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Properties

Label 39.39.832...801.2
Degree $39$
Signature $[39, 0]$
Discriminant $8.320\times 10^{96}$
Root discriminant \(305.58\)
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 - 237*x^37 - 158*x^36 + 24885*x^35 + 31758*x^34 - 1521777*x^33 - 2797074*x^32 + 59981382*x^31 + 142319290*x^30 - 1593173961*x^29 - 4645709076*x^28 + 28932877445*x^27 + 102241716102*x^26 - 355443646614*x^25 - 1553454595933*x^24 + 2810177785458*x^23 + 16432247086167*x^22 - 11967377146298*x^21 - 120635759398347*x^20 - 1823113254630*x^19 + 605746157617644*x^18 + 341698986985275*x^17 - 2017784741258409*x^16 - 2057859861184228*x^15 + 4196664129873231*x^14 + 6285983642002179*x^13 - 4725650025499719*x^12 - 10914822027031086*x^11 + 1465701883750416*x^10 + 10558980638642454*x^9 + 2159465693586741*x^8 - 5293388896015467*x^7 - 2202937031211426*x^6 + 1157156267855625*x^5 + 698944716032751*x^4 - 59632171643934*x^3 - 73740757923315*x^2 - 5146532589771*x + 1075512295747)
 
gp: K = bnfinit(y^39 - 237*y^37 - 158*y^36 + 24885*y^35 + 31758*y^34 - 1521777*y^33 - 2797074*y^32 + 59981382*y^31 + 142319290*y^30 - 1593173961*y^29 - 4645709076*y^28 + 28932877445*y^27 + 102241716102*y^26 - 355443646614*y^25 - 1553454595933*y^24 + 2810177785458*y^23 + 16432247086167*y^22 - 11967377146298*y^21 - 120635759398347*y^20 - 1823113254630*y^19 + 605746157617644*y^18 + 341698986985275*y^17 - 2017784741258409*y^16 - 2057859861184228*y^15 + 4196664129873231*y^14 + 6285983642002179*y^13 - 4725650025499719*y^12 - 10914822027031086*y^11 + 1465701883750416*y^10 + 10558980638642454*y^9 + 2159465693586741*y^8 - 5293388896015467*y^7 - 2202937031211426*y^6 + 1157156267855625*y^5 + 698944716032751*y^4 - 59632171643934*y^3 - 73740757923315*y^2 - 5146532589771*y + 1075512295747, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^39 - 237*x^37 - 158*x^36 + 24885*x^35 + 31758*x^34 - 1521777*x^33 - 2797074*x^32 + 59981382*x^31 + 142319290*x^30 - 1593173961*x^29 - 4645709076*x^28 + 28932877445*x^27 + 102241716102*x^26 - 355443646614*x^25 - 1553454595933*x^24 + 2810177785458*x^23 + 16432247086167*x^22 - 11967377146298*x^21 - 120635759398347*x^20 - 1823113254630*x^19 + 605746157617644*x^18 + 341698986985275*x^17 - 2017784741258409*x^16 - 2057859861184228*x^15 + 4196664129873231*x^14 + 6285983642002179*x^13 - 4725650025499719*x^12 - 10914822027031086*x^11 + 1465701883750416*x^10 + 10558980638642454*x^9 + 2159465693586741*x^8 - 5293388896015467*x^7 - 2202937031211426*x^6 + 1157156267855625*x^5 + 698944716032751*x^4 - 59632171643934*x^3 - 73740757923315*x^2 - 5146532589771*x + 1075512295747);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^39 - 237*x^37 - 158*x^36 + 24885*x^35 + 31758*x^34 - 1521777*x^33 - 2797074*x^32 + 59981382*x^31 + 142319290*x^30 - 1593173961*x^29 - 4645709076*x^28 + 28932877445*x^27 + 102241716102*x^26 - 355443646614*x^25 - 1553454595933*x^24 + 2810177785458*x^23 + 16432247086167*x^22 - 11967377146298*x^21 - 120635759398347*x^20 - 1823113254630*x^19 + 605746157617644*x^18 + 341698986985275*x^17 - 2017784741258409*x^16 - 2057859861184228*x^15 + 4196664129873231*x^14 + 6285983642002179*x^13 - 4725650025499719*x^12 - 10914822027031086*x^11 + 1465701883750416*x^10 + 10558980638642454*x^9 + 2159465693586741*x^8 - 5293388896015467*x^7 - 2202937031211426*x^6 + 1157156267855625*x^5 + 698944716032751*x^4 - 59632171643934*x^3 - 73740757923315*x^2 - 5146532589771*x + 1075512295747)
 

\( x^{39} - 237 x^{37} - 158 x^{36} + 24885 x^{35} + 31758 x^{34} - 1521777 x^{33} - 2797074 x^{32} + \cdots + 1075512295747 \) 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:   \(832\!\cdots\!801\) \(\medspace = 3^{52}\cdot 79^{38}\) 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:  \(305.58\)
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:  $3^{4/3}79^{38/39}\approx 305.58473738571735$
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$$\chi_{711}(256,·)$, $\chi_{711}(1,·)$, $\chi_{711}(643,·)$, $\chi_{711}(4,·)$, $\chi_{711}(10,·)$, $\chi_{711}(16,·)$, $\chi_{711}(529,·)$, $\chi_{711}(25,·)$, $\chi_{711}(640,·)$, $\chi_{711}(541,·)$, $\chi_{711}(286,·)$, $\chi_{711}(31,·)$, $\chi_{711}(160,·)$, $\chi_{711}(289,·)$, $\chi_{711}(418,·)$, $\chi_{711}(40,·)$, $\chi_{711}(427,·)$, $\chi_{711}(562,·)$, $\chi_{711}(46,·)$, $\chi_{711}(433,·)$, $\chi_{711}(178,·)$, $\chi_{711}(310,·)$, $\chi_{711}(439,·)$, $\chi_{711}(184,·)$, $\chi_{711}(313,·)$, $\chi_{711}(445,·)$, $\chi_{711}(64,·)$, $\chi_{711}(694,·)$, $\chi_{711}(460,·)$, $\chi_{711}(334,·)$, $\chi_{711}(400,·)$, $\chi_{711}(100,·)$, $\chi_{711}(358,·)$, $\chi_{711}(367,·)$, $\chi_{711}(496,·)$, $\chi_{711}(625,·)$, $\chi_{711}(115,·)$, $\chi_{711}(250,·)$, $\chi_{711}(124,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{103}a^{32}+\frac{2}{103}a^{31}-\frac{1}{103}a^{30}-\frac{10}{103}a^{29}-\frac{33}{103}a^{28}+\frac{3}{103}a^{27}+\frac{25}{103}a^{26}+\frac{43}{103}a^{25}-\frac{8}{103}a^{24}+\frac{41}{103}a^{23}-\frac{10}{103}a^{22}-\frac{13}{103}a^{21}-\frac{36}{103}a^{20}-\frac{3}{103}a^{19}-\frac{38}{103}a^{18}-\frac{35}{103}a^{17}+\frac{7}{103}a^{16}+\frac{11}{103}a^{15}-\frac{44}{103}a^{14}+\frac{23}{103}a^{13}-\frac{12}{103}a^{12}+\frac{27}{103}a^{11}-\frac{37}{103}a^{10}-\frac{11}{103}a^{9}-\frac{5}{103}a^{8}-\frac{44}{103}a^{7}-\frac{36}{103}a^{6}-\frac{36}{103}a^{5}-\frac{1}{103}a^{4}+\frac{7}{103}a^{3}+\frac{14}{103}a^{2}+\frac{3}{103}a$, $\frac{1}{103}a^{33}-\frac{5}{103}a^{31}-\frac{8}{103}a^{30}-\frac{13}{103}a^{29}-\frac{34}{103}a^{28}+\frac{19}{103}a^{27}-\frac{7}{103}a^{26}+\frac{9}{103}a^{25}-\frac{46}{103}a^{24}+\frac{11}{103}a^{23}+\frac{7}{103}a^{22}-\frac{10}{103}a^{21}-\frac{34}{103}a^{20}-\frac{32}{103}a^{19}+\frac{41}{103}a^{18}-\frac{26}{103}a^{17}-\frac{3}{103}a^{16}+\frac{37}{103}a^{15}+\frac{8}{103}a^{14}+\frac{45}{103}a^{13}+\frac{51}{103}a^{12}+\frac{12}{103}a^{11}-\frac{40}{103}a^{10}+\frac{17}{103}a^{9}-\frac{34}{103}a^{8}-\frac{51}{103}a^{7}+\frac{36}{103}a^{6}-\frac{32}{103}a^{5}+\frac{9}{103}a^{4}-\frac{25}{103}a^{2}-\frac{6}{103}a$, $\frac{1}{103}a^{34}+\frac{2}{103}a^{31}-\frac{18}{103}a^{30}+\frac{19}{103}a^{29}-\frac{43}{103}a^{28}+\frac{8}{103}a^{27}+\frac{31}{103}a^{26}-\frac{37}{103}a^{25}-\frac{29}{103}a^{24}+\frac{6}{103}a^{23}+\frac{43}{103}a^{22}+\frac{4}{103}a^{21}-\frac{6}{103}a^{20}+\frac{26}{103}a^{19}-\frac{10}{103}a^{18}+\frac{28}{103}a^{17}-\frac{31}{103}a^{16}-\frac{40}{103}a^{15}+\frac{31}{103}a^{14}-\frac{40}{103}a^{13}-\frac{48}{103}a^{12}-\frac{8}{103}a^{11}+\frac{38}{103}a^{10}+\frac{14}{103}a^{9}+\frac{27}{103}a^{8}+\frac{22}{103}a^{7}-\frac{6}{103}a^{6}+\frac{35}{103}a^{5}-\frac{5}{103}a^{4}+\frac{10}{103}a^{3}-\frac{39}{103}a^{2}+\frac{15}{103}a$, $\frac{1}{103}a^{35}-\frac{22}{103}a^{31}+\frac{21}{103}a^{30}-\frac{23}{103}a^{29}-\frac{29}{103}a^{28}+\frac{25}{103}a^{27}+\frac{16}{103}a^{26}-\frac{12}{103}a^{25}+\frac{22}{103}a^{24}-\frac{39}{103}a^{23}+\frac{24}{103}a^{22}+\frac{20}{103}a^{21}-\frac{5}{103}a^{20}-\frac{4}{103}a^{19}+\frac{1}{103}a^{18}+\frac{39}{103}a^{17}+\frac{49}{103}a^{16}+\frac{9}{103}a^{15}+\frac{48}{103}a^{14}+\frac{9}{103}a^{13}+\frac{16}{103}a^{12}-\frac{16}{103}a^{11}-\frac{15}{103}a^{10}+\frac{49}{103}a^{9}+\frac{32}{103}a^{8}-\frac{21}{103}a^{7}+\frac{4}{103}a^{6}-\frac{36}{103}a^{5}+\frac{12}{103}a^{4}+\frac{50}{103}a^{3}-\frac{13}{103}a^{2}-\frac{6}{103}a$, $\frac{1}{2369}a^{36}+\frac{8}{2369}a^{34}-\frac{7}{2369}a^{33}+\frac{4}{2369}a^{32}+\frac{845}{2369}a^{31}+\frac{378}{2369}a^{30}+\frac{263}{2369}a^{29}+\frac{812}{2369}a^{28}+\frac{952}{2369}a^{27}+\frac{1038}{2369}a^{26}-\frac{352}{2369}a^{25}-\frac{775}{2369}a^{24}-\frac{999}{2369}a^{23}+\frac{982}{2369}a^{22}+\frac{171}{2369}a^{21}-\frac{235}{2369}a^{20}-\frac{366}{2369}a^{19}-\frac{801}{2369}a^{18}+\frac{369}{2369}a^{17}-\frac{1169}{2369}a^{16}+\frac{888}{2369}a^{15}+\frac{808}{2369}a^{14}-\frac{227}{2369}a^{13}+\frac{270}{2369}a^{12}+\frac{642}{2369}a^{11}-\frac{226}{2369}a^{10}+\frac{20}{103}a^{9}-\frac{315}{2369}a^{8}-\frac{4}{103}a^{7}-\frac{448}{2369}a^{6}-\frac{1038}{2369}a^{5}+\frac{127}{2369}a^{4}-\frac{163}{2369}a^{3}+\frac{839}{2369}a^{2}-\frac{584}{2369}a+\frac{5}{23}$, $\frac{1}{626337541}a^{37}+\frac{124070}{626337541}a^{36}-\frac{1510448}{626337541}a^{35}+\frac{1265655}{626337541}a^{34}-\frac{2428070}{626337541}a^{33}+\frac{2726538}{626337541}a^{32}-\frac{258575974}{626337541}a^{31}-\frac{47913257}{626337541}a^{30}+\frac{55869893}{626337541}a^{29}+\frac{284906470}{626337541}a^{28}+\frac{292579280}{626337541}a^{27}+\frac{54730564}{626337541}a^{26}+\frac{94932175}{626337541}a^{25}-\frac{8605505}{27232067}a^{24}-\frac{188643902}{626337541}a^{23}+\frac{5949948}{27232067}a^{22}+\frac{224001123}{626337541}a^{21}-\frac{220560884}{626337541}a^{20}+\frac{40868605}{626337541}a^{19}+\frac{1307790}{626337541}a^{18}+\frac{214746389}{626337541}a^{17}-\frac{13289685}{27232067}a^{16}-\frac{111192}{264389}a^{15}-\frac{231505852}{626337541}a^{14}-\frac{290024968}{626337541}a^{13}+\frac{290475023}{626337541}a^{12}-\frac{203918104}{626337541}a^{11}-\frac{270748355}{626337541}a^{10}-\frac{130082772}{626337541}a^{9}+\frac{85395744}{626337541}a^{8}-\frac{33493853}{626337541}a^{7}+\frac{8812750}{626337541}a^{6}+\frac{113502435}{626337541}a^{5}-\frac{25737642}{626337541}a^{4}+\frac{129459727}{626337541}a^{3}+\frac{280264969}{626337541}a^{2}+\frac{101141563}{626337541}a-\frac{1798583}{6080947}$, $\frac{1}{27\!\cdots\!23}a^{38}-\frac{21\!\cdots\!34}{27\!\cdots\!23}a^{37}+\frac{51\!\cdots\!24}{27\!\cdots\!23}a^{36}-\frac{10\!\cdots\!25}{27\!\cdots\!23}a^{35}+\frac{43\!\cdots\!40}{27\!\cdots\!23}a^{34}-\frac{11\!\cdots\!86}{27\!\cdots\!23}a^{33}-\frac{10\!\cdots\!70}{27\!\cdots\!23}a^{32}-\frac{10\!\cdots\!76}{27\!\cdots\!23}a^{31}+\frac{40\!\cdots\!90}{12\!\cdots\!01}a^{30}-\frac{10\!\cdots\!72}{27\!\cdots\!23}a^{29}-\frac{18\!\cdots\!80}{27\!\cdots\!23}a^{28}+\frac{37\!\cdots\!45}{27\!\cdots\!23}a^{27}+\frac{11\!\cdots\!10}{27\!\cdots\!23}a^{26}-\frac{31\!\cdots\!40}{27\!\cdots\!23}a^{25}-\frac{13\!\cdots\!32}{27\!\cdots\!23}a^{24}-\frac{10\!\cdots\!76}{27\!\cdots\!23}a^{23}-\frac{10\!\cdots\!04}{27\!\cdots\!23}a^{22}-\frac{89\!\cdots\!88}{44\!\cdots\!33}a^{21}-\frac{70\!\cdots\!38}{27\!\cdots\!23}a^{20}-\frac{98\!\cdots\!88}{27\!\cdots\!23}a^{19}-\frac{34\!\cdots\!94}{27\!\cdots\!23}a^{18}-\frac{28\!\cdots\!24}{27\!\cdots\!23}a^{17}+\frac{48\!\cdots\!97}{27\!\cdots\!23}a^{16}+\frac{49\!\cdots\!07}{27\!\cdots\!23}a^{15}+\frac{49\!\cdots\!57}{27\!\cdots\!23}a^{14}-\frac{55\!\cdots\!30}{27\!\cdots\!23}a^{13}+\frac{10\!\cdots\!78}{27\!\cdots\!23}a^{12}+\frac{52\!\cdots\!65}{27\!\cdots\!23}a^{11}-\frac{90\!\cdots\!76}{27\!\cdots\!23}a^{10}-\frac{67\!\cdots\!11}{27\!\cdots\!23}a^{9}-\frac{50\!\cdots\!52}{27\!\cdots\!23}a^{8}-\frac{64\!\cdots\!57}{27\!\cdots\!23}a^{7}-\frac{13\!\cdots\!55}{27\!\cdots\!23}a^{6}-\frac{74\!\cdots\!17}{27\!\cdots\!23}a^{5}+\frac{55\!\cdots\!97}{12\!\cdots\!01}a^{4}+\frac{55\!\cdots\!75}{27\!\cdots\!23}a^{3}-\frac{91\!\cdots\!13}{27\!\cdots\!23}a^{2}-\frac{15\!\cdots\!41}{27\!\cdots\!23}a-\frac{36\!\cdots\!61}{20\!\cdots\!11}$ 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 - 237*x^37 - 158*x^36 + 24885*x^35 + 31758*x^34 - 1521777*x^33 - 2797074*x^32 + 59981382*x^31 + 142319290*x^30 - 1593173961*x^29 - 4645709076*x^28 + 28932877445*x^27 + 102241716102*x^26 - 355443646614*x^25 - 1553454595933*x^24 + 2810177785458*x^23 + 16432247086167*x^22 - 11967377146298*x^21 - 120635759398347*x^20 - 1823113254630*x^19 + 605746157617644*x^18 + 341698986985275*x^17 - 2017784741258409*x^16 - 2057859861184228*x^15 + 4196664129873231*x^14 + 6285983642002179*x^13 - 4725650025499719*x^12 - 10914822027031086*x^11 + 1465701883750416*x^10 + 10558980638642454*x^9 + 2159465693586741*x^8 - 5293388896015467*x^7 - 2202937031211426*x^6 + 1157156267855625*x^5 + 698944716032751*x^4 - 59632171643934*x^3 - 73740757923315*x^2 - 5146532589771*x + 1075512295747)
 
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 - 237*x^37 - 158*x^36 + 24885*x^35 + 31758*x^34 - 1521777*x^33 - 2797074*x^32 + 59981382*x^31 + 142319290*x^30 - 1593173961*x^29 - 4645709076*x^28 + 28932877445*x^27 + 102241716102*x^26 - 355443646614*x^25 - 1553454595933*x^24 + 2810177785458*x^23 + 16432247086167*x^22 - 11967377146298*x^21 - 120635759398347*x^20 - 1823113254630*x^19 + 605746157617644*x^18 + 341698986985275*x^17 - 2017784741258409*x^16 - 2057859861184228*x^15 + 4196664129873231*x^14 + 6285983642002179*x^13 - 4725650025499719*x^12 - 10914822027031086*x^11 + 1465701883750416*x^10 + 10558980638642454*x^9 + 2159465693586741*x^8 - 5293388896015467*x^7 - 2202937031211426*x^6 + 1157156267855625*x^5 + 698944716032751*x^4 - 59632171643934*x^3 - 73740757923315*x^2 - 5146532589771*x + 1075512295747, 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 - 237*x^37 - 158*x^36 + 24885*x^35 + 31758*x^34 - 1521777*x^33 - 2797074*x^32 + 59981382*x^31 + 142319290*x^30 - 1593173961*x^29 - 4645709076*x^28 + 28932877445*x^27 + 102241716102*x^26 - 355443646614*x^25 - 1553454595933*x^24 + 2810177785458*x^23 + 16432247086167*x^22 - 11967377146298*x^21 - 120635759398347*x^20 - 1823113254630*x^19 + 605746157617644*x^18 + 341698986985275*x^17 - 2017784741258409*x^16 - 2057859861184228*x^15 + 4196664129873231*x^14 + 6285983642002179*x^13 - 4725650025499719*x^12 - 10914822027031086*x^11 + 1465701883750416*x^10 + 10558980638642454*x^9 + 2159465693586741*x^8 - 5293388896015467*x^7 - 2202937031211426*x^6 + 1157156267855625*x^5 + 698944716032751*x^4 - 59632171643934*x^3 - 73740757923315*x^2 - 5146532589771*x + 1075512295747);
 
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 - 237*x^37 - 158*x^36 + 24885*x^35 + 31758*x^34 - 1521777*x^33 - 2797074*x^32 + 59981382*x^31 + 142319290*x^30 - 1593173961*x^29 - 4645709076*x^28 + 28932877445*x^27 + 102241716102*x^26 - 355443646614*x^25 - 1553454595933*x^24 + 2810177785458*x^23 + 16432247086167*x^22 - 11967377146298*x^21 - 120635759398347*x^20 - 1823113254630*x^19 + 605746157617644*x^18 + 341698986985275*x^17 - 2017784741258409*x^16 - 2057859861184228*x^15 + 4196664129873231*x^14 + 6285983642002179*x^13 - 4725650025499719*x^12 - 10914822027031086*x^11 + 1465701883750416*x^10 + 10558980638642454*x^9 + 2159465693586741*x^8 - 5293388896015467*x^7 - 2202937031211426*x^6 + 1157156267855625*x^5 + 698944716032751*x^4 - 59632171643934*x^3 - 73740757923315*x^2 - 5146532589771*x + 1075512295747);
 
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

3.3.505521.2, 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 $39$ ${\href{/padicField/7.13.0.1}{13} }^{3}$ ${\href{/padicField/11.13.0.1}{13} }^{3}$ ${\href{/padicField/13.13.0.1}{13} }^{3}$ ${\href{/padicField/17.13.0.1}{13} }^{3}$ $39$ ${\href{/padicField/23.3.0.1}{3} }^{13}$ ${\href{/padicField/29.13.0.1}{13} }^{3}$ $39$ $39$ $39$ $39$ ${\href{/padicField/47.13.0.1}{13} }^{3}$ $39$ ${\href{/padicField/59.13.0.1}{13} }^{3}$

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$$39$$1$$38$