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Properties

Label 39.39.133...161.1
Degree $39$
Signature $[39, 0]$
Discriminant $1.333\times 10^{93}$
Root discriminant \(244.24\)
Ramified primes $3,79$
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 - 3*x^38 - 144*x^37 + 544*x^36 + 8607*x^35 - 39873*x^34 - 273219*x^33 + 1586748*x^32 + 4768863*x^31 - 38339003*x^30 - 36770526*x^29 + 590401068*x^28 - 186548878*x^27 - 5865182265*x^26 + 7401235119*x^25 + 36650142262*x^24 - 80182575711*x^23 - 129404499477*x^22 + 478911168431*x^21 + 133680753126*x^20 - 1717086418740*x^19 + 841783459401*x^18 + 3624110595294*x^17 - 4051561096485*x^16 - 3908165428785*x^15 + 8158071470571*x^14 + 524362618785*x^13 - 8608241123601*x^12 + 3464219056332*x^11 + 4466815249788*x^10 - 3609335878022*x^9 - 703907774076*x^8 + 1408564281330*x^7 - 178984846489*x^6 - 212099031525*x^5 + 49542152886*x^4 + 13836139901*x^3 - 2992803492*x^2 - 474072537*x + 17616187)
 
gp: K = bnfinit(y^39 - 3*y^38 - 144*y^37 + 544*y^36 + 8607*y^35 - 39873*y^34 - 273219*y^33 + 1586748*y^32 + 4768863*y^31 - 38339003*y^30 - 36770526*y^29 + 590401068*y^28 - 186548878*y^27 - 5865182265*y^26 + 7401235119*y^25 + 36650142262*y^24 - 80182575711*y^23 - 129404499477*y^22 + 478911168431*y^21 + 133680753126*y^20 - 1717086418740*y^19 + 841783459401*y^18 + 3624110595294*y^17 - 4051561096485*y^16 - 3908165428785*y^15 + 8158071470571*y^14 + 524362618785*y^13 - 8608241123601*y^12 + 3464219056332*y^11 + 4466815249788*y^10 - 3609335878022*y^9 - 703907774076*y^8 + 1408564281330*y^7 - 178984846489*y^6 - 212099031525*y^5 + 49542152886*y^4 + 13836139901*y^3 - 2992803492*y^2 - 474072537*y + 17616187, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^39 - 3*x^38 - 144*x^37 + 544*x^36 + 8607*x^35 - 39873*x^34 - 273219*x^33 + 1586748*x^32 + 4768863*x^31 - 38339003*x^30 - 36770526*x^29 + 590401068*x^28 - 186548878*x^27 - 5865182265*x^26 + 7401235119*x^25 + 36650142262*x^24 - 80182575711*x^23 - 129404499477*x^22 + 478911168431*x^21 + 133680753126*x^20 - 1717086418740*x^19 + 841783459401*x^18 + 3624110595294*x^17 - 4051561096485*x^16 - 3908165428785*x^15 + 8158071470571*x^14 + 524362618785*x^13 - 8608241123601*x^12 + 3464219056332*x^11 + 4466815249788*x^10 - 3609335878022*x^9 - 703907774076*x^8 + 1408564281330*x^7 - 178984846489*x^6 - 212099031525*x^5 + 49542152886*x^4 + 13836139901*x^3 - 2992803492*x^2 - 474072537*x + 17616187);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^39 - 3*x^38 - 144*x^37 + 544*x^36 + 8607*x^35 - 39873*x^34 - 273219*x^33 + 1586748*x^32 + 4768863*x^31 - 38339003*x^30 - 36770526*x^29 + 590401068*x^28 - 186548878*x^27 - 5865182265*x^26 + 7401235119*x^25 + 36650142262*x^24 - 80182575711*x^23 - 129404499477*x^22 + 478911168431*x^21 + 133680753126*x^20 - 1717086418740*x^19 + 841783459401*x^18 + 3624110595294*x^17 - 4051561096485*x^16 - 3908165428785*x^15 + 8158071470571*x^14 + 524362618785*x^13 - 8608241123601*x^12 + 3464219056332*x^11 + 4466815249788*x^10 - 3609335878022*x^9 - 703907774076*x^8 + 1408564281330*x^7 - 178984846489*x^6 - 212099031525*x^5 + 49542152886*x^4 + 13836139901*x^3 - 2992803492*x^2 - 474072537*x + 17616187)
 

\( x^{39} - 3 x^{38} - 144 x^{37} + 544 x^{36} + 8607 x^{35} - 39873 x^{34} - 273219 x^{33} + \cdots + 17616187 \) 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:   \(133\!\cdots\!161\) \(\medspace = 3^{52}\cdot 79^{36}\) 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:  \(244.24\)
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:  $3^{4/3}79^{12/13}\approx 244.24037018618242$
Ramified primes:   \(3\), \(79\) 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:  \(711=3^{2}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{711}(640,·)$, $\chi_{711}(1,·)$, $\chi_{711}(259,·)$, $\chi_{711}(520,·)$, $\chi_{711}(10,·)$, $\chi_{711}(526,·)$, $\chi_{711}(403,·)$, $\chi_{711}(22,·)$, $\chi_{711}(538,·)$, $\chi_{711}(283,·)$, $\chi_{711}(541,·)$, $\chi_{711}(670,·)$, $\chi_{711}(289,·)$, $\chi_{711}(166,·)$, $\chi_{711}(301,·)$, $\chi_{711}(46,·)$, $\chi_{711}(304,·)$, $\chi_{711}(433,·)$, $\chi_{711}(52,·)$, $\chi_{711}(694,·)$, $\chi_{711}(697,·)$, $\chi_{711}(571,·)$, $\chi_{711}(574,·)$, $\chi_{711}(64,·)$, $\chi_{711}(67,·)$, $\chi_{711}(196,·)$, $\chi_{711}(457,·)$, $\chi_{711}(460,·)$, $\chi_{711}(334,·)$, $\chi_{711}(337,·)$, $\chi_{711}(100,·)$, $\chi_{711}(475,·)$, $\chi_{711}(220,·)$, $\chi_{711}(223,·)$, $\chi_{711}(97,·)$, $\chi_{711}(484,·)$, $\chi_{711}(238,·)$, $\chi_{711}(496,·)$, $\chi_{711}(247,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{23}a^{30}-\frac{10}{23}a^{29}+\frac{4}{23}a^{28}-\frac{9}{23}a^{27}-\frac{2}{23}a^{26}-\frac{9}{23}a^{25}+\frac{9}{23}a^{24}-\frac{5}{23}a^{23}-\frac{1}{23}a^{21}-\frac{7}{23}a^{20}-\frac{2}{23}a^{19}+\frac{9}{23}a^{18}-\frac{1}{23}a^{17}+\frac{4}{23}a^{16}-\frac{8}{23}a^{15}-\frac{4}{23}a^{14}+\frac{8}{23}a^{13}+\frac{8}{23}a^{12}-\frac{11}{23}a^{11}+\frac{2}{23}a^{10}-\frac{8}{23}a^{9}+\frac{7}{23}a^{8}+\frac{6}{23}a^{7}-\frac{2}{23}a^{6}-\frac{11}{23}a^{5}+\frac{7}{23}a^{4}+\frac{9}{23}a^{3}-\frac{3}{23}a^{2}+\frac{6}{23}a+\frac{4}{23}$, $\frac{1}{23}a^{31}-\frac{4}{23}a^{29}+\frac{8}{23}a^{28}-\frac{6}{23}a^{26}+\frac{11}{23}a^{25}-\frac{7}{23}a^{24}-\frac{4}{23}a^{23}-\frac{1}{23}a^{22}+\frac{6}{23}a^{21}-\frac{3}{23}a^{20}-\frac{11}{23}a^{19}-\frac{3}{23}a^{18}-\frac{6}{23}a^{17}+\frac{9}{23}a^{16}+\frac{8}{23}a^{15}-\frac{9}{23}a^{14}-\frac{4}{23}a^{13}+\frac{7}{23}a^{11}-\frac{11}{23}a^{10}-\frac{4}{23}a^{9}+\frac{7}{23}a^{8}-\frac{11}{23}a^{7}-\frac{8}{23}a^{6}-\frac{11}{23}a^{5}+\frac{10}{23}a^{4}-\frac{5}{23}a^{3}-\frac{1}{23}a^{2}-\frac{5}{23}a-\frac{6}{23}$, $\frac{1}{23}a^{32}-\frac{9}{23}a^{29}-\frac{7}{23}a^{28}+\frac{4}{23}a^{27}+\frac{3}{23}a^{26}+\frac{3}{23}a^{25}+\frac{9}{23}a^{24}+\frac{2}{23}a^{23}+\frac{6}{23}a^{22}-\frac{7}{23}a^{21}+\frac{7}{23}a^{20}-\frac{11}{23}a^{19}+\frac{7}{23}a^{18}+\frac{5}{23}a^{17}+\frac{1}{23}a^{16}+\frac{5}{23}a^{15}+\frac{3}{23}a^{14}+\frac{9}{23}a^{13}-\frac{7}{23}a^{12}-\frac{9}{23}a^{11}+\frac{4}{23}a^{10}-\frac{2}{23}a^{9}-\frac{6}{23}a^{8}-\frac{7}{23}a^{7}+\frac{4}{23}a^{6}-\frac{11}{23}a^{5}-\frac{11}{23}a^{3}+\frac{6}{23}a^{2}-\frac{5}{23}a-\frac{7}{23}$, $\frac{1}{23}a^{33}-\frac{5}{23}a^{29}-\frac{6}{23}a^{28}-\frac{9}{23}a^{27}+\frac{8}{23}a^{26}-\frac{3}{23}a^{25}-\frac{9}{23}a^{24}+\frac{7}{23}a^{23}-\frac{7}{23}a^{22}-\frac{2}{23}a^{21}-\frac{5}{23}a^{20}-\frac{11}{23}a^{19}-\frac{6}{23}a^{18}-\frac{8}{23}a^{17}-\frac{5}{23}a^{16}-\frac{4}{23}a^{14}-\frac{4}{23}a^{13}-\frac{6}{23}a^{12}-\frac{3}{23}a^{11}-\frac{7}{23}a^{10}-\frac{9}{23}a^{9}+\frac{10}{23}a^{8}-\frac{11}{23}a^{7}-\frac{6}{23}a^{6}-\frac{7}{23}a^{5}+\frac{6}{23}a^{4}-\frac{5}{23}a^{3}-\frac{9}{23}a^{2}+\frac{1}{23}a-\frac{10}{23}$, $\frac{1}{23}a^{34}-\frac{10}{23}a^{29}+\frac{11}{23}a^{28}+\frac{9}{23}a^{27}+\frac{10}{23}a^{26}-\frac{8}{23}a^{25}+\frac{6}{23}a^{24}-\frac{9}{23}a^{23}-\frac{2}{23}a^{22}-\frac{10}{23}a^{21}+\frac{7}{23}a^{19}-\frac{9}{23}a^{18}-\frac{10}{23}a^{17}-\frac{3}{23}a^{16}+\frac{2}{23}a^{15}-\frac{1}{23}a^{14}+\frac{11}{23}a^{13}-\frac{9}{23}a^{12}+\frac{7}{23}a^{11}+\frac{1}{23}a^{10}-\frac{7}{23}a^{9}+\frac{1}{23}a^{8}+\frac{1}{23}a^{7}+\frac{6}{23}a^{6}-\frac{3}{23}a^{5}+\frac{7}{23}a^{4}-\frac{10}{23}a^{3}+\frac{9}{23}a^{2}-\frac{3}{23}a-\frac{3}{23}$, $\frac{1}{23}a^{35}+\frac{3}{23}a^{29}+\frac{3}{23}a^{28}-\frac{11}{23}a^{27}-\frac{5}{23}a^{26}+\frac{8}{23}a^{25}-\frac{11}{23}a^{24}-\frac{6}{23}a^{23}-\frac{10}{23}a^{22}-\frac{10}{23}a^{21}+\frac{6}{23}a^{20}-\frac{6}{23}a^{19}+\frac{11}{23}a^{18}+\frac{10}{23}a^{17}-\frac{4}{23}a^{16}+\frac{11}{23}a^{15}-\frac{6}{23}a^{14}+\frac{2}{23}a^{13}-\frac{5}{23}a^{12}+\frac{6}{23}a^{11}-\frac{10}{23}a^{10}-\frac{10}{23}a^{9}+\frac{2}{23}a^{8}-\frac{3}{23}a^{7}-\frac{11}{23}a^{5}-\frac{9}{23}a^{4}+\frac{7}{23}a^{3}-\frac{10}{23}a^{2}+\frac{11}{23}a-\frac{6}{23}$, $\frac{1}{226829381}a^{36}+\frac{321116}{226829381}a^{35}-\frac{4540249}{226829381}a^{34}+\frac{155999}{226829381}a^{33}-\frac{3016489}{226829381}a^{32}+\frac{3645587}{226829381}a^{31}-\frac{3407166}{226829381}a^{30}+\frac{10916726}{226829381}a^{29}-\frac{80722263}{226829381}a^{28}+\frac{78202635}{226829381}a^{27}-\frac{56544752}{226829381}a^{26}+\frac{22074893}{226829381}a^{25}-\frac{34370444}{226829381}a^{24}+\frac{103568165}{226829381}a^{23}+\frac{66446868}{226829381}a^{22}-\frac{25242543}{226829381}a^{21}-\frac{52245798}{226829381}a^{20}+\frac{30088866}{226829381}a^{19}+\frac{8901809}{226829381}a^{18}+\frac{20742133}{226829381}a^{17}-\frac{4688348}{9862147}a^{16}-\frac{95233438}{226829381}a^{15}-\frac{75260794}{226829381}a^{14}-\frac{107180447}{226829381}a^{13}+\frac{14559111}{226829381}a^{12}+\frac{90706519}{226829381}a^{11}-\frac{240298}{2202227}a^{10}+\frac{103434284}{226829381}a^{9}+\frac{68540936}{226829381}a^{8}+\frac{66836954}{226829381}a^{7}+\frac{574362}{2202227}a^{6}-\frac{95560780}{226829381}a^{5}+\frac{7339899}{226829381}a^{4}+\frac{59312784}{226829381}a^{3}+\frac{34709851}{226829381}a^{2}+\frac{26816937}{226829381}a-\frac{307237}{1253201}$, $\frac{1}{226829381}a^{37}-\frac{1416673}{226829381}a^{35}-\frac{4023568}{226829381}a^{34}+\frac{2915387}{226829381}a^{33}-\frac{3688882}{226829381}a^{32}-\frac{3149064}{226829381}a^{31}-\frac{154198}{226829381}a^{30}+\frac{135016}{428789}a^{29}-\frac{13867218}{226829381}a^{28}-\frac{1598519}{226829381}a^{27}-\frac{77634426}{226829381}a^{26}+\frac{28998893}{226829381}a^{25}-\frac{100405617}{226829381}a^{24}-\frac{77966108}{226829381}a^{23}+\frac{109883207}{226829381}a^{22}+\frac{63159331}{226829381}a^{21}+\frac{20114678}{226829381}a^{20}-\frac{12666012}{226829381}a^{19}+\frac{70475533}{226829381}a^{18}-\frac{91082307}{226829381}a^{17}+\frac{93960146}{226829381}a^{16}+\frac{34547563}{226829381}a^{15}-\frac{80107224}{226829381}a^{14}+\frac{55891483}{226829381}a^{13}-\frac{40341771}{226829381}a^{12}+\frac{18751252}{226829381}a^{11}-\frac{95048570}{226829381}a^{10}-\frac{44053176}{226829381}a^{9}-\frac{22975517}{226829381}a^{8}+\frac{111660931}{226829381}a^{7}-\frac{39425883}{226829381}a^{6}-\frac{95828032}{226829381}a^{5}-\frac{73390881}{226829381}a^{4}+\frac{12294510}{226829381}a^{3}-\frac{28056818}{226829381}a^{2}+\frac{46090165}{226829381}a-\frac{404025}{1253201}$, $\frac{1}{64\!\cdots\!91}a^{38}+\frac{66\!\cdots\!91}{64\!\cdots\!91}a^{37}+\frac{84\!\cdots\!42}{64\!\cdots\!91}a^{36}+\frac{23\!\cdots\!12}{64\!\cdots\!91}a^{35}+\frac{57\!\cdots\!73}{64\!\cdots\!91}a^{34}-\frac{13\!\cdots\!25}{64\!\cdots\!91}a^{33}+\frac{50\!\cdots\!41}{64\!\cdots\!91}a^{32}+\frac{12\!\cdots\!95}{64\!\cdots\!91}a^{31}-\frac{11\!\cdots\!57}{64\!\cdots\!91}a^{30}-\frac{16\!\cdots\!11}{64\!\cdots\!91}a^{29}-\frac{23\!\cdots\!83}{64\!\cdots\!91}a^{28}-\frac{20\!\cdots\!47}{64\!\cdots\!91}a^{27}-\frac{82\!\cdots\!03}{64\!\cdots\!91}a^{26}-\frac{10\!\cdots\!85}{64\!\cdots\!91}a^{25}-\frac{10\!\cdots\!59}{64\!\cdots\!91}a^{24}-\frac{23\!\cdots\!95}{64\!\cdots\!91}a^{23}-\frac{14\!\cdots\!84}{64\!\cdots\!91}a^{22}+\frac{11\!\cdots\!55}{64\!\cdots\!91}a^{21}+\frac{16\!\cdots\!27}{64\!\cdots\!91}a^{20}-\frac{34\!\cdots\!29}{27\!\cdots\!17}a^{19}+\frac{39\!\cdots\!37}{64\!\cdots\!91}a^{18}+\frac{15\!\cdots\!41}{64\!\cdots\!91}a^{17}+\frac{14\!\cdots\!30}{64\!\cdots\!91}a^{16}-\frac{31\!\cdots\!59}{64\!\cdots\!91}a^{15}+\frac{22\!\cdots\!58}{64\!\cdots\!91}a^{14}+\frac{28\!\cdots\!18}{64\!\cdots\!91}a^{13}-\frac{13\!\cdots\!17}{64\!\cdots\!91}a^{12}-\frac{21\!\cdots\!05}{64\!\cdots\!91}a^{11}+\frac{15\!\cdots\!93}{64\!\cdots\!91}a^{10}+\frac{17\!\cdots\!45}{64\!\cdots\!91}a^{9}+\frac{61\!\cdots\!03}{64\!\cdots\!91}a^{8}+\frac{18\!\cdots\!76}{27\!\cdots\!17}a^{7}-\frac{36\!\cdots\!70}{64\!\cdots\!91}a^{6}+\frac{94\!\cdots\!53}{64\!\cdots\!91}a^{5}+\frac{20\!\cdots\!45}{64\!\cdots\!91}a^{4}+\frac{34\!\cdots\!44}{64\!\cdots\!91}a^{3}+\frac{26\!\cdots\!63}{64\!\cdots\!91}a^{2}-\frac{22\!\cdots\!11}{64\!\cdots\!91}a+\frac{26\!\cdots\!70}{35\!\cdots\!11}$ 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 - 3*x^38 - 144*x^37 + 544*x^36 + 8607*x^35 - 39873*x^34 - 273219*x^33 + 1586748*x^32 + 4768863*x^31 - 38339003*x^30 - 36770526*x^29 + 590401068*x^28 - 186548878*x^27 - 5865182265*x^26 + 7401235119*x^25 + 36650142262*x^24 - 80182575711*x^23 - 129404499477*x^22 + 478911168431*x^21 + 133680753126*x^20 - 1717086418740*x^19 + 841783459401*x^18 + 3624110595294*x^17 - 4051561096485*x^16 - 3908165428785*x^15 + 8158071470571*x^14 + 524362618785*x^13 - 8608241123601*x^12 + 3464219056332*x^11 + 4466815249788*x^10 - 3609335878022*x^9 - 703907774076*x^8 + 1408564281330*x^7 - 178984846489*x^6 - 212099031525*x^5 + 49542152886*x^4 + 13836139901*x^3 - 2992803492*x^2 - 474072537*x + 17616187)
 
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 - 3*x^38 - 144*x^37 + 544*x^36 + 8607*x^35 - 39873*x^34 - 273219*x^33 + 1586748*x^32 + 4768863*x^31 - 38339003*x^30 - 36770526*x^29 + 590401068*x^28 - 186548878*x^27 - 5865182265*x^26 + 7401235119*x^25 + 36650142262*x^24 - 80182575711*x^23 - 129404499477*x^22 + 478911168431*x^21 + 133680753126*x^20 - 1717086418740*x^19 + 841783459401*x^18 + 3624110595294*x^17 - 4051561096485*x^16 - 3908165428785*x^15 + 8158071470571*x^14 + 524362618785*x^13 - 8608241123601*x^12 + 3464219056332*x^11 + 4466815249788*x^10 - 3609335878022*x^9 - 703907774076*x^8 + 1408564281330*x^7 - 178984846489*x^6 - 212099031525*x^5 + 49542152886*x^4 + 13836139901*x^3 - 2992803492*x^2 - 474072537*x + 17616187, 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 - 3*x^38 - 144*x^37 + 544*x^36 + 8607*x^35 - 39873*x^34 - 273219*x^33 + 1586748*x^32 + 4768863*x^31 - 38339003*x^30 - 36770526*x^29 + 590401068*x^28 - 186548878*x^27 - 5865182265*x^26 + 7401235119*x^25 + 36650142262*x^24 - 80182575711*x^23 - 129404499477*x^22 + 478911168431*x^21 + 133680753126*x^20 - 1717086418740*x^19 + 841783459401*x^18 + 3624110595294*x^17 - 4051561096485*x^16 - 3908165428785*x^15 + 8158071470571*x^14 + 524362618785*x^13 - 8608241123601*x^12 + 3464219056332*x^11 + 4466815249788*x^10 - 3609335878022*x^9 - 703907774076*x^8 + 1408564281330*x^7 - 178984846489*x^6 - 212099031525*x^5 + 49542152886*x^4 + 13836139901*x^3 - 2992803492*x^2 - 474072537*x + 17616187);
 
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 - 3*x^38 - 144*x^37 + 544*x^36 + 8607*x^35 - 39873*x^34 - 273219*x^33 + 1586748*x^32 + 4768863*x^31 - 38339003*x^30 - 36770526*x^29 + 590401068*x^28 - 186548878*x^27 - 5865182265*x^26 + 7401235119*x^25 + 36650142262*x^24 - 80182575711*x^23 - 129404499477*x^22 + 478911168431*x^21 + 133680753126*x^20 - 1717086418740*x^19 + 841783459401*x^18 + 3624110595294*x^17 - 4051561096485*x^16 - 3908165428785*x^15 + 8158071470571*x^14 + 524362618785*x^13 - 8608241123601*x^12 + 3464219056332*x^11 + 4466815249788*x^10 - 3609335878022*x^9 - 703907774076*x^8 + 1408564281330*x^7 - 178984846489*x^6 - 212099031525*x^5 + 49542152886*x^4 + 13836139901*x^3 - 2992803492*x^2 - 474072537*x + 17616187);
 
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 $C_{39}$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 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 $39$ R $39$ $39$ $39$ $39$ ${\href{/padicField/17.13.0.1}{13} }^{3}$ ${\href{/padicField/19.13.0.1}{13} }^{3}$ ${\href{/padicField/23.3.0.1}{3} }^{13}$ $39$ $39$ ${\href{/padicField/37.13.0.1}{13} }^{3}$ $39$ $39$ $39$ ${\href{/padicField/53.13.0.1}{13} }^{3}$ $39$

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
\(3\) Copy content Toggle raw display Deg $39$$3$$13$$52$
\(79\) Copy content Toggle raw display Deg $39$$13$$3$$36$