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Properties

Label 39.39.108...489.1
Degree $39$
Signature $[39, 0]$
Discriminant $1.088\times 10^{91}$
Root discriminant \(215.91\)
Ramified primes $13,53$
Class number not computed
Class group not computed
Galois group $C_{39}$ (as 39T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^39 - 10*x^38 - 81*x^37 + 1050*x^36 + 2269*x^35 - 47944*x^34 - 10490*x^33 + 1262097*x^32 - 921080*x^31 - 21400660*x^30 + 27475516*x^29 + 247428139*x^28 - 406156179*x^27 - 2015287011*x^26 + 3799253180*x^25 + 11792160578*x^24 - 24197312569*x^23 - 50158998143*x^22 + 108120625100*x^21 + 156174553989*x^20 - 342706955337*x^19 - 357124870516*x^18 + 769896542138*x^17 + 599267077900*x^16 - 1212025785075*x^15 - 732019943274*x^14 + 1306427105242*x^13 + 636966693014*x^12 - 927524873227*x^11 - 377238926259*x^10 + 408532943088*x^9 + 139924561470*x^8 - 102630494231*x^7 - 28423351128*x^6 + 13445073140*x^5 + 2704930022*x^4 - 804059407*x^3 - 106744516*x^2 + 15239414*x + 1788197)
 
gp: K = bnfinit(y^39 - 10*y^38 - 81*y^37 + 1050*y^36 + 2269*y^35 - 47944*y^34 - 10490*y^33 + 1262097*y^32 - 921080*y^31 - 21400660*y^30 + 27475516*y^29 + 247428139*y^28 - 406156179*y^27 - 2015287011*y^26 + 3799253180*y^25 + 11792160578*y^24 - 24197312569*y^23 - 50158998143*y^22 + 108120625100*y^21 + 156174553989*y^20 - 342706955337*y^19 - 357124870516*y^18 + 769896542138*y^17 + 599267077900*y^16 - 1212025785075*y^15 - 732019943274*y^14 + 1306427105242*y^13 + 636966693014*y^12 - 927524873227*y^11 - 377238926259*y^10 + 408532943088*y^9 + 139924561470*y^8 - 102630494231*y^7 - 28423351128*y^6 + 13445073140*y^5 + 2704930022*y^4 - 804059407*y^3 - 106744516*y^2 + 15239414*y + 1788197, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^39 - 10*x^38 - 81*x^37 + 1050*x^36 + 2269*x^35 - 47944*x^34 - 10490*x^33 + 1262097*x^32 - 921080*x^31 - 21400660*x^30 + 27475516*x^29 + 247428139*x^28 - 406156179*x^27 - 2015287011*x^26 + 3799253180*x^25 + 11792160578*x^24 - 24197312569*x^23 - 50158998143*x^22 + 108120625100*x^21 + 156174553989*x^20 - 342706955337*x^19 - 357124870516*x^18 + 769896542138*x^17 + 599267077900*x^16 - 1212025785075*x^15 - 732019943274*x^14 + 1306427105242*x^13 + 636966693014*x^12 - 927524873227*x^11 - 377238926259*x^10 + 408532943088*x^9 + 139924561470*x^8 - 102630494231*x^7 - 28423351128*x^6 + 13445073140*x^5 + 2704930022*x^4 - 804059407*x^3 - 106744516*x^2 + 15239414*x + 1788197);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^39 - 10*x^38 - 81*x^37 + 1050*x^36 + 2269*x^35 - 47944*x^34 - 10490*x^33 + 1262097*x^32 - 921080*x^31 - 21400660*x^30 + 27475516*x^29 + 247428139*x^28 - 406156179*x^27 - 2015287011*x^26 + 3799253180*x^25 + 11792160578*x^24 - 24197312569*x^23 - 50158998143*x^22 + 108120625100*x^21 + 156174553989*x^20 - 342706955337*x^19 - 357124870516*x^18 + 769896542138*x^17 + 599267077900*x^16 - 1212025785075*x^15 - 732019943274*x^14 + 1306427105242*x^13 + 636966693014*x^12 - 927524873227*x^11 - 377238926259*x^10 + 408532943088*x^9 + 139924561470*x^8 - 102630494231*x^7 - 28423351128*x^6 + 13445073140*x^5 + 2704930022*x^4 - 804059407*x^3 - 106744516*x^2 + 15239414*x + 1788197)
 

\( x^{39} - 10 x^{38} - 81 x^{37} + 1050 x^{36} + 2269 x^{35} - 47944 x^{34} - 10490 x^{33} + \cdots + 1788197 \) 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:   \(108\!\cdots\!489\) \(\medspace = 13^{26}\cdot 53^{36}\) 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:  \(215.91\)
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:  $13^{2/3}53^{12/13}\approx 215.90771728173132$
Ramified primes:   \(13\), \(53\) 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:  \(689=13\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{689}(256,·)$, $\chi_{689}(1,·)$, $\chi_{689}(386,·)$, $\chi_{689}(261,·)$, $\chi_{689}(646,·)$, $\chi_{689}(521,·)$, $\chi_{689}(523,·)$, $\chi_{689}(399,·)$, $\chi_{689}(16,·)$, $\chi_{689}(152,·)$, $\chi_{689}(664,·)$, $\chi_{689}(672,·)$, $\chi_{689}(289,·)$, $\chi_{689}(425,·)$, $\chi_{689}(42,·)$, $\chi_{689}(172,·)$, $\chi_{689}(685,·)$, $\chi_{689}(183,·)$, $\chi_{689}(66,·)$, $\chi_{689}(651,·)$, $\chi_{689}(68,·)$, $\chi_{689}(417,·)$, $\chi_{689}(328,·)$, $\chi_{689}(599,·)$, $\chi_{689}(334,·)$, $\chi_{689}(81,·)$, $\chi_{689}(471,·)$, $\chi_{689}(222,·)$, $\chi_{689}(607,·)$, $\chi_{689}(354,·)$, $\chi_{689}(100,·)$, $\chi_{689}(360,·)$, $\chi_{689}(490,·)$, $\chi_{689}(107,·)$, $\chi_{689}(365,·)$, $\chi_{689}(367,·)$, $\chi_{689}(625,·)$, $\chi_{689}(627,·)$, $\chi_{689}(248,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{1909}a^{33}+\frac{458}{1909}a^{32}-\frac{490}{1909}a^{31}+\frac{934}{1909}a^{30}+\frac{369}{1909}a^{29}+\frac{510}{1909}a^{28}-\frac{783}{1909}a^{27}+\frac{559}{1909}a^{26}+\frac{779}{1909}a^{25}+\frac{887}{1909}a^{24}-\frac{266}{1909}a^{23}-\frac{545}{1909}a^{22}-\frac{786}{1909}a^{21}-\frac{326}{1909}a^{20}-\frac{324}{1909}a^{19}+\frac{504}{1909}a^{18}+\frac{240}{1909}a^{17}+\frac{8}{1909}a^{16}+\frac{123}{1909}a^{15}+\frac{201}{1909}a^{14}+\frac{886}{1909}a^{13}+\frac{675}{1909}a^{12}-\frac{297}{1909}a^{11}+\frac{111}{1909}a^{10}+\frac{899}{1909}a^{9}-\frac{619}{1909}a^{8}+\frac{40}{1909}a^{7}+\frac{11}{83}a^{6}-\frac{420}{1909}a^{5}-\frac{252}{1909}a^{4}+\frac{394}{1909}a^{3}+\frac{22}{83}a^{2}+\frac{445}{1909}a+\frac{183}{1909}$, $\frac{1}{1909}a^{34}-\frac{264}{1909}a^{32}+\frac{4}{83}a^{31}+\frac{213}{1909}a^{30}-\frac{500}{1909}a^{29}+\frac{444}{1909}a^{28}+\frac{281}{1909}a^{27}+\frac{563}{1909}a^{26}-\frac{821}{1909}a^{25}+\frac{105}{1909}a^{24}-\frac{893}{1909}a^{23}+\frac{654}{1909}a^{22}+\frac{770}{1909}a^{21}+\frac{82}{1909}a^{20}-\frac{6}{1909}a^{19}+\frac{397}{1909}a^{18}+\frac{810}{1909}a^{17}+\frac{277}{1909}a^{16}-\frac{772}{1909}a^{15}+\frac{20}{83}a^{14}-\frac{405}{1909}a^{13}-\frac{189}{1909}a^{12}+\frac{26}{83}a^{11}-\frac{305}{1909}a^{10}-\frac{17}{1909}a^{9}-\frac{899}{1909}a^{8}-\frac{886}{1909}a^{7}+\frac{155}{1909}a^{6}-\frac{701}{1909}a^{5}-\frac{639}{1909}a^{4}-\frac{500}{1909}a^{3}-\frac{314}{1909}a^{2}+\frac{636}{1909}a+\frac{182}{1909}$, $\frac{1}{1909}a^{35}+\frac{737}{1909}a^{32}+\frac{665}{1909}a^{31}-\frac{185}{1909}a^{30}+\frac{501}{1909}a^{29}-\frac{618}{1909}a^{28}+\frac{1}{83}a^{27}-\frac{238}{1909}a^{26}-\frac{411}{1909}a^{25}+\frac{377}{1909}a^{24}-\frac{846}{1909}a^{23}+\frac{65}{1909}a^{22}+\frac{659}{1909}a^{21}-\frac{165}{1909}a^{20}+\frac{766}{1909}a^{19}+\frac{236}{1909}a^{18}+\frac{640}{1909}a^{17}-\frac{569}{1909}a^{16}+\frac{479}{1909}a^{15}-\frac{793}{1909}a^{14}+\frac{817}{1909}a^{13}-\frac{648}{1909}a^{12}-\frac{444}{1909}a^{11}+\frac{652}{1909}a^{10}-\frac{279}{1909}a^{9}-\frac{128}{1909}a^{8}-\frac{739}{1909}a^{7}-\frac{724}{1909}a^{6}-\frac{797}{1909}a^{5}-\frac{213}{1909}a^{4}+\frac{616}{1909}a^{3}+\frac{590}{1909}a^{2}-\frac{696}{1909}a+\frac{587}{1909}$, $\frac{1}{435105007}a^{36}-\frac{67040}{435105007}a^{35}-\frac{30345}{435105007}a^{34}+\frac{107426}{435105007}a^{33}+\frac{14787842}{435105007}a^{32}-\frac{192056280}{435105007}a^{31}-\frac{155487898}{435105007}a^{30}-\frac{101266824}{435105007}a^{29}+\frac{144968881}{435105007}a^{28}+\frac{110940851}{435105007}a^{27}+\frac{187785596}{435105007}a^{26}-\frac{127715305}{435105007}a^{25}-\frac{206542021}{435105007}a^{24}-\frac{213921366}{435105007}a^{23}-\frac{4050441}{435105007}a^{22}-\frac{5704031}{435105007}a^{21}-\frac{67628020}{435105007}a^{20}+\frac{183990767}{435105007}a^{19}+\frac{70419462}{435105007}a^{18}+\frac{49285945}{435105007}a^{17}-\frac{6942345}{18917609}a^{16}+\frac{180802917}{435105007}a^{15}-\frac{147712297}{435105007}a^{14}-\frac{202912644}{435105007}a^{13}-\frac{181298839}{435105007}a^{12}+\frac{38581767}{435105007}a^{11}+\frac{211279934}{435105007}a^{10}+\frac{3099037}{18917609}a^{9}-\frac{176235636}{435105007}a^{8}-\frac{46354611}{435105007}a^{7}+\frac{61759164}{435105007}a^{6}-\frac{52934560}{435105007}a^{5}-\frac{188547735}{435105007}a^{4}-\frac{330328}{1372571}a^{3}+\frac{158365631}{435105007}a^{2}-\frac{4845957}{435105007}a+\frac{69526}{1372571}$, $\frac{1}{461646412427}a^{37}-\frac{495}{461646412427}a^{36}-\frac{26053488}{461646412427}a^{35}-\frac{85273944}{461646412427}a^{34}+\frac{58863179}{461646412427}a^{33}+\frac{191373655611}{461646412427}a^{32}-\frac{186689691871}{461646412427}a^{31}-\frac{80652820791}{461646412427}a^{30}-\frac{152785876980}{461646412427}a^{29}+\frac{198107128519}{461646412427}a^{28}+\frac{116749056020}{461646412427}a^{27}+\frac{919667552}{5562004969}a^{26}-\frac{62087447292}{461646412427}a^{25}-\frac{122180988844}{461646412427}a^{24}+\frac{229964781395}{461646412427}a^{23}+\frac{165746868408}{461646412427}a^{22}-\frac{24168189963}{461646412427}a^{21}+\frac{47006435845}{461646412427}a^{20}+\frac{31996830735}{461646412427}a^{19}+\frac{128666642085}{461646412427}a^{18}-\frac{196429082158}{461646412427}a^{17}-\frac{157312860414}{461646412427}a^{16}+\frac{196897723558}{461646412427}a^{15}-\frac{24505356651}{461646412427}a^{14}+\frac{56523317358}{461646412427}a^{13}-\frac{30362706419}{461646412427}a^{12}+\frac{206179463469}{461646412427}a^{11}-\frac{170881782281}{461646412427}a^{10}-\frac{165307291920}{461646412427}a^{9}+\frac{75772246702}{461646412427}a^{8}-\frac{51496803339}{461646412427}a^{7}-\frac{180036582331}{461646412427}a^{6}-\frac{105347137207}{461646412427}a^{5}-\frac{103267180649}{461646412427}a^{4}+\frac{224816024077}{461646412427}a^{3}-\frac{4504465709}{461646412427}a^{2}+\frac{78349514985}{461646412427}a-\frac{355412354}{1456297831}$, $\frac{1}{34\!\cdots\!33}a^{38}-\frac{16\!\cdots\!05}{34\!\cdots\!33}a^{37}+\frac{38\!\cdots\!19}{34\!\cdots\!33}a^{36}-\frac{42\!\cdots\!72}{34\!\cdots\!33}a^{35}+\frac{56\!\cdots\!85}{34\!\cdots\!33}a^{34}+\frac{52\!\cdots\!10}{34\!\cdots\!33}a^{33}-\frac{13\!\cdots\!81}{34\!\cdots\!33}a^{32}-\frac{11\!\cdots\!32}{34\!\cdots\!33}a^{31}+\frac{92\!\cdots\!95}{34\!\cdots\!33}a^{30}-\frac{80\!\cdots\!02}{34\!\cdots\!33}a^{29}-\frac{68\!\cdots\!81}{34\!\cdots\!33}a^{28}+\frac{16\!\cdots\!76}{34\!\cdots\!33}a^{27}+\frac{52\!\cdots\!32}{34\!\cdots\!33}a^{26}-\frac{60\!\cdots\!93}{34\!\cdots\!33}a^{25}+\frac{14\!\cdots\!72}{34\!\cdots\!33}a^{24}-\frac{16\!\cdots\!34}{34\!\cdots\!33}a^{23}-\frac{81\!\cdots\!84}{34\!\cdots\!33}a^{22}+\frac{28\!\cdots\!15}{34\!\cdots\!33}a^{21}+\frac{41\!\cdots\!44}{34\!\cdots\!33}a^{20}-\frac{39\!\cdots\!29}{34\!\cdots\!33}a^{19}+\frac{63\!\cdots\!10}{34\!\cdots\!33}a^{18}+\frac{48\!\cdots\!21}{34\!\cdots\!33}a^{17}+\frac{77\!\cdots\!26}{34\!\cdots\!33}a^{16}-\frac{16\!\cdots\!53}{34\!\cdots\!33}a^{15}+\frac{21\!\cdots\!18}{34\!\cdots\!33}a^{14}-\frac{16\!\cdots\!04}{34\!\cdots\!33}a^{13}-\frac{11\!\cdots\!59}{34\!\cdots\!33}a^{12}+\frac{10\!\cdots\!48}{34\!\cdots\!33}a^{11}-\frac{24\!\cdots\!98}{34\!\cdots\!33}a^{10}-\frac{10\!\cdots\!84}{34\!\cdots\!33}a^{9}+\frac{68\!\cdots\!75}{15\!\cdots\!71}a^{8}+\frac{17\!\cdots\!66}{34\!\cdots\!33}a^{7}-\frac{14\!\cdots\!11}{34\!\cdots\!33}a^{6}-\frac{47\!\cdots\!58}{34\!\cdots\!33}a^{5}+\frac{16\!\cdots\!49}{15\!\cdots\!71}a^{4}-\frac{15\!\cdots\!07}{34\!\cdots\!33}a^{3}-\frac{95\!\cdots\!18}{34\!\cdots\!33}a^{2}+\frac{91\!\cdots\!43}{34\!\cdots\!33}a+\frac{32\!\cdots\!69}{10\!\cdots\!49}$ 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 - 81*x^37 + 1050*x^36 + 2269*x^35 - 47944*x^34 - 10490*x^33 + 1262097*x^32 - 921080*x^31 - 21400660*x^30 + 27475516*x^29 + 247428139*x^28 - 406156179*x^27 - 2015287011*x^26 + 3799253180*x^25 + 11792160578*x^24 - 24197312569*x^23 - 50158998143*x^22 + 108120625100*x^21 + 156174553989*x^20 - 342706955337*x^19 - 357124870516*x^18 + 769896542138*x^17 + 599267077900*x^16 - 1212025785075*x^15 - 732019943274*x^14 + 1306427105242*x^13 + 636966693014*x^12 - 927524873227*x^11 - 377238926259*x^10 + 408532943088*x^9 + 139924561470*x^8 - 102630494231*x^7 - 28423351128*x^6 + 13445073140*x^5 + 2704930022*x^4 - 804059407*x^3 - 106744516*x^2 + 15239414*x + 1788197)
 
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 - 81*x^37 + 1050*x^36 + 2269*x^35 - 47944*x^34 - 10490*x^33 + 1262097*x^32 - 921080*x^31 - 21400660*x^30 + 27475516*x^29 + 247428139*x^28 - 406156179*x^27 - 2015287011*x^26 + 3799253180*x^25 + 11792160578*x^24 - 24197312569*x^23 - 50158998143*x^22 + 108120625100*x^21 + 156174553989*x^20 - 342706955337*x^19 - 357124870516*x^18 + 769896542138*x^17 + 599267077900*x^16 - 1212025785075*x^15 - 732019943274*x^14 + 1306427105242*x^13 + 636966693014*x^12 - 927524873227*x^11 - 377238926259*x^10 + 408532943088*x^9 + 139924561470*x^8 - 102630494231*x^7 - 28423351128*x^6 + 13445073140*x^5 + 2704930022*x^4 - 804059407*x^3 - 106744516*x^2 + 15239414*x + 1788197, 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 - 81*x^37 + 1050*x^36 + 2269*x^35 - 47944*x^34 - 10490*x^33 + 1262097*x^32 - 921080*x^31 - 21400660*x^30 + 27475516*x^29 + 247428139*x^28 - 406156179*x^27 - 2015287011*x^26 + 3799253180*x^25 + 11792160578*x^24 - 24197312569*x^23 - 50158998143*x^22 + 108120625100*x^21 + 156174553989*x^20 - 342706955337*x^19 - 357124870516*x^18 + 769896542138*x^17 + 599267077900*x^16 - 1212025785075*x^15 - 732019943274*x^14 + 1306427105242*x^13 + 636966693014*x^12 - 927524873227*x^11 - 377238926259*x^10 + 408532943088*x^9 + 139924561470*x^8 - 102630494231*x^7 - 28423351128*x^6 + 13445073140*x^5 + 2704930022*x^4 - 804059407*x^3 - 106744516*x^2 + 15239414*x + 1788197);
 
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 - 81*x^37 + 1050*x^36 + 2269*x^35 - 47944*x^34 - 10490*x^33 + 1262097*x^32 - 921080*x^31 - 21400660*x^30 + 27475516*x^29 + 247428139*x^28 - 406156179*x^27 - 2015287011*x^26 + 3799253180*x^25 + 11792160578*x^24 - 24197312569*x^23 - 50158998143*x^22 + 108120625100*x^21 + 156174553989*x^20 - 342706955337*x^19 - 357124870516*x^18 + 769896542138*x^17 + 599267077900*x^16 - 1212025785075*x^15 - 732019943274*x^14 + 1306427105242*x^13 + 636966693014*x^12 - 927524873227*x^11 - 377238926259*x^10 + 408532943088*x^9 + 139924561470*x^8 - 102630494231*x^7 - 28423351128*x^6 + 13445073140*x^5 + 2704930022*x^4 - 804059407*x^3 - 106744516*x^2 + 15239414*x + 1788197);
 
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, 13.13.491258904256726154641.1

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 $39$ $39$ ${\href{/padicField/23.3.0.1}{3} }^{13}$ $39$ ${\href{/padicField/31.13.0.1}{13} }^{3}$ $39$ $39$ $39$ ${\href{/padicField/47.13.0.1}{13} }^{3}$ R $39$

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$
\(53\) Copy content Toggle raw display 53.13.12.1$x^{13} + 53$$13$$1$$12$$C_{13}$$[\ ]_{13}$
53.13.12.1$x^{13} + 53$$13$$1$$12$$C_{13}$$[\ ]_{13}$
53.13.12.1$x^{13} + 53$$13$$1$$12$$C_{13}$$[\ ]_{13}$