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Properties

Label 39.39.120...489.1
Degree $39$
Signature $[39, 0]$
Discriminant $1.209\times 10^{94}$
Root discriminant \(258.45\)
Ramified primes $7,79$
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 - x^38 - 196*x^37 + 37*x^36 + 17177*x^35 + 9403*x^34 - 882381*x^33 - 1067223*x^32 + 29336001*x^31 + 53273410*x^30 - 658772565*x^29 - 1583577890*x^28 + 10123695780*x^27 + 30773254905*x^26 - 105181421187*x^25 - 406462457224*x^24 + 701275142050*x^23 + 3703719049533*x^22 - 2495293357664*x^21 - 23296951876563*x^20 - 582451600575*x^19 + 100148494787668*x^18 + 52233018535617*x^17 - 288162921569191*x^16 - 262264312267859*x^15 + 535079835460586*x^14 + 679366178486570*x^13 - 600819998721797*x^12 - 1028790083129260*x^11 + 359936453258218*x^10 + 916306649955834*x^9 - 86249482058917*x^8 - 469190872234743*x^7 + 4041363119341*x^6 + 135236928048853*x^5 - 4445757797975*x^4 - 19527462450618*x^3 + 2244998319481*x^2 + 789598736860*x - 116423735903)
 
gp: K = bnfinit(y^39 - y^38 - 196*y^37 + 37*y^36 + 17177*y^35 + 9403*y^34 - 882381*y^33 - 1067223*y^32 + 29336001*y^31 + 53273410*y^30 - 658772565*y^29 - 1583577890*y^28 + 10123695780*y^27 + 30773254905*y^26 - 105181421187*y^25 - 406462457224*y^24 + 701275142050*y^23 + 3703719049533*y^22 - 2495293357664*y^21 - 23296951876563*y^20 - 582451600575*y^19 + 100148494787668*y^18 + 52233018535617*y^17 - 288162921569191*y^16 - 262264312267859*y^15 + 535079835460586*y^14 + 679366178486570*y^13 - 600819998721797*y^12 - 1028790083129260*y^11 + 359936453258218*y^10 + 916306649955834*y^9 - 86249482058917*y^8 - 469190872234743*y^7 + 4041363119341*y^6 + 135236928048853*y^5 - 4445757797975*y^4 - 19527462450618*y^3 + 2244998319481*y^2 + 789598736860*y - 116423735903, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^39 - x^38 - 196*x^37 + 37*x^36 + 17177*x^35 + 9403*x^34 - 882381*x^33 - 1067223*x^32 + 29336001*x^31 + 53273410*x^30 - 658772565*x^29 - 1583577890*x^28 + 10123695780*x^27 + 30773254905*x^26 - 105181421187*x^25 - 406462457224*x^24 + 701275142050*x^23 + 3703719049533*x^22 - 2495293357664*x^21 - 23296951876563*x^20 - 582451600575*x^19 + 100148494787668*x^18 + 52233018535617*x^17 - 288162921569191*x^16 - 262264312267859*x^15 + 535079835460586*x^14 + 679366178486570*x^13 - 600819998721797*x^12 - 1028790083129260*x^11 + 359936453258218*x^10 + 916306649955834*x^9 - 86249482058917*x^8 - 469190872234743*x^7 + 4041363119341*x^6 + 135236928048853*x^5 - 4445757797975*x^4 - 19527462450618*x^3 + 2244998319481*x^2 + 789598736860*x - 116423735903);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^39 - x^38 - 196*x^37 + 37*x^36 + 17177*x^35 + 9403*x^34 - 882381*x^33 - 1067223*x^32 + 29336001*x^31 + 53273410*x^30 - 658772565*x^29 - 1583577890*x^28 + 10123695780*x^27 + 30773254905*x^26 - 105181421187*x^25 - 406462457224*x^24 + 701275142050*x^23 + 3703719049533*x^22 - 2495293357664*x^21 - 23296951876563*x^20 - 582451600575*x^19 + 100148494787668*x^18 + 52233018535617*x^17 - 288162921569191*x^16 - 262264312267859*x^15 + 535079835460586*x^14 + 679366178486570*x^13 - 600819998721797*x^12 - 1028790083129260*x^11 + 359936453258218*x^10 + 916306649955834*x^9 - 86249482058917*x^8 - 469190872234743*x^7 + 4041363119341*x^6 + 135236928048853*x^5 - 4445757797975*x^4 - 19527462450618*x^3 + 2244998319481*x^2 + 789598736860*x - 116423735903)
 

\( x^{39} - x^{38} - 196 x^{37} + 37 x^{36} + 17177 x^{35} + 9403 x^{34} - 882381 x^{33} + \cdots - 116423735903 \) 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:   \(120\!\cdots\!489\) \(\medspace = 7^{26}\cdot 79^{38}\) 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:  \(258.45\)
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:  $7^{2/3}79^{38/39}\approx 258.44532445081165$
Ramified primes:   \(7\), \(79\) 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:  \(553=7\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{553}(512,·)$, $\chi_{553}(1,·)$, $\chi_{553}(2,·)$, $\chi_{553}(4,·)$, $\chi_{553}(389,·)$, $\chi_{553}(263,·)$, $\chi_{553}(8,·)$, $\chi_{553}(128,·)$, $\chi_{553}(11,·)$, $\chi_{553}(141,·)$, $\chi_{553}(526,·)$, $\chi_{553}(16,·)$, $\chi_{553}(277,·)$, $\chi_{553}(22,·)$, $\chi_{553}(151,·)$, $\chi_{553}(408,·)$, $\chi_{553}(282,·)$, $\chi_{553}(415,·)$, $\chi_{553}(32,·)$, $\chi_{553}(44,·)$, $\chi_{553}(302,·)$, $\chi_{553}(176,·)$, $\chi_{553}(51,·)$, $\chi_{553}(445,·)$, $\chi_{553}(64,·)$, $\chi_{553}(450,·)$, $\chi_{553}(256,·)$, $\chi_{553}(204,·)$, $\chi_{553}(337,·)$, $\chi_{553}(471,·)$, $\chi_{553}(88,·)$, $\chi_{553}(347,·)$, $\chi_{553}(352,·)$, $\chi_{553}(225,·)$, $\chi_{553}(484,·)$, $\chi_{553}(102,·)$, $\chi_{553}(242,·)$, $\chi_{553}(499,·)$, $\chi_{553}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{23}a^{21}+\frac{9}{23}a^{20}-\frac{2}{23}a^{19}-\frac{6}{23}a^{18}-\frac{3}{23}a^{17}+\frac{11}{23}a^{16}+\frac{3}{23}a^{15}+\frac{11}{23}a^{14}+\frac{11}{23}a^{13}-\frac{9}{23}a^{12}-\frac{5}{23}a^{11}-\frac{11}{23}a^{10}-\frac{6}{23}a^{9}+\frac{8}{23}a^{8}-\frac{5}{23}a^{7}+\frac{4}{23}a^{6}-\frac{9}{23}a^{5}+\frac{1}{23}a^{4}-\frac{3}{23}a^{3}+\frac{5}{23}a^{2}-\frac{5}{23}a$, $\frac{1}{23}a^{22}+\frac{9}{23}a^{20}-\frac{11}{23}a^{19}+\frac{5}{23}a^{18}-\frac{8}{23}a^{17}-\frac{4}{23}a^{16}+\frac{7}{23}a^{15}+\frac{4}{23}a^{14}+\frac{7}{23}a^{13}+\frac{7}{23}a^{12}+\frac{11}{23}a^{11}+\frac{1}{23}a^{10}-\frac{7}{23}a^{9}-\frac{8}{23}a^{8}+\frac{3}{23}a^{7}+\frac{1}{23}a^{6}-\frac{10}{23}a^{5}+\frac{11}{23}a^{4}+\frac{9}{23}a^{3}-\frac{4}{23}a^{2}-\frac{1}{23}a$, $\frac{1}{23}a^{23}-\frac{1}{23}a$, $\frac{1}{23}a^{24}-\frac{1}{23}a^{2}$, $\frac{1}{23}a^{25}-\frac{1}{23}a^{3}$, $\frac{1}{23}a^{26}-\frac{1}{23}a^{4}$, $\frac{1}{23}a^{27}-\frac{1}{23}a^{5}$, $\frac{1}{23}a^{28}-\frac{1}{23}a^{6}$, $\frac{1}{23}a^{29}-\frac{1}{23}a^{7}$, $\frac{1}{23}a^{30}-\frac{1}{23}a^{8}$, $\frac{1}{23}a^{31}-\frac{1}{23}a^{9}$, $\frac{1}{23}a^{32}-\frac{1}{23}a^{10}$, $\frac{1}{529}a^{33}+\frac{1}{529}a^{32}-\frac{6}{529}a^{31}-\frac{2}{529}a^{30}-\frac{5}{529}a^{29}-\frac{6}{529}a^{28}-\frac{5}{529}a^{27}-\frac{9}{529}a^{26}+\frac{11}{529}a^{25}-\frac{1}{529}a^{24}-\frac{4}{529}a^{23}+\frac{10}{529}a^{22}+\frac{251}{529}a^{20}+\frac{212}{529}a^{19}+\frac{4}{529}a^{18}-\frac{11}{529}a^{17}-\frac{224}{529}a^{16}+\frac{70}{529}a^{15}+\frac{201}{529}a^{14}-\frac{206}{529}a^{13}+\frac{254}{529}a^{12}+\frac{201}{529}a^{11}-\frac{14}{529}a^{10}+\frac{74}{529}a^{9}+\frac{14}{529}a^{8}+\frac{127}{529}a^{7}+\frac{246}{529}a^{6}+\frac{20}{529}a^{5}+\frac{188}{529}a^{4}-\frac{174}{529}a^{3}-\frac{108}{529}a^{2}-\frac{167}{529}a+\frac{5}{23}$, $\frac{1}{529}a^{34}-\frac{7}{529}a^{32}+\frac{4}{529}a^{31}-\frac{3}{529}a^{30}-\frac{1}{529}a^{29}+\frac{1}{529}a^{28}-\frac{4}{529}a^{27}-\frac{3}{529}a^{26}+\frac{11}{529}a^{25}-\frac{3}{529}a^{24}-\frac{9}{529}a^{23}-\frac{10}{529}a^{22}-\frac{2}{529}a^{21}-\frac{200}{529}a^{20}-\frac{231}{529}a^{19}-\frac{84}{529}a^{18}+\frac{17}{529}a^{17}+\frac{156}{529}a^{16}-\frac{99}{529}a^{15}-\frac{16}{529}a^{14}-\frac{9}{23}a^{13}+\frac{108}{529}a^{12}-\frac{8}{529}a^{11}+\frac{226}{529}a^{10}-\frac{129}{529}a^{9}+\frac{205}{529}a^{8}-\frac{203}{529}a^{7}-\frac{180}{529}a^{6}-\frac{200}{529}a^{5}-\frac{63}{529}a^{4}-\frac{256}{529}a^{3}+\frac{263}{529}a^{2}-\frac{17}{529}a-\frac{5}{23}$, $\frac{1}{529}a^{35}+\frac{11}{529}a^{32}+\frac{1}{529}a^{31}+\frac{8}{529}a^{30}-\frac{11}{529}a^{29}+\frac{8}{529}a^{27}-\frac{6}{529}a^{26}+\frac{5}{529}a^{25}+\frac{7}{529}a^{24}+\frac{8}{529}a^{23}-\frac{1}{529}a^{22}+\frac{7}{529}a^{21}+\frac{123}{529}a^{20}+\frac{158}{529}a^{19}+\frac{45}{529}a^{18}+\frac{10}{529}a^{17}-\frac{172}{529}a^{16}+\frac{83}{529}a^{15}+\frac{27}{529}a^{14}-\frac{3}{23}a^{13}-\frac{47}{529}a^{12}-\frac{7}{23}a^{11}+\frac{72}{529}a^{10}-\frac{82}{529}a^{9}-\frac{36}{529}a^{8}-\frac{27}{529}a^{7}+\frac{119}{529}a^{6}-\frac{84}{529}a^{5}-\frac{67}{529}a^{4}-\frac{12}{529}a^{3}-\frac{14}{529}a^{2}-\frac{180}{529}a-\frac{11}{23}$, $\frac{1}{12167}a^{36}+\frac{5}{12167}a^{35}-\frac{6}{12167}a^{34}-\frac{5}{12167}a^{33}+\frac{59}{12167}a^{32}-\frac{122}{12167}a^{31}-\frac{243}{12167}a^{30}+\frac{31}{12167}a^{29}+\frac{190}{12167}a^{28}-\frac{8}{529}a^{27}-\frac{70}{12167}a^{26}-\frac{49}{12167}a^{25}-\frac{199}{12167}a^{24}-\frac{27}{12167}a^{23}-\frac{75}{12167}a^{22}-\frac{14}{12167}a^{21}-\frac{4550}{12167}a^{20}+\frac{5292}{12167}a^{19}+\frac{2423}{12167}a^{18}-\frac{738}{12167}a^{17}+\frac{5045}{12167}a^{16}+\frac{4286}{12167}a^{15}+\frac{4536}{12167}a^{14}+\frac{4928}{12167}a^{13}+\frac{4644}{12167}a^{12}+\frac{3620}{12167}a^{11}+\frac{2803}{12167}a^{10}-\frac{3938}{12167}a^{9}-\frac{879}{12167}a^{8}+\frac{159}{12167}a^{7}-\frac{1034}{12167}a^{6}-\frac{5794}{12167}a^{5}+\frac{5234}{12167}a^{4}+\frac{4844}{12167}a^{3}+\frac{3396}{12167}a^{2}+\frac{1644}{12167}a+\frac{125}{529}$, $\frac{1}{12167}a^{37}-\frac{8}{12167}a^{35}+\frac{2}{12167}a^{34}-\frac{8}{12167}a^{33}-\frac{95}{12167}a^{32}-\frac{208}{12167}a^{31}+\frac{96}{12167}a^{30}-\frac{264}{12167}a^{29}-\frac{76}{12167}a^{28}-\frac{1}{12167}a^{27}+\frac{2}{12167}a^{26}-\frac{2}{529}a^{25}+\frac{232}{12167}a^{24}-\frac{239}{12167}a^{23}+\frac{177}{12167}a^{22}-\frac{41}{12167}a^{21}-\frac{5607}{12167}a^{20}-\frac{209}{12167}a^{19}+\frac{3500}{12167}a^{18}+\frac{4825}{12167}a^{17}+\frac{60}{12167}a^{16}-\frac{2749}{12167}a^{15}+\frac{1246}{12167}a^{14}+\frac{3717}{12167}a^{13}+\frac{4251}{12167}a^{12}-\frac{3981}{12167}a^{11}-\frac{5395}{12167}a^{10}-\frac{2786}{12167}a^{9}+\frac{200}{529}a^{8}-\frac{4175}{12167}a^{7}+\frac{549}{12167}a^{6}+\frac{5408}{12167}a^{5}-\frac{3271}{12167}a^{4}+\frac{3970}{12167}a^{3}-\frac{3836}{12167}a^{2}-\frac{2194}{12167}a-\frac{165}{529}$, $\frac{1}{49\!\cdots\!77}a^{38}+\frac{39\!\cdots\!23}{49\!\cdots\!77}a^{37}+\frac{40\!\cdots\!97}{49\!\cdots\!77}a^{36}-\frac{36\!\cdots\!22}{49\!\cdots\!77}a^{35}-\frac{21\!\cdots\!66}{49\!\cdots\!77}a^{34}-\frac{21\!\cdots\!36}{49\!\cdots\!77}a^{33}+\frac{10\!\cdots\!10}{49\!\cdots\!77}a^{32}-\frac{10\!\cdots\!63}{49\!\cdots\!77}a^{31}+\frac{68\!\cdots\!46}{49\!\cdots\!77}a^{30}-\frac{62\!\cdots\!71}{49\!\cdots\!77}a^{29}+\frac{26\!\cdots\!25}{49\!\cdots\!77}a^{28}+\frac{85\!\cdots\!88}{49\!\cdots\!77}a^{27}+\frac{65\!\cdots\!80}{49\!\cdots\!77}a^{26}+\frac{61\!\cdots\!45}{49\!\cdots\!77}a^{25}-\frac{10\!\cdots\!35}{49\!\cdots\!77}a^{24}-\frac{19\!\cdots\!78}{49\!\cdots\!77}a^{23}-\frac{50\!\cdots\!04}{49\!\cdots\!77}a^{22}+\frac{65\!\cdots\!96}{49\!\cdots\!77}a^{21}-\frac{96\!\cdots\!97}{49\!\cdots\!77}a^{20}-\frac{10\!\cdots\!39}{49\!\cdots\!77}a^{19}+\frac{98\!\cdots\!39}{49\!\cdots\!77}a^{18}-\frac{20\!\cdots\!38}{49\!\cdots\!77}a^{17}+\frac{97\!\cdots\!12}{49\!\cdots\!77}a^{16}+\frac{20\!\cdots\!66}{49\!\cdots\!77}a^{15}-\frac{15\!\cdots\!44}{49\!\cdots\!77}a^{14}+\frac{20\!\cdots\!95}{49\!\cdots\!77}a^{13}-\frac{10\!\cdots\!99}{49\!\cdots\!77}a^{12}-\frac{64\!\cdots\!44}{49\!\cdots\!77}a^{11}+\frac{90\!\cdots\!86}{49\!\cdots\!77}a^{10}+\frac{89\!\cdots\!83}{49\!\cdots\!77}a^{9}-\frac{82\!\cdots\!13}{49\!\cdots\!77}a^{8}+\frac{18\!\cdots\!66}{49\!\cdots\!77}a^{7}-\frac{23\!\cdots\!31}{49\!\cdots\!77}a^{6}+\frac{84\!\cdots\!34}{49\!\cdots\!77}a^{5}+\frac{16\!\cdots\!76}{49\!\cdots\!77}a^{4}-\frac{24\!\cdots\!92}{49\!\cdots\!77}a^{3}-\frac{17\!\cdots\!06}{49\!\cdots\!77}a^{2}+\frac{19\!\cdots\!31}{49\!\cdots\!77}a-\frac{51\!\cdots\!13}{34\!\cdots\!29}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $23$

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 - x^38 - 196*x^37 + 37*x^36 + 17177*x^35 + 9403*x^34 - 882381*x^33 - 1067223*x^32 + 29336001*x^31 + 53273410*x^30 - 658772565*x^29 - 1583577890*x^28 + 10123695780*x^27 + 30773254905*x^26 - 105181421187*x^25 - 406462457224*x^24 + 701275142050*x^23 + 3703719049533*x^22 - 2495293357664*x^21 - 23296951876563*x^20 - 582451600575*x^19 + 100148494787668*x^18 + 52233018535617*x^17 - 288162921569191*x^16 - 262264312267859*x^15 + 535079835460586*x^14 + 679366178486570*x^13 - 600819998721797*x^12 - 1028790083129260*x^11 + 359936453258218*x^10 + 916306649955834*x^9 - 86249482058917*x^8 - 469190872234743*x^7 + 4041363119341*x^6 + 135236928048853*x^5 - 4445757797975*x^4 - 19527462450618*x^3 + 2244998319481*x^2 + 789598736860*x - 116423735903)
 
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 - x^38 - 196*x^37 + 37*x^36 + 17177*x^35 + 9403*x^34 - 882381*x^33 - 1067223*x^32 + 29336001*x^31 + 53273410*x^30 - 658772565*x^29 - 1583577890*x^28 + 10123695780*x^27 + 30773254905*x^26 - 105181421187*x^25 - 406462457224*x^24 + 701275142050*x^23 + 3703719049533*x^22 - 2495293357664*x^21 - 23296951876563*x^20 - 582451600575*x^19 + 100148494787668*x^18 + 52233018535617*x^17 - 288162921569191*x^16 - 262264312267859*x^15 + 535079835460586*x^14 + 679366178486570*x^13 - 600819998721797*x^12 - 1028790083129260*x^11 + 359936453258218*x^10 + 916306649955834*x^9 - 86249482058917*x^8 - 469190872234743*x^7 + 4041363119341*x^6 + 135236928048853*x^5 - 4445757797975*x^4 - 19527462450618*x^3 + 2244998319481*x^2 + 789598736860*x - 116423735903, 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 - x^38 - 196*x^37 + 37*x^36 + 17177*x^35 + 9403*x^34 - 882381*x^33 - 1067223*x^32 + 29336001*x^31 + 53273410*x^30 - 658772565*x^29 - 1583577890*x^28 + 10123695780*x^27 + 30773254905*x^26 - 105181421187*x^25 - 406462457224*x^24 + 701275142050*x^23 + 3703719049533*x^22 - 2495293357664*x^21 - 23296951876563*x^20 - 582451600575*x^19 + 100148494787668*x^18 + 52233018535617*x^17 - 288162921569191*x^16 - 262264312267859*x^15 + 535079835460586*x^14 + 679366178486570*x^13 - 600819998721797*x^12 - 1028790083129260*x^11 + 359936453258218*x^10 + 916306649955834*x^9 - 86249482058917*x^8 - 469190872234743*x^7 + 4041363119341*x^6 + 135236928048853*x^5 - 4445757797975*x^4 - 19527462450618*x^3 + 2244998319481*x^2 + 789598736860*x - 116423735903);
 
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 - x^38 - 196*x^37 + 37*x^36 + 17177*x^35 + 9403*x^34 - 882381*x^33 - 1067223*x^32 + 29336001*x^31 + 53273410*x^30 - 658772565*x^29 - 1583577890*x^28 + 10123695780*x^27 + 30773254905*x^26 - 105181421187*x^25 - 406462457224*x^24 + 701275142050*x^23 + 3703719049533*x^22 - 2495293357664*x^21 - 23296951876563*x^20 - 582451600575*x^19 + 100148494787668*x^18 + 52233018535617*x^17 - 288162921569191*x^16 - 262264312267859*x^15 + 535079835460586*x^14 + 679366178486570*x^13 - 600819998721797*x^12 - 1028790083129260*x^11 + 359936453258218*x^10 + 916306649955834*x^9 - 86249482058917*x^8 - 469190872234743*x^7 + 4041363119341*x^6 + 135236928048853*x^5 - 4445757797975*x^4 - 19527462450618*x^3 + 2244998319481*x^2 + 789598736860*x - 116423735903);
 
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

3.3.305809.2, 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 $39$ ${\href{/padicField/3.13.0.1}{13} }^{3}$ ${\href{/padicField/5.13.0.1}{13} }^{3}$ R $39$ $39$ $39$ ${\href{/padicField/19.13.0.1}{13} }^{3}$ ${\href{/padicField/23.1.0.1}{1} }^{39}$ $39$ $39$ $39$ ${\href{/padicField/41.13.0.1}{13} }^{3}$ $39$ ${\href{/padicField/47.13.0.1}{13} }^{3}$ $39$ ${\href{/padicField/59.13.0.1}{13} }^{3}$

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
\(7\) Copy content Toggle raw display Deg $39$$3$$13$$26$
\(79\) Copy content Toggle raw display Deg $39$$39$$1$$38$