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Properties

Label 39.39.189...089.1
Degree $39$
Signature $[39, 0]$
Discriminant $1.893\times 10^{97}$
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: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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(y^39 - 10*y^38 - 117*y^37 + 1248*y^36 + 6574*y^35 - 70508*y^34 - 238767*y^33 + 2376429*y^32 + 6246879*y^31 - 53084512*y^30 - 122075225*y^29 + 827367236*y^28 + 1790293625*y^27 - 9246368917*y^26 - 19595581406*y^25 + 75169877668*y^24 + 158901467338*y^23 - 447662704038*y^22 - 947228275523*y^21 + 1958725720367*y^20 + 4113180518714*y^19 - 6305779770878*y^18 - 12855958984538*y^17 + 14959051199327*y^16 + 28441505512102*y^15 - 26161675308530*y^14 - 43445251987869*y^13 + 33478011120120*y^12 + 44089907326138*y^11 - 30444285645115*y^10 - 27975777878380*y^9 + 18404264085338*y^8 + 10144928480062*y^7 - 6624071325699*y^6 - 1906724813303*y^5 + 1267027871999*y^4 + 154439956714*y^3 - 106661114208*y^2 - 3009281044*y + 2198272487, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{39} - 10 x^{38} - 117 x^{37} + 1248 x^{36} + 6574 x^{35} - 70508 x^{34} - 238767 x^{33} + \cdots + 2198272487 \) Copy content Toggle raw display

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

Invariants

Degree:  $39$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[39, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(189\!\cdots\!089\) \(\medspace = 13^{26}\cdot 79^{36}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(312.09\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $13^{2/3}79^{12/13}\approx 312.09346726335514$
Ramified primes:   \(13\), \(79\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $39$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{23}a^{30}-\frac{4}{23}a^{28}-\frac{7}{23}a^{27}-\frac{10}{23}a^{26}+\frac{10}{23}a^{25}+\frac{1}{23}a^{23}+\frac{1}{23}a^{22}-\frac{4}{23}a^{21}+\frac{3}{23}a^{20}-\frac{6}{23}a^{19}+\frac{1}{23}a^{18}+\frac{7}{23}a^{17}+\frac{6}{23}a^{16}-\frac{2}{23}a^{15}+\frac{11}{23}a^{14}-\frac{11}{23}a^{13}-\frac{9}{23}a^{12}+\frac{7}{23}a^{11}-\frac{3}{23}a^{10}+\frac{6}{23}a^{9}+\frac{8}{23}a^{8}+\frac{2}{23}a^{7}-\frac{4}{23}a^{6}+\frac{10}{23}a^{5}+\frac{6}{23}a^{3}-\frac{4}{23}a^{2}+\frac{1}{23}a+\frac{5}{23}$, $\frac{1}{2369}a^{31}+\frac{43}{2369}a^{30}-\frac{188}{2369}a^{29}-\frac{984}{2369}a^{28}-\frac{817}{2369}a^{27}-\frac{1156}{2369}a^{26}+\frac{154}{2369}a^{25}-\frac{597}{2369}a^{24}-\frac{1106}{2369}a^{23}-\frac{9}{23}a^{22}-\frac{169}{2369}a^{21}-\frac{245}{2369}a^{20}-\frac{464}{2369}a^{19}-\frac{456}{2369}a^{18}-\frac{498}{2369}a^{17}-\frac{940}{2369}a^{16}+\frac{730}{2369}a^{15}+\frac{94}{2369}a^{14}+\frac{438}{2369}a^{13}+\frac{1069}{2369}a^{12}-\frac{323}{2369}a^{11}+\frac{153}{2369}a^{10}-\frac{1183}{2369}a^{9}-\frac{1080}{2369}a^{8}+\frac{565}{2369}a^{7}+\frac{689}{2369}a^{6}-\frac{766}{2369}a^{5}-\frac{1167}{2369}a^{4}+\frac{599}{2369}a^{3}-\frac{332}{2369}a^{2}-\frac{550}{2369}a-\frac{521}{2369}$, $\frac{1}{2369}a^{32}+\frac{1}{103}a^{30}-\frac{7}{2369}a^{29}+\frac{89}{2369}a^{28}+\frac{603}{2369}a^{27}+\frac{834}{2369}a^{26}-\frac{833}{2369}a^{25}+\frac{875}{2369}a^{24}-\frac{46}{103}a^{23}-\frac{890}{2369}a^{22}+\frac{1151}{2369}a^{21}-\frac{332}{2369}a^{20}+\frac{29}{2369}a^{19}-\frac{151}{2369}a^{18}-\frac{641}{2369}a^{17}-\frac{977}{2369}a^{16}+\frac{119}{2369}a^{15}+\frac{104}{2369}a^{14}-\frac{152}{2369}a^{13}-\frac{867}{2369}a^{12}+\frac{34}{2369}a^{11}+\frac{272}{2369}a^{10}+\frac{555}{2369}a^{9}-\frac{478}{2369}a^{8}-\frac{534}{2369}a^{7}-\frac{729}{2369}a^{6}+\frac{11}{103}a^{5}+\frac{1031}{2369}a^{4}+\frac{485}{2369}a^{3}+\frac{748}{2369}a^{2}-\frac{870}{2369}a-\frac{463}{2369}$, $\frac{1}{2369}a^{33}+\frac{34}{2369}a^{30}-\frac{325}{2369}a^{29}+\frac{163}{2369}a^{28}+\frac{570}{2369}a^{27}-\frac{1128}{2369}a^{26}+\frac{526}{2369}a^{25}+\frac{36}{103}a^{24}-\frac{481}{2369}a^{23}-\frac{188}{2369}a^{22}-\frac{565}{2369}a^{21}-\frac{722}{2369}a^{20}-\frac{397}{2369}a^{19}-\frac{968}{2369}a^{18}+\frac{48}{103}a^{17}-\frac{509}{2369}a^{16}+\frac{2}{23}a^{15}-\frac{20}{103}a^{14}-\frac{950}{2369}a^{13}-\frac{657}{2369}a^{12}+\frac{697}{2369}a^{11}+\frac{1053}{2369}a^{10}-\frac{255}{2369}a^{9}-\frac{620}{2369}a^{8}+\frac{181}{2369}a^{7}-\frac{762}{2369}a^{6}+\frac{521}{2369}a^{5}-\frac{1102}{2369}a^{4}+\frac{258}{2369}a^{3}+\frac{277}{2369}a^{2}-\frac{997}{2369}a+\frac{550}{2369}$, $\frac{1}{371933}a^{34}-\frac{1}{16171}a^{33}+\frac{36}{371933}a^{32}-\frac{38}{371933}a^{31}-\frac{1109}{371933}a^{30}-\frac{73838}{371933}a^{29}+\frac{106820}{371933}a^{28}+\frac{36218}{371933}a^{27}+\frac{112328}{371933}a^{26}+\frac{88764}{371933}a^{25}-\frac{49277}{371933}a^{24}-\frac{54289}{371933}a^{23}-\frac{80090}{371933}a^{22}-\frac{26102}{371933}a^{21}+\frac{151883}{371933}a^{20}+\frac{71661}{371933}a^{19}+\frac{107517}{371933}a^{18}-\frac{132292}{371933}a^{17}+\frac{107766}{371933}a^{16}-\frac{1698}{16171}a^{15}+\frac{19687}{371933}a^{14}-\frac{137870}{371933}a^{13}+\frac{27005}{371933}a^{12}-\frac{145204}{371933}a^{11}-\frac{96459}{371933}a^{10}+\frac{36344}{371933}a^{9}+\frac{152346}{371933}a^{8}+\frac{141068}{371933}a^{7}+\frac{77640}{371933}a^{6}-\frac{46984}{371933}a^{5}+\frac{23556}{371933}a^{4}-\frac{67478}{371933}a^{3}-\frac{67467}{371933}a^{2}-\frac{91530}{371933}a+\frac{31369}{371933}$, $\frac{1}{117902761}a^{35}-\frac{94}{117902761}a^{34}-\frac{10263}{117902761}a^{33}-\frac{16567}{117902761}a^{32}+\frac{12893}{117902761}a^{31}+\frac{2383608}{117902761}a^{30}-\frac{8049376}{117902761}a^{29}+\frac{11921568}{117902761}a^{28}-\frac{51954185}{117902761}a^{27}+\frac{30672}{117902761}a^{26}+\frac{40711113}{117902761}a^{25}-\frac{1234536}{117902761}a^{24}+\frac{58674660}{117902761}a^{23}-\frac{47028912}{117902761}a^{22}+\frac{31973285}{117902761}a^{21}+\frac{22508854}{117902761}a^{20}+\frac{46451844}{117902761}a^{19}+\frac{26419024}{117902761}a^{18}-\frac{479043}{5126207}a^{17}-\frac{7048624}{117902761}a^{16}+\frac{8750985}{117902761}a^{15}+\frac{5228070}{117902761}a^{14}+\frac{91764}{1144687}a^{13}-\frac{12942502}{117902761}a^{12}-\frac{44145085}{117902761}a^{11}+\frac{34601242}{117902761}a^{10}-\frac{53890009}{117902761}a^{9}-\frac{13512645}{117902761}a^{8}-\frac{58842275}{117902761}a^{7}+\frac{486230}{1144687}a^{6}+\frac{39412743}{117902761}a^{5}-\frac{29578409}{117902761}a^{4}-\frac{8305959}{117902761}a^{3}-\frac{2461829}{117902761}a^{2}-\frac{2397492}{117902761}a+\frac{20138079}{117902761}$, $\frac{1}{6424167738607}a^{36}+\frac{5794}{6424167738607}a^{35}-\frac{430595}{6424167738607}a^{34}-\frac{1261994620}{6424167738607}a^{33}-\frac{39723063}{279311640809}a^{32}+\frac{1136901340}{6424167738607}a^{31}-\frac{117156623376}{6424167738607}a^{30}+\frac{2043852574507}{6424167738607}a^{29}+\frac{1331600725420}{6424167738607}a^{28}+\frac{2864008903449}{6424167738607}a^{27}+\frac{946994328726}{6424167738607}a^{26}-\frac{2455694162924}{6424167738607}a^{25}+\frac{1979771322857}{6424167738607}a^{24}+\frac{1352314283770}{6424167738607}a^{23}+\frac{133985228962}{279311640809}a^{22}-\frac{2549816911736}{6424167738607}a^{21}+\frac{1995941401527}{6424167738607}a^{20}-\frac{389321732725}{6424167738607}a^{19}+\frac{2823698313594}{6424167738607}a^{18}+\frac{2716874164672}{6424167738607}a^{17}+\frac{2745299819512}{6424167738607}a^{16}+\frac{2994763049267}{6424167738607}a^{15}+\frac{69834492875}{279311640809}a^{14}-\frac{14099557679}{40918265851}a^{13}+\frac{2488189736701}{6424167738607}a^{12}-\frac{1215922835661}{6424167738607}a^{11}-\frac{1143303397305}{6424167738607}a^{10}+\frac{623206572622}{6424167738607}a^{9}+\frac{817457752719}{6424167738607}a^{8}+\frac{635056100214}{6424167738607}a^{7}-\frac{141656182839}{6424167738607}a^{6}+\frac{710283494053}{6424167738607}a^{5}-\frac{629613292640}{6424167738607}a^{4}-\frac{2171282188113}{6424167738607}a^{3}+\frac{3036980292597}{6424167738607}a^{2}-\frac{1355935340982}{6424167738607}a+\frac{1643198350323}{6424167738607}$, $\frac{1}{11\!\cdots\!67}a^{37}-\frac{4}{11\!\cdots\!67}a^{36}+\frac{4443615}{11\!\cdots\!67}a^{35}+\frac{762192900}{11\!\cdots\!67}a^{34}+\frac{53354402113}{11\!\cdots\!67}a^{33}+\frac{85599368089}{11\!\cdots\!67}a^{32}+\frac{167696379877}{11\!\cdots\!67}a^{31}+\frac{6270985183349}{11\!\cdots\!67}a^{30}-\frac{276747045663262}{11\!\cdots\!67}a^{29}+\frac{202632763268072}{11\!\cdots\!67}a^{28}+\frac{325047711780}{6424167738607}a^{27}+\frac{100275867371381}{11\!\cdots\!67}a^{26}-\frac{580961030918838}{11\!\cdots\!67}a^{25}+\frac{387701198275009}{11\!\cdots\!67}a^{24}-\frac{461849106955381}{11\!\cdots\!67}a^{23}-\frac{119320779859610}{11\!\cdots\!67}a^{22}-\frac{230206581280144}{11\!\cdots\!67}a^{21}-\frac{564566500665439}{11\!\cdots\!67}a^{20}+\frac{351616281755970}{11\!\cdots\!67}a^{19}+\frac{494610867864956}{11\!\cdots\!67}a^{18}-\frac{553759451377192}{11\!\cdots\!67}a^{17}-\frac{2607692214704}{7406206119031}a^{16}-\frac{541732092482030}{11\!\cdots\!67}a^{15}-\frac{452584279339555}{11\!\cdots\!67}a^{14}+\frac{376810287442870}{11\!\cdots\!67}a^{13}+\frac{297775281984268}{11\!\cdots\!67}a^{12}+\frac{391475452407519}{11\!\cdots\!67}a^{11}-\frac{430136625492531}{11\!\cdots\!67}a^{10}-\frac{545625901785727}{11\!\cdots\!67}a^{9}+\frac{111700382330158}{11\!\cdots\!67}a^{8}+\frac{259028023032020}{11\!\cdots\!67}a^{7}-\frac{94004315669045}{11\!\cdots\!67}a^{6}+\frac{419989060725365}{11\!\cdots\!67}a^{5}-\frac{483193347466651}{11\!\cdots\!67}a^{4}-\frac{294500964719736}{11\!\cdots\!67}a^{3}+\frac{1805566025666}{50555406986429}a^{2}-\frac{556093264544324}{11\!\cdots\!67}a-\frac{526596825879676}{11\!\cdots\!67}$, $\frac{1}{31\!\cdots\!03}a^{38}+\frac{94\!\cdots\!52}{31\!\cdots\!03}a^{37}-\frac{21\!\cdots\!27}{31\!\cdots\!03}a^{36}+\frac{37\!\cdots\!01}{31\!\cdots\!03}a^{35}-\frac{33\!\cdots\!70}{31\!\cdots\!03}a^{34}+\frac{33\!\cdots\!70}{31\!\cdots\!03}a^{33}+\frac{31\!\cdots\!94}{31\!\cdots\!03}a^{32}+\frac{50\!\cdots\!14}{31\!\cdots\!03}a^{31}+\frac{48\!\cdots\!50}{31\!\cdots\!03}a^{30}-\frac{77\!\cdots\!03}{31\!\cdots\!03}a^{29}-\frac{11\!\cdots\!81}{31\!\cdots\!03}a^{28}+\frac{37\!\cdots\!85}{31\!\cdots\!03}a^{27}-\frac{30\!\cdots\!53}{31\!\cdots\!03}a^{26}+\frac{42\!\cdots\!57}{31\!\cdots\!03}a^{25}-\frac{60\!\cdots\!85}{31\!\cdots\!03}a^{24}+\frac{69\!\cdots\!72}{31\!\cdots\!03}a^{23}-\frac{47\!\cdots\!11}{31\!\cdots\!03}a^{22}-\frac{10\!\cdots\!51}{31\!\cdots\!03}a^{21}+\frac{37\!\cdots\!76}{31\!\cdots\!03}a^{20}-\frac{36\!\cdots\!66}{31\!\cdots\!03}a^{19}+\frac{12\!\cdots\!60}{31\!\cdots\!03}a^{18}+\frac{14\!\cdots\!13}{31\!\cdots\!03}a^{17}+\frac{11\!\cdots\!55}{31\!\cdots\!03}a^{16}-\frac{80\!\cdots\!82}{31\!\cdots\!03}a^{15}-\frac{10\!\cdots\!73}{31\!\cdots\!03}a^{14}-\frac{89\!\cdots\!00}{31\!\cdots\!03}a^{13}+\frac{60\!\cdots\!72}{31\!\cdots\!03}a^{12}+\frac{58\!\cdots\!59}{31\!\cdots\!03}a^{11}+\frac{90\!\cdots\!68}{31\!\cdots\!03}a^{10}+\frac{16\!\cdots\!47}{31\!\cdots\!03}a^{9}-\frac{12\!\cdots\!89}{31\!\cdots\!03}a^{8}-\frac{14\!\cdots\!49}{31\!\cdots\!03}a^{7}+\frac{14\!\cdots\!97}{31\!\cdots\!03}a^{6}+\frac{29\!\cdots\!31}{31\!\cdots\!03}a^{5}-\frac{23\!\cdots\!48}{13\!\cdots\!61}a^{4}-\frac{97\!\cdots\!21}{31\!\cdots\!03}a^{3}-\frac{14\!\cdots\!09}{31\!\cdots\!03}a^{2}+\frac{12\!\cdots\!72}{31\!\cdots\!03}a-\frac{26\!\cdots\!81}{54\!\cdots\!39}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $38$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 39
The 39 conjugacy class representatives for $C_{39}$
Character table for $C_{39}$

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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $39$ $39$ ${\href{/padicField/5.13.0.1}{13} }^{3}$ $39$ $39$ R $39$ $39$ ${\href{/padicField/23.3.0.1}{3} }^{13}$ $39$ ${\href{/padicField/31.13.0.1}{13} }^{3}$ $39$ $39$ $39$ ${\href{/padicField/47.13.0.1}{13} }^{3}$ ${\href{/padicField/53.13.0.1}{13} }^{3}$ $39$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(13\) Copy content Toggle raw display Deg $39$$3$$13$$26$
\(79\) Copy content Toggle raw display 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}$