a^{21} + \frac{214}{1909} a^{20} + \frac{511}{1909} a^{19} + \frac{7}{23} a^{18} + \frac{74}{1909} a^{17} + \frac{21}{83} a^{16} + \frac{579}{1909} a^{15} - \frac{767}{1909} a^{14} - \frac{541}{1909} a^{13} - \frac{123}{1909} a^{12} - \frac{710}{1909} a^{11} + \frac{426}{1909} a^{10} - \frac{28}{1909} a^{9} + \frac{674}{1909} a^{8} + \frac{5}{1909} a^{7} + \frac{710}{1909} a^{6} + \frac{114}{1909} a^{5} - \frac{681}{1909} a^{4} + \frac{633}{1909} a^{3} - \frac{349}{1909} a^{2} + \frac{678}{1909} a - \frac{3}{23}$, $\frac{1}{1909} a^{34} + \frac{11}{1909} a^{32} + \frac{9}{1909} a^{31} + \frac{91}{1909} a^{30} - \frac{911}{1909} a^{29} + \frac{9}{23} a^{28} - \frac{879}{1909} a^{27} + \frac{277}{1909} a^{26} + \frac{353}{1909} a^{25} + \frac{339}{1909} a^{24} - \frac{871}{1909} a^{23} + \frac{571}{1909} a^{22} - \frac{165}{1909} a^{21} - \frac{211}{1909} a^{20} - \frac{835}{1909} a^{19} - \frac{949}{1909} a^{18} + \frac{562}{1909} a^{17} - \frac{893}{1909} a^{16} - \frac{31}{83} a^{15} + \frac{64}{1909} a^{14} - \frac{515}{1909} a^{13} - \frac{128}{1909} a^{12} - \frac{810}{1909} a^{11} - \frac{487}{1909} a^{10} - \frac{877}{1909} a^{9} + \frac{341}{1909} a^{8} - \frac{407}{1909} a^{7} - \frac{536}{1909} a^{6} + \frac{927}{1909} a^{5} + \frac{76}{1909} a^{4} + \frac{342}{1909} a^{3} + \frac{923}{1909} a^{2} + \frac{486}{1909} a - \frac{6}{23}$, $\frac{1}{1909} a^{35} + \frac{8}{1909} a^{32} - \frac{5}{1909} a^{31} - \frac{496}{1909} a^{30} + \frac{135}{1909} a^{29} + \frac{246}{1909} a^{28} + \frac{298}{1909} a^{27} - \frac{853}{1909} a^{26} + \frac{509}{1909} a^{25} - \frac{448}{1909} a^{24} - \frac{508}{1909} a^{23} + \frac{764}{1909} a^{22} - \frac{530}{1909} a^{21} + \frac{215}{1909} a^{20} + \frac{606}{1909} a^{19} - \frac{516}{1909} a^{18} - \frac{143}{1909} a^{17} + \frac{28}{83} a^{16} - \frac{716}{1909} a^{15} + \frac{56}{1909} a^{14} + \frac{165}{1909} a^{13} + \frac{911}{1909} a^{12} - \frac{382}{1909} a^{11} + \frac{854}{1909} a^{10} + \frac{212}{1909} a^{9} - \frac{921}{1909} a^{8} - \frac{476}{1909} a^{7} - \frac{282}{1909} a^{6} - \frac{281}{1909} a^{5} + \frac{703}{1909} a^{4} + \frac{791}{1909} a^{3} + \frac{760}{1909} a^{2} - \frac{159}{1909} a + \frac{10}{23}$, $\frac{1}{435105007} a^{36} - \frac{67040}{435105007} a^{35} - \frac{30321}{435105007} a^{34} + \frac{31397}{435105007} a^{33} - \frac{1039630}{435105007} a^{32} - \frac{1023346}{435105007} a^{31} - \frac{9195197}{435105007} a^{30} + \frac{215300484}{435105007} a^{29} + \frac{169782691}{435105007} a^{28} - \frac{68429824}{435105007} a^{27} - \frac{140508366}{435105007} a^{26} - \frac{137646052}{435105007} a^{25} - \frac{106071776}{435105007} a^{24} + \frac{2539541}{435105007} a^{23} + \frac{71864732}{435105007} a^{22} - \frac{99371206}{435105007} a^{21} - \frac{153190109}{435105007} a^{20} - \frac{187434909}{435105007} a^{19} + \frac{82001253}{435105007} a^{18} + \frac{210533319}{435105007} a^{17} + \frac{64883935}{435105007} a^{16} + \frac{24542742}{435105007} a^{15} + \frac{96463632}{435105007} a^{14} + \frac{26757242}{435105007} a^{13} - \frac{143868370}{435105007} a^{12} + \frac{197499317}{435105007} a^{11} + \frac{108238735}{435105007} a^{10} + \frac{124499006}{435105007} a^{9} - \frac{45506248}{435105007} a^{8} + \frac{193725523}{435105007} a^{7} - \frac{73753079}{435105007} a^{6} + \frac{202595629}{435105007} a^{5} + \frac{48451495}{435105007} a^{4} + \frac{30075163}{435105007} a^{3} - \frac{131065835}{435105007} a^{2} - \frac{177748284}{435105007} a + \frac{876341}{5242229}$, $\frac{1}{435105007} a^{37} + \frac{21716}{435105007} a^{35} - \frac{71129}{435105007} a^{34} + \frac{85960}{435105007} a^{33} + \frac{351085}{435105007} a^{32} - \frac{2194501}{435105007} a^{31} + \frac{146020964}{435105007} a^{30} + \frac{3523052}{435105007} a^{29} - \frac{74177341}{435105007} a^{28} + \frac{40742800}{435105007} a^{27} + \frac{79769131}{435105007} a^{26} + \frac{79451432}{435105007} a^{25} + \frac{130042661}{435105007} a^{24} - \frac{189693850}{435105007} a^{23} + \frac{94922077}{435105007} a^{22} - \frac{80550796}{435105007} a^{21} - \frac{113190639}{435105007} a^{20} + \frac{200126464}{435105007} a^{19} + \frac{202374318}{435105007} a^{18} - \frac{20992884}{435105007} a^{17} - \frac{95400233}{435105007} a^{16} - \frac{158070074}{435105007} a^{15} - \frac{99808568}{435105007} a^{14} - \frac{4630692}{18917609} a^{13} - \frac{149613492}{435105007} a^{12} + \frac{146431275}{435105007} a^{11} - \frac{61982782}{435105007} a^{10} + \frac{94721748}{435105007} a^{9} - \frac{183482420}{435105007} a^{8} - \frac{153104798}{435105007} a^{7} + \frac{70786412}{435105007} a^{6} - \frac{17337296}{435105007} a^{5} + \frac{106772495}{435105007} a^{4} - \frac{172615444}{435105007} a^{3} - \frac{169930300}{435105007} a^{2} + \frac{72535567}{435105007} a + \frac{12314}{5242229}$, $\frac{1}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{38} - \frac{386028661199537242156773090573609557200472100022646907920103073504241323080912233569676254552090695505105110657332860711}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{37} + \frac{148860925459295549847669431003512011588281994613416997278893908953548933812529785854047989263836365096979768483235272670}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{36} - \frac{45845272755150023310755794672958271572978127938285321248333636153168472300242220086543854059201726211498840326142676158066935}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{35} + \frac{6757680518276387751385531212839125161592965241838450539728142096034548350288298818930007005203757411451375441793414467311656}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{34} + \frac{94622104656136821869288287443987997698304954192551711764976292175020506038356641329548627395536183760510444187758437291339384}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{33} + \frac{2069660538156451414674737789586971624934676657141011263513150050824920177840507615538616715434372762868862428909215652157981063}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{32} + \frac{247542604556212972623893648765817893364189974078337909420976190105688444516855619368719311564973212352036687571822667771180533}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{31} + \frac{7201509929244065361858116052770485313540426011534647949241685185534800997041568668255578364241118604450928406170370488893633467}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{30} - \frac{60861966324107810823565059662462458036120130331007429009848394109647562495733975690627028629418351553461638980835430768261215270}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{29} + \frac{35572639260886930388130793071399723805476117088688811292622302013703111388748318791760284954698362698451824806305782186962513366}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{28} - \frac{144347520910282127451033242525068307515545515709432890419363373021087889114006563682131319596177546982627470858534338939494136321}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{27} - \frac{152070195601709351842590950760022109545784931320840084200060952352037931568665980849335853524265805837256480472412294490186607212}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{26} - \frac{138039055655184797563712812634819975875541552684488336832471105413990364484177393886356885361423731859430469515071182263040334943}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{25} - \frac{95745151511521575966882006680422322325473261612308769380384264291958645746251700256161083544052076197382621660577531903902517170}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{24} + \frac{25942047184467590738451440771065199548445641269852678479069473628065142844147679958841938070953525504534906301773880961620898644}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{23} + \frac{80457967657079706700217168337259448645994609374420995763358435340582280315904191251605699221846983156460697181855779153483103613}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{22} - \frac{6018938502782253335973459295977662588612636673580406304750005110520048649510791247689802487363680686779562023098366700614805503}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{21} + \frac{135469047472425976532150606010729840601990033531222446837186469791185877681635423828100714599331699216523234490149166852890935100}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{20} + \frac{91992599639869311614048827355970408483555175359713820747005902205558826325222597460430105346366171084118104569307111592242987499}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{19} - \frac{92752365357979388954418658685713910831910184374775220838230748568801743341675221468604901033385526710312660765936287080784539235}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{18} + \frac{6860473780346965853553832020871749558475539954461455282598911915295738415331649817665324107706290601803960231207921012838992103}{15751482884249243360609021176572836648214998463027380112496703312730900831311897731684525871772560459425425841669501839490762477} a^{17} - \frac{122353726468476294827479640527051656966546985980568444679349974561997867839718055051398964026938022370880852897270619623186043281}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{16} - \frac{105228085601499298450943462752222624039098470400072957958461402478312639310634650423187950903322150352442681403306830382909238693}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{15} + \frac{139649764316299193165492663180306540445271054518523270009754219557618218498296269643048012416562633135933292682759323147371963370}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{14} - \frac{4527077839293823612659658775679045066448218066090521552969636218762732171858298528388277281157346096032970911917822832851615087}{15751482884249243360609021176572836648214998463027380112496703312730900831311897731684525871772560459425425841669501839490762477} a^{13} - \frac{165523921816389588806224514804534737344778750721842582056865230675088378491677258077395023411740389532466484906568019031828340575}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{12} - \frac{96898841984212915317873800178775470311676124106040929742490085104673842576801967000263463215263586817191562724964068361054477467}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{11} - \frac{127652532642691913631313207173412528485981270738034785255931925856388356244629837444350022735874288127482711514863363700763243837}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{10} - \frac{174978766152196609096826955441854723980731647520524741993140377555183683811412737623334473759422136295287763225603566928274470908}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{9} + \frac{25663112635242618488688543065634260804011914478255743731388281628669982513720907861047914634332738878159831013368625060882483424}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{8} - \frac{83940461597517706385932098923571659258471372956152496955400559158207891032407968484907688829930239505934576367805405177502118332}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{7} - \frac{21759027806119535735385419134944587064713562179967779230229934085425343796176157860716329186030335969768901488262539854696495213}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{6} + \frac{99757131108890260495305235201466432840890686734463579137647853587905856218649082240317262262665325285161464130736641765341898914}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{5} - \frac{72510577245208245142160372848061169019501270255840804256314227155269002331909131037776340468176520111764050589933693462427858097}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{4} + \frac{123308910684890109298971122131078461355581017206700379499584944087091420086496355307885299257951614620534209101886186535348822268}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{3} - \frac{79115959259784225848405067775661323642309634557989499845222205681921040005446979575521656922535872112354810314974389844206322626}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a^{2} + \frac{130933577929662047065522763855639703611650501637725809440744684549454397835842538113971033232221140874342992928039209461562402026}{362284106337732597294007487061175242908944964649629742587424176192810719120173647828744095050768890566784794358398542308287536971} a - \frac{1790644148395849042308604160714169562038653699322634294129776696340017116302677530662784195057173113701678979398439049263932483}{4364868751057019244506114301941870396493312827103972802258122604732659266508116238900531265671914344178130052510825810943223337}$

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:  \( 688762923023490700000000000000 \) (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

\(\Q(\zeta_{7})^+\), 13.13.491258904256726154641.1

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 $39$ $39$ $39$ R $39$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{3}$ $39$ $39$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{13}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{3}$ $39$ $39$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{3}$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{3}$ $39$ R $39$

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
7Data not computed
53Data not computed