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Properties

Label 39.39.1892790059...8089.1
Degree $39$
Signature $[39, 0]$
Discriminant $13^{26}\cdot 79^{36}$
Root discriminant $312.09$
Ramified primes $13, 79$
Class number Not computed
Class group Not computed
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![2198272487, -3009281044, -106661114208, 154439956714, 1267027871999, -1906724813303, -6624071325699, 10144928480062, 18404264085338, -27975777878380, -30444285645115, 44089907326138, 33478011120120, -43445251987869, -26161675308530, 28441505512102, 14959051199327, -12855958984538, -6305779770878, 4113180518714, 1958725720367, -947228275523, -447662704038, 158901467338, 75169877668, -19595581406, -9246368917, 1790293625, 827367236, -122075225, -53084512, 6246879, 2376429, -238767, -70508, 6574, 1248, -117, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^39 - 10*x^38 - 117*x^37 + 1248*x^36 + 6574*x^35 - 70508*x^34 - 238767*x^33 + 2376429*x^32 + 6246879*x^31 - 53084512*x^30 - 122075225*x^29 + 827367236*x^28 + 1790293625*x^27 - 9246368917*x^26 - 19595581406*x^25 + 75169877668*x^24 + 158901467338*x^23 - 447662704038*x^22 - 947228275523*x^21 + 1958725720367*x^20 + 4113180518714*x^19 - 6305779770878*x^18 - 12855958984538*x^17 + 14959051199327*x^16 + 28441505512102*x^15 - 26161675308530*x^14 - 43445251987869*x^13 + 33478011120120*x^12 + 44089907326138*x^11 - 30444285645115*x^10 - 27975777878380*x^9 + 18404264085338*x^8 + 10144928480062*x^7 - 6624071325699*x^6 - 1906724813303*x^5 + 1267027871999*x^4 + 154439956714*x^3 - 106661114208*x^2 - 3009281044*x + 2198272487)
 
gp: K = bnfinit(x^39 - 10*x^38 - 117*x^37 + 1248*x^36 + 6574*x^35 - 70508*x^34 - 238767*x^33 + 2376429*x^32 + 6246879*x^31 - 53084512*x^30 - 122075225*x^29 + 827367236*x^28 + 1790293625*x^27 - 9246368917*x^26 - 19595581406*x^25 + 75169877668*x^24 + 158901467338*x^23 - 447662704038*x^22 - 947228275523*x^21 + 1958725720367*x^20 + 4113180518714*x^19 - 6305779770878*x^18 - 12855958984538*x^17 + 14959051199327*x^16 + 28441505512102*x^15 - 26161675308530*x^14 - 43445251987869*x^13 + 33478011120120*x^12 + 44089907326138*x^11 - 30444285645115*x^10 - 27975777878380*x^9 + 18404264085338*x^8 + 10144928480062*x^7 - 6624071325699*x^6 - 1906724813303*x^5 + 1267027871999*x^4 + 154439956714*x^3 - 106661114208*x^2 - 3009281044*x + 2198272487, 1)
 

Normalized defining polynomial

\( x^{39} - 10 x^{38} - 117 x^{37} + 1248 x^{36} + 6574 x^{35} - 70508 x^{34} - 238767 x^{33} + 2376429 x^{32} + 6246879 x^{31} - 53084512 x^{30} - 122075225 x^{29} + 827367236 x^{28} + 1790293625 x^{27} - 9246368917 x^{26} - 19595581406 x^{25} + 75169877668 x^{24} + 158901467338 x^{23} - 447662704038 x^{22} - 947228275523 x^{21} + 1958725720367 x^{20} + 4113180518714 x^{19} - 6305779770878 x^{18} - 12855958984538 x^{17} + 14959051199327 x^{16} + 28441505512102 x^{15} - 26161675308530 x^{14} - 43445251987869 x^{13} + 33478011120120 x^{12} + 44089907326138 x^{11} - 30444285645115 x^{10} - 27975777878380 x^{9} + 18404264085338 x^{8} + 10144928480062 x^{7} - 6624071325699 x^{6} - 1906724813303 x^{5} + 1267027871999 x^{4} + 154439956714 x^{3} - 106661114208 x^{2} - 3009281044 x + 2198272487 \)

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:  \(18927900593212290278037469352150105186673660971663667045096692101161408140762563765836642973478089=13^{26}\cdot 79^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $312.09$
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:  $13, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1027=13\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{1027}(640,·)$, $\chi_{1027}(1,·)$, $\chi_{1027}(131,·)$, $\chi_{1027}(776,·)$, $\chi_{1027}(599,·)$, $\chi_{1027}(653,·)$, $\chi_{1027}(144,·)$, $\chi_{1027}(146,·)$, $\chi_{1027}(22,·)$, $\chi_{1027}(536,·)$, $\chi_{1027}(159,·)$, $\chi_{1027}(289,·)$, $\chi_{1027}(302,·)$, $\chi_{1027}(828,·)$, $\chi_{1027}(958,·)$, $\chi_{1027}(575,·)$, $\chi_{1027}(100,·)$, $\chi_{1027}(196,·)$, $\chi_{1027}(326,·)$, $\chi_{1027}(417,·)$, $\chi_{1027}(887,·)$, $\chi_{1027}(204,·)$, $\chi_{1027}(354,·)$, $\chi_{1027}(334,·)$, $\chi_{1027}(854,·)$, $\chi_{1027}(87,·)$, $\chi_{1027}(729,·)$, $\chi_{1027}(222,·)$, $\chi_{1027}(482,·)$, $\chi_{1027}(484,·)$, $\chi_{1027}(620,·)$, $\chi_{1027}(495,·)$, $\chi_{1027}(1010,·)$, $\chi_{1027}(757,·)$, $\chi_{1027}(1015,·)$, $\chi_{1027}(633,·)$, $\chi_{1027}(378,·)$, $\chi_{1027}(763,·)$, $\chi_{1027}(380,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{23} a^{30} - \frac{4}{23} a^{28} - \frac{7}{23} a^{27} - \frac{10}{23} a^{26} + \frac{10}{23} a^{25} + \frac{1}{23} a^{23} + \frac{1}{23} a^{22} - \frac{4}{23} a^{21} + \frac{3}{23} a^{20} - \frac{6}{23} a^{19} + \frac{1}{23} a^{18} + \frac{7}{23} a^{17} + \frac{6}{23} a^{16} - \frac{2}{23} a^{15} + \frac{11}{23} a^{14} - \frac{11}{23} a^{13} - \frac{9}{23} a^{12} + \frac{7}{23} a^{11} - \frac{3}{23} a^{10} + \frac{6}{23} a^{9} + \frac{8}{23} a^{8} + \frac{2}{23} a^{7} - \frac{4}{23} a^{6} + \frac{10}{23} a^{5} + \frac{6}{23} a^{3} - \frac{4}{23} a^{2} + \frac{1}{23} a + \frac{5}{23}$, $\frac{1}{2369} a^{31} + \frac{43}{2369} a^{30} - \frac{188}{2369} a^{29} - \frac{984}{2369} a^{28} - \frac{817}{2369} a^{27} - \frac{1156}{2369} a^{26} + \frac{154}{2369} a^{25} - \frac{597}{2369} a^{24} - \frac{1106}{2369} a^{23} - \frac{9}{23} a^{22} - \frac{169}{2369} a^{21} - \frac{245}{2369} a^{20} - \frac{464}{2369} a^{19} - \frac{456}{2369} a^{18} - \frac{498}{2369} a^{17} - \frac{940}{2369} a^{16} + \frac{730}{2369} a^{15} + \frac{94}{2369} a^{14} + \frac{438}{2369} a^{13} + \frac{1069}{2369} a^{12} - \frac{323}{2369} a^{11} + \frac{153}{2369} a^{10} - \frac{1183}{2369} a^{9} - \frac{1080}{2369} a^{8} + \frac{565}{2369} a^{7} + \frac{689}{2369} a^{6} - \frac{766}{2369} a^{5} - \frac{1167}{2369} a^{4} + \frac{599}{2369} a^{3} - \frac{332}{2369} a^{2} - \frac{550}{2369} a - \frac{521}{2369}$, $\frac{1}{2369} a^{32} + \frac{1}{103} a^{30} - \frac{7}{2369} a^{29} + \frac{89}{2369} a^{28} + \frac{603}{2369} a^{27} + \frac{834}{2369} a^{26} - \frac{833}{2369} a^{25} + \frac{875}{2369} a^{24} - \frac{46}{103} a^{23} - \frac{890}{2369} a^{22} + \frac{1151}{2369} a^{21} - \frac{332}{2369} a^{20} + \frac{29}{2369} a^{19} - \frac{151}{2369} a^{18} - \frac{641}{2369} a^{17} - \frac{977}{2369} a^{16} + \frac{119}{2369} a^{15} + \frac{104}{2369} a^{14} - \frac{152}{2369} a^{13} - \frac{867}{2369} a^{12} + \frac{34}{2369} a^{11} + \frac{272}{2369} a^{10} + \frac{555}{2369} a^{9} - \frac{478}{2369} a^{8} - \frac{534}{2369} a^{7} - \frac{729}{2369} a^{6} + \frac{11}{103} a^{5} + \frac{1031}{2369} a^{4} + \frac{485}{2369} a^{3} + \frac{748}{2369} a^{2} - \frac{870}{2369} a - \frac{463}{2369}$, $\frac{1}{2369} a^{33} + \frac{34}{2369} a^{30} - \frac{325}{2369} a^{29} + \frac{163}{2369} a^{28} + \frac{570}{2369} a^{27} - \frac{1128}{2369} a^{26} + \frac{526}{2369} a^{25} + \frac{36}{103} a^{24} - \frac{481}{2369} a^{23} - \frac{188}{2369} a^{22} - \frac{565}{2369} a^{21} - \frac{722}{2369} a^{20} - \frac{397}{2369} a^{19} - \frac{968}{2369} a^{18} + \frac{48}{103} a^{17} - \frac{509}{2369} a^{16} + \frac{2}{23} a^{15} - \frac{20}{103} a^{14} - \frac{950}{2369} a^{13} - \frac{657}{2369} a^{12} + \frac{697}{2369} a^{11} + \frac{1053}{2369} a^{10} - \frac{255}{2369} a^{9} - \frac{620}{2369} a^{8} + \frac{181}{2369} a^{7} - \frac{762}{2369} a^{6} + \frac{521}{2369} a^{5} - \frac{1102}{2369} a^{4} + \frac{258}{2369} a^{3} + \frac{277}{2369} a^{2} - \frac{997}{2369} a + \frac{550}{2369}$, $\frac{1}{371933} a^{34} - \frac{1}{16171} a^{33} + \frac{36}{371933} a^{32} - \frac{38}{371933} a^{31} - \frac{1109}{371933} a^{30} - \frac{73838}{371933} a^{29} + \frac{106820}{371933} a^{28} + \frac{36218}{371933} a^{27} + \frac{112328}{371933} a^{26} + \frac{88764}{371933} a^{25} - \frac{49277}{371933} a^{24} - \frac{54289}{371933} a^{23} - \frac{80090}{371933} a^{22} - \frac{26102}{371933} a^{21} + \frac{151883}{371933} a^{20} + \frac{71661}{371933} a^{19} + \frac{107517}{371933} a^{18} - \frac{132292}{371933} a^{17} + \frac{107766}{371933} a^{16} - \frac{1698}{16171} a^{15} + \frac{19687}{371933} a^{14} - \frac{137870}{371933} a^{13} + \frac{27005}{371933} a^{12} - \frac{145204}{371933} a^{11} - \frac{96459}{371933} a^{10} + \frac{36344}{371933} a^{9} + \frac{152346}{371933} a^{8} + \frac{141068}{371933} a^{7} + \frac{77640}{371933} a^{6} - \frac{46984}{371933} a^{5} + \frac{23556}{371933} a^{4} - \frac{67478}{371933} a^{3} - \frac{67467}{371933} a^{2} - \frac{91530}{371933} a + \frac{31369}{371933}$, $\frac{1}{117902761} a^{35} - \frac{94}{117902761} a^{34} - \frac{10263}{117902761} a^{33} - \frac{16567}{117902761} a^{32} + \frac{12893}{117902761} a^{31} + \frac{2383608}{117902761} a^{30} - \frac{8049376}{117902761} a^{29} + \frac{11921568}{117902761} a^{28} - \frac{51954185}{117902761} a^{27} + \frac{30672}{117902761} a^{26} + \frac{40711113}{117902761} a^{25} - \frac{1234536}{117902761} a^{24} + \frac{58674660}{117902761} a^{23} - \frac{47028912}{117902761} a^{22} + \frac{31973285}{117902761} a^{21} + \frac{22508854}{117902761} a^{20} + \frac{46451844}{117902761} a^{19} + \frac{26419024}{117902761} a^{18} - \frac{479043}{5126207} a^{17} - \frac{7048624}{117902761} a^{16} + \frac{8750985}{117902761} a^{15} + \frac{5228070}{117902761} a^{14} + \frac{91764}{1144687} a^{13} - \frac{12942502}{117902761} a^{12} - \frac{44145085}{117902761} a^{11} + \frac{34601242}{117902761} a^{10} - \frac{53890009}{117902761} a^{9} - \frac{13512645}{117902761} a^{8} - \frac{58842275}{117902761} a^{7} + \frac{486230}{1144687} a^{6} + \frac{39412743}{117902761} a^{5} - \frac{29578409}{117902761} a^{4} - \frac{8305959}{117902761} a^{3} - \frac{2461829}{117902761} a^{2} - \frac{2397492}{117902761} a + \frac{20138079}{117902761}$, $\frac{1}{6424167738607} a^{36} + \frac{5794}{6424167738607} a^{35} - \frac{430595}{6424167738607} a^{34} - \frac{1261994620}{6424167738607} a^{33} - \frac{39723063}{279311640809} a^{32} + \frac{1136901340}{6424167738607} a^{31} - \frac{117156623376}{6424167738607} a^{30} + \frac{2043852574507}{6424167738607} a^{29} + \frac{1331600725420}{6424167738607} a^{28} + \frac{2864008903449}{6424167738607} a^{27} + \frac{946994328726}{6424167738607} a^{26} - \frac{2455694162924}{6424167738607} a^{25} + \frac{1979771322857}{6424167738607} a^{24} + \frac{1352314283770}{6424167738607} a^{23} + \frac{133985228962}{279311640809} a^{22} - \frac{2549816911736}{6424167738607} a^{21} + \frac{1995941401527}{6424167738607} a^{20} - \frac{389321732725}{6424167738607} a^{19} + \frac{2823698313594}{6424167738607} a^{18} + \frac{2716874164672}{6424167738607} a^{17} + \frac{2745299819512}{6424167738607} a^{16} + \frac{2994763049267}{6424167738607} a^{15} + \frac{69834492875}{279311640809} a^{14} - \frac{14099557679}{40918265851} a^{13} + \frac{2488189736701}{6424167738607} a^{12} - \frac{1215922835661}{6424167738607} a^{11} - \frac{1143303397305}{6424167738607} a^{10} + \frac{623206572622}{6424167738607} a^{9} + \frac{817457752719}{6424167738607} a^{8} + \frac{635056100214}{6424167738607} a^{7} - \frac{141656182839}{6424167738607} a^{6} + \frac{710283494053}{6424167738607} a^{5} - \frac{629613292640}{6424167738607} a^{4} - \frac{2171282188113}{6424167738607} a^{3} + \frac{3036980292597}{6424167738607} a^{2} - \frac{1355935340982}{6424167738607} a + \frac{1643198350323}{6424167738607}$, $\frac{1}{1162774360687867} a^{37} - \frac{4}{1162774360687867} a^{36} + \frac{4443615}{1162774360687867} a^{35} + \frac{762192900}{1162774360687867} a^{34} + \frac{53354402113}{1162774360687867} a^{33} + \frac{85599368089}{1162774360687867} a^{32} + \frac{167696379877}{1162774360687867} a^{31} + \frac{6270985183349}{1162774360687867} a^{30} - \frac{276747045663262}{1162774360687867} a^{29} + \frac{202632763268072}{1162774360687867} a^{28} + \frac{325047711780}{6424167738607} a^{27} + \frac{100275867371381}{1162774360687867} a^{26} - \frac{580961030918838}{1162774360687867} a^{25} + \frac{387701198275009}{1162774360687867} a^{24} - \frac{461849106955381}{1162774360687867} a^{23} - \frac{119320779859610}{1162774360687867} a^{22} - \frac{230206581280144}{1162774360687867} a^{21} - \frac{564566500665439}{1162774360687867} a^{20} + \frac{351616281755970}{1162774360687867} a^{19} + \frac{494610867864956}{1162774360687867} a^{18} - \frac{553759451377192}{1162774360687867} a^{17} - \frac{2607692214704}{7406206119031} a^{16} - \frac{541732092482030}{1162774360687867} a^{15} - \frac{452584279339555}{1162774360687867} a^{14} + \frac{376810287442870}{1162774360687867} a^{13} + \frac{297775281984268}{1162774360687867} a^{12} + \frac{391475452407519}{1162774360687867} a^{11} - \frac{430136625492531}{1162774360687867} a^{10} - \frac{545625901785727}{1162774360687867} a^{9} + \frac{111700382330158}{1162774360687867} a^{8} + \frac{259028023032020}{1162774360687867} a^{7} - \frac{94004315669045}{1162774360687867} a^{6} + \frac{419989060725365}{1162774360687867} a^{5} - \frac{483193347466651}{1162774360687867} a^{4} - \frac{294500964719736}{1162774360687867} a^{3} + \frac{1805566025666}{50555406986429} a^{2} - \frac{556093264544324}{1162774360687867} a - \frac{526596825879676}{1162774360687867}$, $\frac{1}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{38} + \frac{948205647861725104730339610463159868779726352166985178831121881562321520435594456297294147140758264432387894715010363132235184754879975598933716469052}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{37} - \frac{211058068846167807079929346971221873272696532562943941172309348443995013911380298449979676391112871725863951841397705873413842829991753904589640836529327}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{36} + \frac{3780739218374240758248695924042434423436259117206143911172469822833370039695851110029176739837034211754784871906662377900420910254899548937641511508746248101}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{35} - \frac{330522377725057903810525123043946404653942682487716800500542909521902654660899168019558886102804490042362682468143140610565988320179623106766569128498185540570}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{34} + \frac{337718124317850573653779143071220969436872077537591576961800205408967871806405041471737025187261064849508917929346470023529103404934394315652600547250873941570070}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{33} + \frac{316380016355249566526755193405483040286683475557933422544796690759892903020701714095927437109430371528384255724628887106580166419889712841937846014463599968779694}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{32} + \frac{509484461675897065715958682667953053846042969283195271191784285077141289226688711035007738115776813055509818397297274969703848604495441183130697144183858110267414}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{31} + \frac{48123934604397131325737517629689429453252460093933884456881562966192117596110457508597567101290298122365118347662035746982781628652606852735131405983887400834722950}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{30} - \frac{770288073542169377505062217126778902653953336404678845427133876321336998290811051281141093903678040204236767207438168826853266347877844212566236520576760912520655203}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{29} - \frac{1134403293360817383342370997034334547243470061505960834529767325980265175850850401022509760814423293679644862637895575184474927748327192807917754785941426550450953281}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{28} + \frac{375434360188064712866481936840670596063627518054923975838289493163049642456600322496969674157187391893512757136281982083581190346346902785688130350249315648181600285}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{27} - \frac{304008484762570010955887426922976539175407917351897320434295119114908008176880683350398092318211364499268903457228327879482558245557256065960129913063081304656677853}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{26} + \frac{425956748614975859105720770242114092718023333259049703645270506098260846480859784714925212041539141028223528478929975532745896880252573001845142233502300560808336357}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{25} - \frac{600822902800032708248241388243405935060727001004313014711676283098020174451471721629841768779246830631971707928761819033459340979061807604805815741675564253809873785}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{24} + \frac{69162597904807109787399403685254960033142445144056426664435548258878937911431415093322509787142185797269211025966550512427355119849811135083227500863576458982467772}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{23} - \frac{470767623855039504218884348343745831988567365274207200331867718647037095952728858470888919255094312589351794112798107552666061791882300710369932777779734772295105011}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{22} - \frac{1088848705217681816134802648568015602070604420553728174111489314993631707281869945648823966257428365425126705262908420126227478294517595582832169568759296988267658551}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{21} + \frac{375424167866979300907766736394900445576593646438204391873448224777276780228412965347606446483171486033273229100969241932042456965542295306194434589278445668946605076}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{20} - \frac{363419408572381766420064676523514524437855768040598889996674197131297746343160041951939346677472865879965413957030795403881870372100232910553790152894740719850143266}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{19} + \frac{1296377356596654951935345841597630547758261180664628959042116860857304470325620781411084024838319022783914481322106772684277859385931817971219620479618047633236490060}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{18} + \frac{1442713683939935720467547619783773136030317265153882844179561424432889518767344066996857983838440850439778884042238338164703857969509708962441645438458811912856938313}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{17} + \frac{1117217835434002401586925069715286264176606388500533533979622194079063487447110878079530474918705743567320314379641878929146625821785415825476878037560254898501687855}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{16} - \frac{807816408853566911523419133707167956041116043905088335419719268531558182922866566655706716220726042298795127526661165997166998559527301847907549986250974820599157282}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{15} - \frac{1042862608371045077516011817029416207804096047255949437085593020940575420757864412157871695197382801088842255995520253609287261058681549180884747029858342879430236273}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{14} - \frac{897622945207198162311472805555226730409243886046469335016865728698684344195104196420545407527162394313137264582887046118105447516417715657809516876887743505919124400}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{13} + \frac{600924219288591428071409782071587015343295188414896263011535708234566100799942853943386826311975582007528673036183004797316872102811273999785052028729130484017590272}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{12} + \frac{585891722404381123333216757393269242100694391249052692086517233661876966364173051764193703074538902410246717537830720436901128484497883334213092633752365180673828559}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{11} + \frac{904961610934591113074317801603054423061928511196379387372353711072852564208853844701925483183589753090402423382341835037974961196343982707587038262565343307172517968}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{10} + \frac{167475040265447499335698300170534846157170422358935071089985501579802636118777605318706458514730937633264593754682841050581472100429209543871983174464113156417416547}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{9} - \frac{1231662474760729581991606509239606398170145969613156599167542523204708927990124249165008884739992159199143588128775111890826383340047122974295696534082935333527973789}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{8} - \frac{1495692952515233885809054483935038160559697602227063499271594435118897109975618647327992699634074754419693695928945013460167023746856346560763810322298398020048909149}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{7} + \frac{1480185197000291534130655868766496029927947769818041240949549766895633413045435501556655436349185739901927515952530557251430330120713491572473781809621442334527361497}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{6} + \frac{299321233378044205376596367377791673793807200081602577075988909263284160731654431311546289152792144306148249546432841208370259642282412708550755997921311441620831031}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{5} - \frac{23308309470221321415219830417621681458621455300363913981832590866594242856687293533208169714680187617173007131777540983971756769107259716444983132528063818770679948}{135765692397706281886253314129225619569340779741931202131425194110363851631176074112358730184828909798189774581577417654674155378464451222100833097600473219354580761} a^{4} - \frac{97189320648490964020612744190160704924221287816219564107532542231168438784797918823198654067985075780531679824747711688710789926074879703518398028743817312376132621}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{3} - \frac{1422823426483282377232723434462198575147218767474576347286603783309804457333317923460236124369423290048131888653278434677929764220419663964177323062619631243037346809}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a^{2} + \frac{124991748614228236968125800201650454940062249621780810867337934472622197137557321585931764876882949975856483481867373166132360733696209823117124055168454123661746272}{3122610925147244483383826224972189250094837934064417649022779464538368587517049704584250794251064925358364815376280606057505573704682378108319161244810884045155357503} a - \frac{2614208647930652238719835905010507324269630808292920621202979617611128513307877595314806718683315678384600920208814322378018169708776202795822942489334273773724981}{5411804029717928047458970927161506499297812710683566116157330094520569475766117338967505709273942678264063804811578173409888342642430464659131995224975535606855039}$

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:  $38$
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:  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

$C_{39}$ (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, 13.13.59091511031674153381441.1

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

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
$79$79.13.12.1$x^{13} - 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$
79.13.12.1$x^{13} - 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$
79.13.12.1$x^{13} - 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$