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Properties

Label 39.39.120...489.2
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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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 - 658679661*x^29 - 1582974014*x^28 + 10115860876*x^27 + 30713447955*x^26 - 104944361700*x^25 - 404054826285*x^24 + 698746620334*x^23 + 3652383956122*x^22 - 2514455451909*x^21 - 22664595762180*x^20 + 201147288071*x^19 + 95542005646477*x^18 + 43535191283083*x^17 - 268603953306336*x^16 - 212548013703570*x^15 + 489275966728363*x^14 + 517783387282365*x^13 - 549595953418310*x^12 - 723715965685304*x^11 + 343123809068303*x^10 + 580854022082499*x^9 - 85745300820944*x^8 - 252040837202603*x^7 - 10510715956375*x^6 + 52566975147352*x^5 + 6905591385122*x^4 - 5057933882725*x^3 - 892913278781*x^2 + 181524069429*x + 36571406327)
 
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 - 658679661*y^29 - 1582974014*y^28 + 10115860876*y^27 + 30713447955*y^26 - 104944361700*y^25 - 404054826285*y^24 + 698746620334*y^23 + 3652383956122*y^22 - 2514455451909*y^21 - 22664595762180*y^20 + 201147288071*y^19 + 95542005646477*y^18 + 43535191283083*y^17 - 268603953306336*y^16 - 212548013703570*y^15 + 489275966728363*y^14 + 517783387282365*y^13 - 549595953418310*y^12 - 723715965685304*y^11 + 343123809068303*y^10 + 580854022082499*y^9 - 85745300820944*y^8 - 252040837202603*y^7 - 10510715956375*y^6 + 52566975147352*y^5 + 6905591385122*y^4 - 5057933882725*y^3 - 892913278781*y^2 + 181524069429*y + 36571406327, 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 - 658679661*x^29 - 1582974014*x^28 + 10115860876*x^27 + 30713447955*x^26 - 104944361700*x^25 - 404054826285*x^24 + 698746620334*x^23 + 3652383956122*x^22 - 2514455451909*x^21 - 22664595762180*x^20 + 201147288071*x^19 + 95542005646477*x^18 + 43535191283083*x^17 - 268603953306336*x^16 - 212548013703570*x^15 + 489275966728363*x^14 + 517783387282365*x^13 - 549595953418310*x^12 - 723715965685304*x^11 + 343123809068303*x^10 + 580854022082499*x^9 - 85745300820944*x^8 - 252040837202603*x^7 - 10510715956375*x^6 + 52566975147352*x^5 + 6905591385122*x^4 - 5057933882725*x^3 - 892913278781*x^2 + 181524069429*x + 36571406327);
 
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 - 658679661*x^29 - 1582974014*x^28 + 10115860876*x^27 + 30713447955*x^26 - 104944361700*x^25 - 404054826285*x^24 + 698746620334*x^23 + 3652383956122*x^22 - 2514455451909*x^21 - 22664595762180*x^20 + 201147288071*x^19 + 95542005646477*x^18 + 43535191283083*x^17 - 268603953306336*x^16 - 212548013703570*x^15 + 489275966728363*x^14 + 517783387282365*x^13 - 549595953418310*x^12 - 723715965685304*x^11 + 343123809068303*x^10 + 580854022082499*x^9 - 85745300820944*x^8 - 252040837202603*x^7 - 10510715956375*x^6 + 52566975147352*x^5 + 6905591385122*x^4 - 5057933882725*x^3 - 892913278781*x^2 + 181524069429*x + 36571406327)
 

\( x^{39} - x^{38} - 196 x^{37} + 37 x^{36} + 17177 x^{35} + 9403 x^{34} - 882381 x^{33} + \cdots + 36571406327 \) 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$$\chi_{553}(512,·)$, $\chi_{553}(1,·)$, $\chi_{553}(130,·)$, $\chi_{553}(8,·)$, $\chi_{553}(9,·)$, $\chi_{553}(268,·)$, $\chi_{553}(141,·)$, $\chi_{553}(526,·)$, $\chi_{553}(529,·)$, $\chi_{553}(366,·)$, $\chi_{553}(22,·)$, $\chi_{553}(23,·)$, $\chi_{553}(25,·)$, $\chi_{553}(163,·)$, $\chi_{553}(550,·)$, $\chi_{553}(302,·)$, $\chi_{553}(431,·)$, $\chi_{553}(176,·)$, $\chi_{553}(177,·)$, $\chi_{553}(310,·)$, $\chi_{553}(184,·)$, $\chi_{553}(64,·)$, $\chi_{553}(198,·)$, $\chi_{553}(72,·)$, $\chi_{553}(204,·)$, $\chi_{553}(207,·)$, $\chi_{553}(337,·)$, $\chi_{553}(478,·)$, $\chi_{553}(95,·)$, $\chi_{553}(225,·)$, $\chi_{553}(484,·)$, $\chi_{553}(485,·)$, $\chi_{553}(81,·)$, $\chi_{553}(361,·)$, $\chi_{553}(487,·)$, $\chi_{553}(494,·)$, $\chi_{553}(506,·)$, $\chi_{553}(123,·)$, $\chi_{553}(200,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{103}a^{31}+\frac{11}{103}a^{30}+\frac{40}{103}a^{29}+\frac{24}{103}a^{28}-\frac{3}{103}a^{27}+\frac{11}{103}a^{26}-\frac{21}{103}a^{25}-\frac{44}{103}a^{24}-\frac{38}{103}a^{23}+\frac{17}{103}a^{22}+\frac{42}{103}a^{21}-\frac{47}{103}a^{20}-\frac{36}{103}a^{19}-\frac{49}{103}a^{18}+\frac{10}{103}a^{17}+\frac{36}{103}a^{16}-\frac{14}{103}a^{15}+\frac{32}{103}a^{14}+\frac{43}{103}a^{13}-\frac{31}{103}a^{12}-\frac{19}{103}a^{11}+\frac{28}{103}a^{10}-\frac{8}{103}a^{9}-\frac{51}{103}a^{8}+\frac{45}{103}a^{7}+\frac{29}{103}a^{6}+\frac{16}{103}a^{5}+\frac{29}{103}a^{4}+\frac{28}{103}a^{3}-\frac{26}{103}a^{2}-\frac{31}{103}a+\frac{14}{103}$, $\frac{1}{103}a^{32}+\frac{22}{103}a^{30}-\frac{4}{103}a^{29}+\frac{42}{103}a^{28}+\frac{44}{103}a^{27}-\frac{39}{103}a^{26}-\frac{19}{103}a^{25}+\frac{34}{103}a^{24}+\frac{23}{103}a^{23}-\frac{42}{103}a^{22}+\frac{6}{103}a^{21}-\frac{34}{103}a^{20}+\frac{38}{103}a^{19}+\frac{34}{103}a^{18}+\frac{29}{103}a^{17}+\frac{2}{103}a^{16}-\frac{20}{103}a^{15}+\frac{11}{103}a^{13}+\frac{13}{103}a^{12}+\frac{31}{103}a^{11}-\frac{7}{103}a^{10}+\frac{37}{103}a^{9}-\frac{12}{103}a^{8}+\frac{49}{103}a^{7}+\frac{6}{103}a^{6}-\frac{44}{103}a^{5}+\frac{18}{103}a^{4}-\frac{25}{103}a^{3}+\frac{49}{103}a^{2}+\frac{46}{103}a-\frac{51}{103}$, $\frac{1}{103}a^{33}-\frac{40}{103}a^{30}-\frac{14}{103}a^{29}+\frac{31}{103}a^{28}+\frac{27}{103}a^{27}+\frac{48}{103}a^{26}-\frac{19}{103}a^{25}-\frac{39}{103}a^{24}-\frac{30}{103}a^{23}+\frac{44}{103}a^{22}-\frac{31}{103}a^{21}+\frac{42}{103}a^{20}+\frac{2}{103}a^{19}-\frac{26}{103}a^{18}-\frac{12}{103}a^{17}+\frac{12}{103}a^{16}-\frac{1}{103}a^{15}+\frac{28}{103}a^{14}-\frac{6}{103}a^{13}-\frac{8}{103}a^{12}-\frac{1}{103}a^{11}+\frac{39}{103}a^{10}-\frac{42}{103}a^{9}+\frac{38}{103}a^{8}+\frac{46}{103}a^{7}+\frac{39}{103}a^{6}-\frac{25}{103}a^{5}-\frac{45}{103}a^{4}+\frac{51}{103}a^{3}+\frac{13}{103}a+\frac{1}{103}$, $\frac{1}{103}a^{34}+\frac{14}{103}a^{30}-\frac{17}{103}a^{29}-\frac{43}{103}a^{28}+\frac{31}{103}a^{27}+\frac{9}{103}a^{26}+\frac{48}{103}a^{25}-\frac{39}{103}a^{24}-\frac{34}{103}a^{23}+\frac{31}{103}a^{22}-\frac{29}{103}a^{21}-\frac{24}{103}a^{20}-\frac{24}{103}a^{19}-\frac{15}{103}a^{18}-\frac{3}{103}a^{16}-\frac{17}{103}a^{15}+\frac{38}{103}a^{14}-\frac{39}{103}a^{13}-\frac{5}{103}a^{12}+\frac{48}{103}a^{10}+\frac{27}{103}a^{9}-\frac{37}{103}a^{8}-\frac{15}{103}a^{7}+\frac{2}{103}a^{6}-\frac{23}{103}a^{5}-\frac{25}{103}a^{4}-\frac{13}{103}a^{3}+\frac{3}{103}a^{2}-\frac{3}{103}a+\frac{45}{103}$, $\frac{1}{103}a^{35}+\frac{35}{103}a^{30}+\frac{15}{103}a^{29}+\frac{4}{103}a^{28}+\frac{51}{103}a^{27}-\frac{3}{103}a^{26}+\frac{49}{103}a^{25}-\frac{36}{103}a^{24}+\frac{48}{103}a^{23}+\frac{42}{103}a^{22}+\frac{6}{103}a^{21}+\frac{16}{103}a^{20}-\frac{26}{103}a^{19}-\frac{35}{103}a^{18}-\frac{40}{103}a^{17}-\frac{6}{103}a^{16}+\frac{28}{103}a^{15}+\frac{28}{103}a^{14}+\frac{11}{103}a^{13}+\frac{22}{103}a^{12}+\frac{5}{103}a^{11}+\frac{47}{103}a^{10}-\frac{28}{103}a^{9}-\frac{22}{103}a^{8}-\frac{10}{103}a^{7}-\frac{17}{103}a^{6}-\frac{43}{103}a^{5}-\frac{7}{103}a^{4}+\frac{23}{103}a^{3}-\frac{51}{103}a^{2}-\frac{36}{103}a+\frac{10}{103}$, $\frac{1}{103}a^{36}+\frac{42}{103}a^{30}+\frac{46}{103}a^{29}+\frac{35}{103}a^{28}-\frac{1}{103}a^{27}-\frac{27}{103}a^{26}-\frac{22}{103}a^{25}+\frac{43}{103}a^{24}+\frac{33}{103}a^{23}+\frac{29}{103}a^{22}-\frac{12}{103}a^{21}-\frac{29}{103}a^{20}-\frac{11}{103}a^{19}+\frac{27}{103}a^{18}-\frac{47}{103}a^{17}+\frac{4}{103}a^{16}+\frac{3}{103}a^{15}+\frac{24}{103}a^{14}-\frac{41}{103}a^{13}-\frac{43}{103}a^{12}-\frac{9}{103}a^{11}+\frac{22}{103}a^{10}-\frac{51}{103}a^{9}+\frac{24}{103}a^{8}-\frac{47}{103}a^{7}-\frac{28}{103}a^{6}+\frac{51}{103}a^{5}+\frac{38}{103}a^{4}-\frac{1}{103}a^{3}+\frac{50}{103}a^{2}-\frac{38}{103}a+\frac{25}{103}$, $\frac{1}{276761}a^{37}-\frac{1154}{276761}a^{36}-\frac{289}{276761}a^{35}+\frac{1115}{276761}a^{34}+\frac{1075}{276761}a^{33}+\frac{1039}{276761}a^{32}-\frac{728}{276761}a^{31}-\frac{101924}{276761}a^{30}-\frac{25508}{276761}a^{29}-\frac{86114}{276761}a^{28}+\frac{131685}{276761}a^{27}-\frac{131895}{276761}a^{26}-\frac{98359}{276761}a^{25}-\frac{111833}{276761}a^{24}-\frac{85923}{276761}a^{23}-\frac{104218}{276761}a^{22}+\frac{51541}{276761}a^{21}-\frac{86330}{276761}a^{20}+\frac{7413}{276761}a^{19}+\frac{69709}{276761}a^{18}-\frac{22002}{276761}a^{17}-\frac{63256}{276761}a^{16}-\frac{111600}{276761}a^{15}+\frac{108100}{276761}a^{14}+\frac{41383}{276761}a^{13}-\frac{57967}{276761}a^{12}-\frac{86795}{276761}a^{11}+\frac{68893}{276761}a^{10}+\frac{90348}{276761}a^{9}-\frac{18163}{276761}a^{8}-\frac{125664}{276761}a^{7}-\frac{97920}{276761}a^{6}-\frac{76090}{276761}a^{5}-\frac{271}{276761}a^{4}-\frac{117193}{276761}a^{3}+\frac{29217}{276761}a^{2}+\frac{77968}{276761}a-\frac{53529}{276761}$, $\frac{1}{31\!\cdots\!57}a^{38}-\frac{48\!\cdots\!38}{31\!\cdots\!57}a^{37}+\frac{10\!\cdots\!96}{31\!\cdots\!57}a^{36}+\frac{93\!\cdots\!90}{31\!\cdots\!57}a^{35}+\frac{14\!\cdots\!01}{31\!\cdots\!57}a^{34}+\frac{90\!\cdots\!14}{31\!\cdots\!57}a^{33}-\frac{11\!\cdots\!62}{31\!\cdots\!57}a^{32}-\frac{33\!\cdots\!91}{31\!\cdots\!57}a^{31}+\frac{11\!\cdots\!42}{31\!\cdots\!57}a^{30}-\frac{43\!\cdots\!99}{31\!\cdots\!57}a^{29}-\frac{23\!\cdots\!12}{31\!\cdots\!57}a^{28}-\frac{76\!\cdots\!43}{31\!\cdots\!57}a^{27}-\frac{11\!\cdots\!43}{31\!\cdots\!57}a^{26}-\frac{73\!\cdots\!47}{31\!\cdots\!57}a^{25}-\frac{48\!\cdots\!22}{31\!\cdots\!57}a^{24}+\frac{36\!\cdots\!22}{31\!\cdots\!57}a^{23}+\frac{38\!\cdots\!03}{31\!\cdots\!57}a^{22}-\frac{97\!\cdots\!58}{31\!\cdots\!57}a^{21}+\frac{14\!\cdots\!26}{31\!\cdots\!57}a^{20}+\frac{14\!\cdots\!16}{31\!\cdots\!57}a^{19}-\frac{10\!\cdots\!81}{31\!\cdots\!57}a^{18}+\frac{11\!\cdots\!86}{31\!\cdots\!57}a^{17}+\frac{10\!\cdots\!15}{31\!\cdots\!57}a^{16}-\frac{76\!\cdots\!19}{31\!\cdots\!57}a^{15}+\frac{15\!\cdots\!13}{31\!\cdots\!57}a^{14}-\frac{13\!\cdots\!40}{31\!\cdots\!57}a^{13}-\frac{14\!\cdots\!06}{31\!\cdots\!57}a^{12}+\frac{62\!\cdots\!90}{31\!\cdots\!57}a^{11}+\frac{13\!\cdots\!39}{31\!\cdots\!57}a^{10}+\frac{47\!\cdots\!73}{31\!\cdots\!57}a^{9}-\frac{98\!\cdots\!08}{31\!\cdots\!57}a^{8}+\frac{29\!\cdots\!23}{31\!\cdots\!57}a^{7}-\frac{53\!\cdots\!13}{31\!\cdots\!57}a^{6}-\frac{49\!\cdots\!08}{31\!\cdots\!57}a^{5}-\frac{14\!\cdots\!83}{31\!\cdots\!57}a^{4}-\frac{67\!\cdots\!88}{31\!\cdots\!57}a^{3}-\frac{30\!\cdots\!56}{31\!\cdots\!57}a^{2}-\frac{27\!\cdots\!20}{31\!\cdots\!57}a+\frac{11\!\cdots\!93}{31\!\cdots\!57}$ 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 - 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 - 658679661*x^29 - 1582974014*x^28 + 10115860876*x^27 + 30713447955*x^26 - 104944361700*x^25 - 404054826285*x^24 + 698746620334*x^23 + 3652383956122*x^22 - 2514455451909*x^21 - 22664595762180*x^20 + 201147288071*x^19 + 95542005646477*x^18 + 43535191283083*x^17 - 268603953306336*x^16 - 212548013703570*x^15 + 489275966728363*x^14 + 517783387282365*x^13 - 549595953418310*x^12 - 723715965685304*x^11 + 343123809068303*x^10 + 580854022082499*x^9 - 85745300820944*x^8 - 252040837202603*x^7 - 10510715956375*x^6 + 52566975147352*x^5 + 6905591385122*x^4 - 5057933882725*x^3 - 892913278781*x^2 + 181524069429*x + 36571406327)
 
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 - 658679661*x^29 - 1582974014*x^28 + 10115860876*x^27 + 30713447955*x^26 - 104944361700*x^25 - 404054826285*x^24 + 698746620334*x^23 + 3652383956122*x^22 - 2514455451909*x^21 - 22664595762180*x^20 + 201147288071*x^19 + 95542005646477*x^18 + 43535191283083*x^17 - 268603953306336*x^16 - 212548013703570*x^15 + 489275966728363*x^14 + 517783387282365*x^13 - 549595953418310*x^12 - 723715965685304*x^11 + 343123809068303*x^10 + 580854022082499*x^9 - 85745300820944*x^8 - 252040837202603*x^7 - 10510715956375*x^6 + 52566975147352*x^5 + 6905591385122*x^4 - 5057933882725*x^3 - 892913278781*x^2 + 181524069429*x + 36571406327, 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 - 658679661*x^29 - 1582974014*x^28 + 10115860876*x^27 + 30713447955*x^26 - 104944361700*x^25 - 404054826285*x^24 + 698746620334*x^23 + 3652383956122*x^22 - 2514455451909*x^21 - 22664595762180*x^20 + 201147288071*x^19 + 95542005646477*x^18 + 43535191283083*x^17 - 268603953306336*x^16 - 212548013703570*x^15 + 489275966728363*x^14 + 517783387282365*x^13 - 549595953418310*x^12 - 723715965685304*x^11 + 343123809068303*x^10 + 580854022082499*x^9 - 85745300820944*x^8 - 252040837202603*x^7 - 10510715956375*x^6 + 52566975147352*x^5 + 6905591385122*x^4 - 5057933882725*x^3 - 892913278781*x^2 + 181524069429*x + 36571406327);
 
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 - 658679661*x^29 - 1582974014*x^28 + 10115860876*x^27 + 30713447955*x^26 - 104944361700*x^25 - 404054826285*x^24 + 698746620334*x^23 + 3652383956122*x^22 - 2514455451909*x^21 - 22664595762180*x^20 + 201147288071*x^19 + 95542005646477*x^18 + 43535191283083*x^17 - 268603953306336*x^16 - 212548013703570*x^15 + 489275966728363*x^14 + 517783387282365*x^13 - 549595953418310*x^12 - 723715965685304*x^11 + 343123809068303*x^10 + 580854022082499*x^9 - 85745300820944*x^8 - 252040837202603*x^7 - 10510715956375*x^6 + 52566975147352*x^5 + 6905591385122*x^4 - 5057933882725*x^3 - 892913278781*x^2 + 181524069429*x + 36571406327);
 
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

3.3.305809.1, 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 ${\href{/padicField/2.13.0.1}{13} }^{3}$ $39$ $39$ R ${\href{/padicField/11.13.0.1}{13} }^{3}$ $39$ $39$ $39$ ${\href{/padicField/23.3.0.1}{3} }^{13}$ $39$ ${\href{/padicField/31.13.0.1}{13} }^{3}$ ${\href{/padicField/37.13.0.1}{13} }^{3}$ ${\href{/padicField/41.13.0.1}{13} }^{3}$ $39$ $39$ ${\href{/padicField/53.13.0.1}{13} }^{3}$ $39$

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