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Properties

Label 39.39.193...529.1
Degree $39$
Signature $[39, 0]$
Discriminant $1.937\times 10^{90}$
Root discriminant \(206.56\)
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 - 10*x^38 - 91*x^37 + 1036*x^36 + 3946*x^35 - 48394*x^34 - 113775*x^33 + 1340155*x^32 + 2502275*x^31 - 24308912*x^30 - 43149237*x^29 + 301161228*x^28 + 566666743*x^27 - 2578642861*x^26 - 5471311188*x^25 + 15078823676*x^24 + 37876474410*x^23 - 57546616634*x^22 - 184436616929*x^21 + 124709319279*x^20 + 618742359228*x^19 - 54324200102*x^18 - 1388110537758*x^17 - 483581361681*x^16 + 1979142956060*x^15 + 1427263115288*x^14 - 1609144180429*x^13 - 1887910006886*x^12 + 509773426670*x^11 + 1295640832119*x^10 + 171082251210*x^9 - 429221222404*x^8 - 160917420416*x^7 + 56100044133*x^6 + 37303198473*x^5 - 296394467*x^4 - 3173168574*x^3 - 345469686*x^2 + 87223662*x + 14066053)
 
gp: K = bnfinit(y^39 - 10*y^38 - 91*y^37 + 1036*y^36 + 3946*y^35 - 48394*y^34 - 113775*y^33 + 1340155*y^32 + 2502275*y^31 - 24308912*y^30 - 43149237*y^29 + 301161228*y^28 + 566666743*y^27 - 2578642861*y^26 - 5471311188*y^25 + 15078823676*y^24 + 37876474410*y^23 - 57546616634*y^22 - 184436616929*y^21 + 124709319279*y^20 + 618742359228*y^19 - 54324200102*y^18 - 1388110537758*y^17 - 483581361681*y^16 + 1979142956060*y^15 + 1427263115288*y^14 - 1609144180429*y^13 - 1887910006886*y^12 + 509773426670*y^11 + 1295640832119*y^10 + 171082251210*y^9 - 429221222404*y^8 - 160917420416*y^7 + 56100044133*y^6 + 37303198473*y^5 - 296394467*y^4 - 3173168574*y^3 - 345469686*y^2 + 87223662*y + 14066053, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^39 - 10*x^38 - 91*x^37 + 1036*x^36 + 3946*x^35 - 48394*x^34 - 113775*x^33 + 1340155*x^32 + 2502275*x^31 - 24308912*x^30 - 43149237*x^29 + 301161228*x^28 + 566666743*x^27 - 2578642861*x^26 - 5471311188*x^25 + 15078823676*x^24 + 37876474410*x^23 - 57546616634*x^22 - 184436616929*x^21 + 124709319279*x^20 + 618742359228*x^19 - 54324200102*x^18 - 1388110537758*x^17 - 483581361681*x^16 + 1979142956060*x^15 + 1427263115288*x^14 - 1609144180429*x^13 - 1887910006886*x^12 + 509773426670*x^11 + 1295640832119*x^10 + 171082251210*x^9 - 429221222404*x^8 - 160917420416*x^7 + 56100044133*x^6 + 37303198473*x^5 - 296394467*x^4 - 3173168574*x^3 - 345469686*x^2 + 87223662*x + 14066053);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^39 - 10*x^38 - 91*x^37 + 1036*x^36 + 3946*x^35 - 48394*x^34 - 113775*x^33 + 1340155*x^32 + 2502275*x^31 - 24308912*x^30 - 43149237*x^29 + 301161228*x^28 + 566666743*x^27 - 2578642861*x^26 - 5471311188*x^25 + 15078823676*x^24 + 37876474410*x^23 - 57546616634*x^22 - 184436616929*x^21 + 124709319279*x^20 + 618742359228*x^19 - 54324200102*x^18 - 1388110537758*x^17 - 483581361681*x^16 + 1979142956060*x^15 + 1427263115288*x^14 - 1609144180429*x^13 - 1887910006886*x^12 + 509773426670*x^11 + 1295640832119*x^10 + 171082251210*x^9 - 429221222404*x^8 - 160917420416*x^7 + 56100044133*x^6 + 37303198473*x^5 - 296394467*x^4 - 3173168574*x^3 - 345469686*x^2 + 87223662*x + 14066053)
 

\( x^{39} - 10 x^{38} - 91 x^{37} + 1036 x^{36} + 3946 x^{35} - 48394 x^{34} - 113775 x^{33} + \cdots + 14066053 \) 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:   \(193\!\cdots\!529\) \(\medspace = 7^{26}\cdot 79^{36}\) 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:  \(206.56\)
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:  $7^{2/3}79^{12/13}\approx 206.5639215386565$
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$$\chi_{553}(512,·)$, $\chi_{553}(1,·)$, $\chi_{553}(520,·)$, $\chi_{553}(396,·)$, $\chi_{553}(141,·)$, $\chi_{553}(526,·)$, $\chi_{553}(144,·)$, $\chi_{553}(18,·)$, $\chi_{553}(275,·)$, $\chi_{553}(46,·)$, $\chi_{553}(22,·)$, $\chi_{553}(536,·)$, $\chi_{553}(541,·)$, $\chi_{553}(289,·)$, $\chi_{553}(302,·)$, $\chi_{553}(176,·)$, $\chi_{553}(8,·)$, $\chi_{553}(179,·)$, $\chi_{553}(317,·)$, $\chi_{553}(64,·)$, $\chi_{553}(65,·)$, $\chi_{553}(67,·)$, $\chi_{553}(324,·)$, $\chi_{553}(326,·)$, $\chi_{553}(417,·)$, $\chi_{553}(457,·)$, $\chi_{553}(459,·)$, $\chi_{553}(204,·)$, $\chi_{553}(337,·)$, $\chi_{553}(338,·)$, $\chi_{553}(100,·)$, $\chi_{553}(225,·)$, $\chi_{553}(354,·)$, $\chi_{553}(484,·)$, $\chi_{553}(492,·)$, $\chi_{553}(368,·)$, $\chi_{553}(403,·)$, $\chi_{553}(247,·)$, $\chi_{553}(380,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{23}a^{30}-\frac{7}{23}a^{28}-\frac{11}{23}a^{27}+\frac{4}{23}a^{26}-\frac{6}{23}a^{25}-\frac{11}{23}a^{24}+\frac{7}{23}a^{23}-\frac{6}{23}a^{22}-\frac{11}{23}a^{21}-\frac{2}{23}a^{20}+\frac{9}{23}a^{19}-\frac{11}{23}a^{18}-\frac{11}{23}a^{17}-\frac{10}{23}a^{16}+\frac{5}{23}a^{14}-\frac{9}{23}a^{13}-\frac{8}{23}a^{12}-\frac{7}{23}a^{11}+\frac{8}{23}a^{10}-\frac{9}{23}a^{9}+\frac{11}{23}a^{7}+\frac{4}{23}a^{6}+\frac{5}{23}a^{5}-\frac{7}{23}a^{4}-\frac{6}{23}a^{3}+\frac{4}{23}a^{2}-\frac{1}{23}a+\frac{7}{23}$, $\frac{1}{23}a^{31}-\frac{7}{23}a^{29}-\frac{11}{23}a^{28}+\frac{4}{23}a^{27}-\frac{6}{23}a^{26}-\frac{11}{23}a^{25}+\frac{7}{23}a^{24}-\frac{6}{23}a^{23}-\frac{11}{23}a^{22}-\frac{2}{23}a^{21}+\frac{9}{23}a^{20}-\frac{11}{23}a^{19}-\frac{11}{23}a^{18}-\frac{10}{23}a^{17}+\frac{5}{23}a^{15}-\frac{9}{23}a^{14}-\frac{8}{23}a^{13}-\frac{7}{23}a^{12}+\frac{8}{23}a^{11}-\frac{9}{23}a^{10}+\frac{11}{23}a^{8}+\frac{4}{23}a^{7}+\frac{5}{23}a^{6}-\frac{7}{23}a^{5}-\frac{6}{23}a^{4}+\frac{4}{23}a^{3}-\frac{1}{23}a^{2}+\frac{7}{23}a$, $\frac{1}{23}a^{32}-\frac{11}{23}a^{29}+\frac{1}{23}a^{28}+\frac{9}{23}a^{27}-\frac{6}{23}a^{26}+\frac{11}{23}a^{25}+\frac{9}{23}a^{24}-\frac{8}{23}a^{23}+\frac{2}{23}a^{22}+\frac{1}{23}a^{21}-\frac{2}{23}a^{20}+\frac{6}{23}a^{19}+\frac{5}{23}a^{18}-\frac{8}{23}a^{17}+\frac{4}{23}a^{16}-\frac{9}{23}a^{15}+\frac{4}{23}a^{14}-\frac{1}{23}a^{13}-\frac{2}{23}a^{12}+\frac{11}{23}a^{11}+\frac{10}{23}a^{10}-\frac{6}{23}a^{9}+\frac{4}{23}a^{8}-\frac{10}{23}a^{7}-\frac{2}{23}a^{6}+\frac{6}{23}a^{5}+\frac{1}{23}a^{4}+\frac{3}{23}a^{3}-\frac{11}{23}a^{2}-\frac{7}{23}a+\frac{3}{23}$, $\frac{1}{23}a^{33}+\frac{1}{23}a^{29}+\frac{1}{23}a^{28}+\frac{11}{23}a^{27}+\frac{9}{23}a^{26}-\frac{11}{23}a^{25}+\frac{9}{23}a^{24}+\frac{10}{23}a^{23}+\frac{4}{23}a^{22}-\frac{8}{23}a^{21}+\frac{7}{23}a^{20}-\frac{11}{23}a^{19}+\frac{9}{23}a^{18}-\frac{2}{23}a^{17}-\frac{4}{23}a^{16}+\frac{4}{23}a^{15}+\frac{8}{23}a^{14}-\frac{9}{23}a^{13}-\frac{8}{23}a^{12}+\frac{2}{23}a^{11}-\frac{10}{23}a^{10}-\frac{3}{23}a^{9}-\frac{10}{23}a^{8}+\frac{4}{23}a^{7}+\frac{4}{23}a^{6}+\frac{10}{23}a^{5}-\frac{5}{23}a^{4}-\frac{8}{23}a^{3}-\frac{9}{23}a^{2}-\frac{8}{23}a+\frac{8}{23}$, $\frac{1}{23}a^{34}+\frac{1}{23}a^{29}-\frac{5}{23}a^{28}-\frac{3}{23}a^{27}+\frac{8}{23}a^{26}-\frac{8}{23}a^{25}-\frac{2}{23}a^{24}-\frac{3}{23}a^{23}-\frac{2}{23}a^{22}-\frac{5}{23}a^{21}-\frac{9}{23}a^{20}+\frac{9}{23}a^{18}+\frac{7}{23}a^{17}-\frac{9}{23}a^{16}+\frac{8}{23}a^{15}+\frac{9}{23}a^{14}+\frac{1}{23}a^{13}+\frac{10}{23}a^{12}-\frac{3}{23}a^{11}-\frac{11}{23}a^{10}-\frac{1}{23}a^{9}+\frac{4}{23}a^{8}-\frac{7}{23}a^{7}+\frac{6}{23}a^{6}-\frac{10}{23}a^{5}-\frac{1}{23}a^{4}-\frac{3}{23}a^{3}+\frac{11}{23}a^{2}+\frac{9}{23}a-\frac{7}{23}$, $\frac{1}{4163}a^{35}-\frac{53}{4163}a^{34}+\frac{28}{4163}a^{33}+\frac{68}{4163}a^{32}+\frac{86}{4163}a^{31}+\frac{37}{4163}a^{30}-\frac{19}{181}a^{29}-\frac{1484}{4163}a^{28}+\frac{80}{181}a^{27}+\frac{765}{4163}a^{26}+\frac{1816}{4163}a^{25}-\frac{1}{181}a^{24}-\frac{877}{4163}a^{23}-\frac{2032}{4163}a^{22}+\frac{912}{4163}a^{21}+\frac{1538}{4163}a^{20}-\frac{2031}{4163}a^{19}+\frac{1816}{4163}a^{18}+\frac{1053}{4163}a^{17}-\frac{842}{4163}a^{16}-\frac{1727}{4163}a^{15}-\frac{1609}{4163}a^{14}+\frac{166}{4163}a^{13}+\frac{356}{4163}a^{12}-\frac{107}{4163}a^{11}+\frac{565}{4163}a^{10}-\frac{2}{181}a^{9}-\frac{1259}{4163}a^{8}+\frac{1193}{4163}a^{7}+\frac{153}{4163}a^{6}-\frac{1114}{4163}a^{5}-\frac{1687}{4163}a^{4}-\frac{1539}{4163}a^{3}+\frac{439}{4163}a^{2}-\frac{1308}{4163}a+\frac{10}{23}$, $\frac{1}{66461008633}a^{36}-\frac{5715341}{66461008633}a^{35}-\frac{987680524}{66461008633}a^{34}+\frac{1089105494}{66461008633}a^{33}-\frac{966819925}{66461008633}a^{32}+\frac{6939409}{367187893}a^{31}-\frac{200769355}{66461008633}a^{30}+\frac{25444747788}{66461008633}a^{29}+\frac{21204731649}{66461008633}a^{28}-\frac{11240756320}{66461008633}a^{27}+\frac{30179681312}{66461008633}a^{26}-\frac{1232286949}{66461008633}a^{25}-\frac{18021002148}{66461008633}a^{24}-\frac{240604411}{2889609071}a^{23}+\frac{11658439379}{66461008633}a^{22}-\frac{9018339614}{66461008633}a^{21}-\frac{13363574286}{66461008633}a^{20}+\frac{5258001330}{66461008633}a^{19}-\frac{28794846413}{66461008633}a^{18}+\frac{3990368025}{66461008633}a^{17}-\frac{15609345460}{66461008633}a^{16}-\frac{29623183016}{66461008633}a^{15}+\frac{19487216287}{66461008633}a^{14}+\frac{22969978329}{66461008633}a^{13}+\frac{33157487463}{66461008633}a^{12}-\frac{24516973264}{66461008633}a^{11}-\frac{358039729}{2889609071}a^{10}-\frac{10729311452}{66461008633}a^{9}+\frac{28684351874}{66461008633}a^{8}+\frac{25785160075}{66461008633}a^{7}-\frac{1569049217}{66461008633}a^{6}+\frac{26714874304}{66461008633}a^{5}-\frac{27677915143}{66461008633}a^{4}-\frac{1281928813}{2889609071}a^{3}-\frac{13928141288}{66461008633}a^{2}-\frac{9008082708}{66461008633}a-\frac{22057646}{367187893}$, $\frac{1}{66461008633}a^{37}-\frac{5831228}{66461008633}a^{35}-\frac{612863846}{66461008633}a^{34}+\frac{360028399}{66461008633}a^{33}+\frac{375095819}{66461008633}a^{32}-\frac{24360459}{2889609071}a^{31}+\frac{1178876231}{66461008633}a^{30}-\frac{19505380840}{66461008633}a^{29}+\frac{3545556535}{66461008633}a^{28}-\frac{13908252657}{66461008633}a^{27}+\frac{31302011613}{66461008633}a^{26}+\frac{10521057270}{66461008633}a^{25}+\frac{28792902313}{66461008633}a^{24}+\frac{27499594228}{66461008633}a^{23}+\frac{1612386824}{66461008633}a^{22}-\frac{1201854918}{66461008633}a^{21}-\frac{23090557611}{66461008633}a^{20}+\frac{3315108425}{66461008633}a^{19}+\frac{11458284368}{66461008633}a^{18}-\frac{1015746810}{2889609071}a^{17}-\frac{32285159427}{66461008633}a^{16}+\frac{24555295407}{66461008633}a^{15}+\frac{10002446486}{66461008633}a^{14}+\frac{5834518551}{66461008633}a^{13}+\frac{946732442}{66461008633}a^{12}-\frac{19065687634}{66461008633}a^{11}-\frac{22764104554}{66461008633}a^{10}-\frac{23529827560}{66461008633}a^{9}+\frac{27005795175}{66461008633}a^{8}-\frac{27354168791}{66461008633}a^{7}-\frac{9005336677}{66461008633}a^{6}-\frac{25624969708}{66461008633}a^{5}+\frac{19537846068}{66461008633}a^{4}+\frac{19041122623}{66461008633}a^{3}-\frac{1608601315}{66461008633}a^{2}+\frac{3707802548}{66461008633}a+\frac{65838370}{367187893}$, $\frac{1}{91\!\cdots\!07}a^{38}+\frac{56\!\cdots\!07}{91\!\cdots\!07}a^{37}+\frac{44\!\cdots\!29}{91\!\cdots\!07}a^{36}-\frac{22\!\cdots\!83}{91\!\cdots\!07}a^{35}+\frac{12\!\cdots\!53}{91\!\cdots\!07}a^{34}+\frac{10\!\cdots\!00}{91\!\cdots\!07}a^{33}-\frac{98\!\cdots\!71}{91\!\cdots\!07}a^{32}+\frac{19\!\cdots\!86}{91\!\cdots\!07}a^{31}-\frac{14\!\cdots\!48}{91\!\cdots\!07}a^{30}+\frac{24\!\cdots\!87}{91\!\cdots\!07}a^{29}+\frac{29\!\cdots\!30}{91\!\cdots\!07}a^{28}+\frac{55\!\cdots\!73}{91\!\cdots\!07}a^{27}-\frac{82\!\cdots\!14}{91\!\cdots\!07}a^{26}+\frac{30\!\cdots\!83}{91\!\cdots\!07}a^{25}+\frac{35\!\cdots\!96}{91\!\cdots\!07}a^{24}+\frac{26\!\cdots\!74}{91\!\cdots\!07}a^{23}-\frac{41\!\cdots\!59}{91\!\cdots\!07}a^{22}-\frac{16\!\cdots\!26}{91\!\cdots\!07}a^{21}+\frac{33\!\cdots\!97}{91\!\cdots\!07}a^{20}-\frac{39\!\cdots\!44}{91\!\cdots\!07}a^{19}-\frac{42\!\cdots\!13}{91\!\cdots\!07}a^{18}+\frac{32\!\cdots\!45}{91\!\cdots\!07}a^{17}-\frac{28\!\cdots\!68}{91\!\cdots\!07}a^{16}-\frac{40\!\cdots\!97}{91\!\cdots\!07}a^{15}+\frac{31\!\cdots\!60}{91\!\cdots\!07}a^{14}+\frac{24\!\cdots\!33}{91\!\cdots\!07}a^{13}+\frac{21\!\cdots\!19}{91\!\cdots\!07}a^{12}+\frac{39\!\cdots\!81}{91\!\cdots\!07}a^{11}-\frac{43\!\cdots\!13}{91\!\cdots\!07}a^{10}-\frac{33\!\cdots\!42}{91\!\cdots\!07}a^{9}-\frac{10\!\cdots\!12}{91\!\cdots\!07}a^{8}-\frac{38\!\cdots\!29}{91\!\cdots\!07}a^{7}-\frac{19\!\cdots\!42}{91\!\cdots\!07}a^{6}+\frac{40\!\cdots\!91}{91\!\cdots\!07}a^{5}+\frac{41\!\cdots\!42}{91\!\cdots\!07}a^{4}-\frac{16\!\cdots\!49}{91\!\cdots\!07}a^{3}+\frac{18\!\cdots\!43}{91\!\cdots\!07}a^{2}+\frac{13\!\cdots\!91}{91\!\cdots\!07}a+\frac{29\!\cdots\!60}{50\!\cdots\!47}$ 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 - 91*x^37 + 1036*x^36 + 3946*x^35 - 48394*x^34 - 113775*x^33 + 1340155*x^32 + 2502275*x^31 - 24308912*x^30 - 43149237*x^29 + 301161228*x^28 + 566666743*x^27 - 2578642861*x^26 - 5471311188*x^25 + 15078823676*x^24 + 37876474410*x^23 - 57546616634*x^22 - 184436616929*x^21 + 124709319279*x^20 + 618742359228*x^19 - 54324200102*x^18 - 1388110537758*x^17 - 483581361681*x^16 + 1979142956060*x^15 + 1427263115288*x^14 - 1609144180429*x^13 - 1887910006886*x^12 + 509773426670*x^11 + 1295640832119*x^10 + 171082251210*x^9 - 429221222404*x^8 - 160917420416*x^7 + 56100044133*x^6 + 37303198473*x^5 - 296394467*x^4 - 3173168574*x^3 - 345469686*x^2 + 87223662*x + 14066053)
 
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 - 91*x^37 + 1036*x^36 + 3946*x^35 - 48394*x^34 - 113775*x^33 + 1340155*x^32 + 2502275*x^31 - 24308912*x^30 - 43149237*x^29 + 301161228*x^28 + 566666743*x^27 - 2578642861*x^26 - 5471311188*x^25 + 15078823676*x^24 + 37876474410*x^23 - 57546616634*x^22 - 184436616929*x^21 + 124709319279*x^20 + 618742359228*x^19 - 54324200102*x^18 - 1388110537758*x^17 - 483581361681*x^16 + 1979142956060*x^15 + 1427263115288*x^14 - 1609144180429*x^13 - 1887910006886*x^12 + 509773426670*x^11 + 1295640832119*x^10 + 171082251210*x^9 - 429221222404*x^8 - 160917420416*x^7 + 56100044133*x^6 + 37303198473*x^5 - 296394467*x^4 - 3173168574*x^3 - 345469686*x^2 + 87223662*x + 14066053, 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 - 91*x^37 + 1036*x^36 + 3946*x^35 - 48394*x^34 - 113775*x^33 + 1340155*x^32 + 2502275*x^31 - 24308912*x^30 - 43149237*x^29 + 301161228*x^28 + 566666743*x^27 - 2578642861*x^26 - 5471311188*x^25 + 15078823676*x^24 + 37876474410*x^23 - 57546616634*x^22 - 184436616929*x^21 + 124709319279*x^20 + 618742359228*x^19 - 54324200102*x^18 - 1388110537758*x^17 - 483581361681*x^16 + 1979142956060*x^15 + 1427263115288*x^14 - 1609144180429*x^13 - 1887910006886*x^12 + 509773426670*x^11 + 1295640832119*x^10 + 171082251210*x^9 - 429221222404*x^8 - 160917420416*x^7 + 56100044133*x^6 + 37303198473*x^5 - 296394467*x^4 - 3173168574*x^3 - 345469686*x^2 + 87223662*x + 14066053);
 
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 - 91*x^37 + 1036*x^36 + 3946*x^35 - 48394*x^34 - 113775*x^33 + 1340155*x^32 + 2502275*x^31 - 24308912*x^30 - 43149237*x^29 + 301161228*x^28 + 566666743*x^27 - 2578642861*x^26 - 5471311188*x^25 + 15078823676*x^24 + 37876474410*x^23 - 57546616634*x^22 - 184436616929*x^21 + 124709319279*x^20 + 618742359228*x^19 - 54324200102*x^18 - 1388110537758*x^17 - 483581361681*x^16 + 1979142956060*x^15 + 1427263115288*x^14 - 1609144180429*x^13 - 1887910006886*x^12 + 509773426670*x^11 + 1295640832119*x^10 + 171082251210*x^9 - 429221222404*x^8 - 160917420416*x^7 + 56100044133*x^6 + 37303198473*x^5 - 296394467*x^4 - 3173168574*x^3 - 345469686*x^2 + 87223662*x + 14066053);
 
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

\(\Q(\zeta_{7})^+\), 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 $39$ $39$ $39$ R $39$ ${\href{/padicField/13.13.0.1}{13} }^{3}$ $39$ $39$ ${\href{/padicField/23.3.0.1}{3} }^{13}$ ${\href{/padicField/29.13.0.1}{13} }^{3}$ $39$ $39$ ${\href{/padicField/41.13.0.1}{13} }^{3}$ ${\href{/padicField/43.13.0.1}{13} }^{3}$ $39$ $39$ $39$

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
\(7\) Copy content Toggle raw display Deg $39$$3$$13$$26$
\(79\) Copy content Toggle raw display Deg $39$$13$$3$$36$