Properties

Label 15.15.138...697.2
Degree $15$
Signature $[15, 0]$
Discriminant $1.386\times 10^{26}$
Root discriminant \(55.31\)
Ramified primes $3,11,23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^4:C_{10}$ (as 15T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 60*x^13 - 63*x^12 + 1341*x^11 + 2727*x^10 - 12674*x^9 - 40869*x^8 + 31254*x^7 + 243271*x^6 + 196857*x^5 - 393162*x^4 - 847458*x^3 - 566559*x^2 - 111090*x + 12167)
 
gp: K = bnfinit(y^15 - 60*y^13 - 63*y^12 + 1341*y^11 + 2727*y^10 - 12674*y^9 - 40869*y^8 + 31254*y^7 + 243271*y^6 + 196857*y^5 - 393162*y^4 - 847458*y^3 - 566559*y^2 - 111090*y + 12167, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 60*x^13 - 63*x^12 + 1341*x^11 + 2727*x^10 - 12674*x^9 - 40869*x^8 + 31254*x^7 + 243271*x^6 + 196857*x^5 - 393162*x^4 - 847458*x^3 - 566559*x^2 - 111090*x + 12167);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 60*x^13 - 63*x^12 + 1341*x^11 + 2727*x^10 - 12674*x^9 - 40869*x^8 + 31254*x^7 + 243271*x^6 + 196857*x^5 - 393162*x^4 - 847458*x^3 - 566559*x^2 - 111090*x + 12167)
 

\( x^{15} - 60 x^{13} - 63 x^{12} + 1341 x^{11} + 2727 x^{10} - 12674 x^{9} - 40869 x^{8} + 31254 x^{7} + 243271 x^{6} + 196857 x^{5} - 393162 x^{4} - 847458 x^{3} + \cdots + 12167 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[15, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(138622929316925604547946697\) \(\medspace = 3^{15}\cdot 11^{13}\cdot 23^{4}\) 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:  \(55.31\)
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:  $3^{241/162}11^{9/10}23^{2/3}\approx 358.80911756371285$
Ramified primes:   \(3\), \(11\), \(23\) 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:  \(\Q(\sqrt{33}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{46}a^{11}+\frac{9}{46}a^{9}-\frac{17}{46}a^{8}-\frac{8}{23}a^{7}+\frac{13}{46}a^{6}+\frac{11}{23}a^{5}-\frac{21}{46}a^{4}-\frac{3}{46}a^{3}-\frac{1}{2}a$, $\frac{1}{46}a^{12}+\frac{9}{46}a^{10}-\frac{17}{46}a^{9}-\frac{8}{23}a^{8}+\frac{13}{46}a^{7}+\frac{11}{23}a^{6}-\frac{21}{46}a^{5}-\frac{3}{46}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{46}a^{13}+\frac{3}{23}a^{10}-\frac{5}{46}a^{9}+\frac{5}{46}a^{8}+\frac{5}{46}a^{7}+\frac{3}{23}a^{5}+\frac{5}{46}a^{4}-\frac{19}{46}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{12\!\cdots\!06}a^{14}+\frac{1370582028003}{28\!\cdots\!61}a^{13}+\frac{11\!\cdots\!01}{12\!\cdots\!06}a^{12}-\frac{476042798426446}{64\!\cdots\!03}a^{11}+\frac{11\!\cdots\!80}{64\!\cdots\!03}a^{10}-\frac{30\!\cdots\!15}{64\!\cdots\!03}a^{9}-\frac{48\!\cdots\!31}{12\!\cdots\!06}a^{8}+\frac{45\!\cdots\!15}{12\!\cdots\!06}a^{7}-\frac{15\!\cdots\!50}{64\!\cdots\!03}a^{6}+\frac{11\!\cdots\!65}{28\!\cdots\!61}a^{5}-\frac{829197257205122}{28\!\cdots\!61}a^{4}+\frac{24\!\cdots\!87}{56\!\cdots\!22}a^{3}-\frac{12653543406775}{122541950361907}a^{2}-\frac{122129403538203}{245083900723814}a+\frac{13370030390824}{122541950361907}$ 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

Trivial group, which has order $1$ (assuming GRH)

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:  $14$
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:   $\frac{39\!\cdots\!85}{12\!\cdots\!06}a^{14}-\frac{402746262548505}{56\!\cdots\!22}a^{13}-\frac{21\!\cdots\!41}{12\!\cdots\!06}a^{12}+\frac{12\!\cdots\!67}{64\!\cdots\!03}a^{11}+\frac{23\!\cdots\!36}{64\!\cdots\!03}a^{10}-\frac{28\!\cdots\!25}{12\!\cdots\!06}a^{9}-\frac{24\!\cdots\!26}{64\!\cdots\!03}a^{8}-\frac{22\!\cdots\!96}{64\!\cdots\!03}a^{7}+\frac{11\!\cdots\!82}{64\!\cdots\!03}a^{6}+\frac{90\!\cdots\!37}{28\!\cdots\!61}a^{5}-\frac{90\!\cdots\!79}{56\!\cdots\!22}a^{4}-\frac{22\!\cdots\!90}{28\!\cdots\!61}a^{3}-\frac{16\!\cdots\!25}{245083900723814}a^{2}-\frac{18\!\cdots\!37}{122541950361907}a+\frac{39\!\cdots\!01}{245083900723814}$, $\frac{368748726176169}{64\!\cdots\!03}a^{14}-\frac{47850023790447}{56\!\cdots\!22}a^{13}-\frac{21\!\cdots\!66}{64\!\cdots\!03}a^{12}+\frac{88\!\cdots\!58}{64\!\cdots\!03}a^{11}+\frac{48\!\cdots\!20}{64\!\cdots\!03}a^{10}+\frac{55\!\cdots\!51}{12\!\cdots\!06}a^{9}-\frac{10\!\cdots\!87}{12\!\cdots\!06}a^{8}-\frac{14\!\cdots\!15}{12\!\cdots\!06}a^{7}+\frac{22\!\cdots\!67}{64\!\cdots\!03}a^{6}+\frac{23\!\cdots\!21}{28\!\cdots\!61}a^{5}-\frac{10\!\cdots\!35}{56\!\cdots\!22}a^{4}-\frac{11\!\cdots\!83}{56\!\cdots\!22}a^{3}-\frac{44\!\cdots\!57}{245083900723814}a^{2}-\frac{10\!\cdots\!69}{245083900723814}a+\frac{11\!\cdots\!71}{245083900723814}$, $\frac{97681176961776}{64\!\cdots\!03}a^{14}+\frac{13628846444295}{28\!\cdots\!61}a^{13}-\frac{77\!\cdots\!22}{64\!\cdots\!03}a^{12}-\frac{36\!\cdots\!67}{12\!\cdots\!06}a^{11}+\frac{20\!\cdots\!67}{64\!\cdots\!03}a^{10}+\frac{90\!\cdots\!41}{12\!\cdots\!06}a^{9}-\frac{44\!\cdots\!91}{12\!\cdots\!06}a^{8}-\frac{57\!\cdots\!83}{64\!\cdots\!03}a^{7}+\frac{18\!\cdots\!49}{12\!\cdots\!06}a^{6}+\frac{14\!\cdots\!22}{28\!\cdots\!61}a^{5}+\frac{18\!\cdots\!67}{56\!\cdots\!22}a^{4}-\frac{61\!\cdots\!19}{56\!\cdots\!22}a^{3}-\frac{13\!\cdots\!77}{122541950361907}a^{2}-\frac{69\!\cdots\!73}{245083900723814}a+\frac{30\!\cdots\!02}{122541950361907}$, $\frac{466429903137945}{64\!\cdots\!03}a^{14}-\frac{20592330901857}{56\!\cdots\!22}a^{13}-\frac{29\!\cdots\!88}{64\!\cdots\!03}a^{12}-\frac{18\!\cdots\!51}{12\!\cdots\!06}a^{11}+\frac{68\!\cdots\!87}{64\!\cdots\!03}a^{10}+\frac{72\!\cdots\!96}{64\!\cdots\!03}a^{9}-\frac{73\!\cdots\!39}{64\!\cdots\!03}a^{8}-\frac{26\!\cdots\!81}{12\!\cdots\!06}a^{7}+\frac{64\!\cdots\!83}{12\!\cdots\!06}a^{6}+\frac{38\!\cdots\!43}{28\!\cdots\!61}a^{5}-\frac{44\!\cdots\!34}{28\!\cdots\!61}a^{4}-\frac{37\!\cdots\!87}{122541950361907}a^{3}-\frac{71\!\cdots\!11}{245083900723814}a^{2}-\frac{88\!\cdots\!71}{122541950361907}a+\frac{17\!\cdots\!75}{245083900723814}$, $\frac{432580983011511}{64\!\cdots\!03}a^{14}-\frac{39165327120835}{56\!\cdots\!22}a^{13}-\frac{52\!\cdots\!49}{12\!\cdots\!06}a^{12}+\frac{51\!\cdots\!99}{12\!\cdots\!06}a^{11}+\frac{12\!\cdots\!53}{12\!\cdots\!06}a^{10}+\frac{90\!\cdots\!11}{12\!\cdots\!06}a^{9}-\frac{64\!\cdots\!93}{64\!\cdots\!03}a^{8}-\frac{98\!\cdots\!48}{64\!\cdots\!03}a^{7}+\frac{57\!\cdots\!09}{12\!\cdots\!06}a^{6}+\frac{61\!\cdots\!61}{56\!\cdots\!22}a^{5}-\frac{13\!\cdots\!21}{56\!\cdots\!22}a^{4}-\frac{30\!\cdots\!47}{122541950361907}a^{3}-\frac{27\!\cdots\!27}{122541950361907}a^{2}-\frac{67\!\cdots\!80}{122541950361907}a+\frac{13\!\cdots\!25}{245083900723814}$, $\frac{25\!\cdots\!89}{64\!\cdots\!03}a^{14}-\frac{481488339031855}{56\!\cdots\!22}a^{13}-\frac{14\!\cdots\!09}{64\!\cdots\!03}a^{12}+\frac{14\!\cdots\!05}{64\!\cdots\!03}a^{11}+\frac{31\!\cdots\!98}{64\!\cdots\!03}a^{10}+\frac{62\!\cdots\!59}{12\!\cdots\!06}a^{9}-\frac{67\!\cdots\!47}{12\!\cdots\!06}a^{8}-\frac{69\!\cdots\!87}{12\!\cdots\!06}a^{7}+\frac{15\!\cdots\!80}{64\!\cdots\!03}a^{6}+\frac{13\!\cdots\!97}{28\!\cdots\!61}a^{5}-\frac{11\!\cdots\!55}{56\!\cdots\!22}a^{4}-\frac{64\!\cdots\!29}{56\!\cdots\!22}a^{3}-\frac{23\!\cdots\!01}{245083900723814}a^{2}-\frac{54\!\cdots\!15}{245083900723814}a+\frac{56\!\cdots\!53}{245083900723814}$, $\frac{334899806049735}{64\!\cdots\!03}a^{14}-\frac{66423020009425}{56\!\cdots\!22}a^{13}-\frac{36\!\cdots\!05}{12\!\cdots\!06}a^{12}+\frac{20\!\cdots\!33}{64\!\cdots\!03}a^{11}+\frac{80\!\cdots\!19}{12\!\cdots\!06}a^{10}+\frac{11\!\cdots\!85}{64\!\cdots\!03}a^{9}-\frac{84\!\cdots\!95}{12\!\cdots\!06}a^{8}-\frac{40\!\cdots\!65}{64\!\cdots\!03}a^{7}+\frac{19\!\cdots\!30}{64\!\cdots\!03}a^{6}+\frac{32\!\cdots\!17}{56\!\cdots\!22}a^{5}-\frac{75\!\cdots\!94}{28\!\cdots\!61}a^{4}-\frac{79\!\cdots\!43}{56\!\cdots\!22}a^{3}-\frac{14\!\cdots\!50}{122541950361907}a^{2}-\frac{65\!\cdots\!87}{245083900723814}a+\frac{72\!\cdots\!21}{245083900723814}$, $\frac{70\!\cdots\!29}{12\!\cdots\!06}a^{14}-\frac{314702514561712}{28\!\cdots\!61}a^{13}-\frac{39\!\cdots\!79}{12\!\cdots\!06}a^{12}+\frac{18\!\cdots\!15}{64\!\cdots\!03}a^{11}+\frac{86\!\cdots\!75}{12\!\cdots\!06}a^{10}+\frac{70\!\cdots\!97}{64\!\cdots\!03}a^{9}-\frac{45\!\cdots\!82}{64\!\cdots\!03}a^{8}-\frac{49\!\cdots\!09}{64\!\cdots\!03}a^{7}+\frac{21\!\cdots\!88}{64\!\cdots\!03}a^{6}+\frac{36\!\cdots\!83}{56\!\cdots\!22}a^{5}-\frac{73\!\cdots\!06}{28\!\cdots\!61}a^{4}-\frac{44\!\cdots\!61}{28\!\cdots\!61}a^{3}-\frac{33\!\cdots\!99}{245083900723814}a^{2}-\frac{76\!\cdots\!39}{245083900723814}a+\frac{79\!\cdots\!99}{245083900723814}$, $\frac{98\!\cdots\!41}{12\!\cdots\!06}a^{14}-\frac{651807138082563}{56\!\cdots\!22}a^{13}-\frac{57\!\cdots\!49}{12\!\cdots\!06}a^{12}+\frac{12\!\cdots\!88}{64\!\cdots\!03}a^{11}+\frac{64\!\cdots\!41}{64\!\cdots\!03}a^{10}+\frac{71\!\cdots\!75}{12\!\cdots\!06}a^{9}-\frac{68\!\cdots\!98}{64\!\cdots\!03}a^{8}-\frac{96\!\cdots\!95}{64\!\cdots\!03}a^{7}+\frac{30\!\cdots\!70}{64\!\cdots\!03}a^{6}+\frac{31\!\cdots\!13}{28\!\cdots\!61}a^{5}-\frac{15\!\cdots\!03}{56\!\cdots\!22}a^{4}-\frac{73\!\cdots\!27}{28\!\cdots\!61}a^{3}-\frac{58\!\cdots\!41}{245083900723814}a^{2}-\frac{69\!\cdots\!32}{122541950361907}a+\frac{14\!\cdots\!53}{245083900723814}$, $\frac{43\!\cdots\!76}{64\!\cdots\!03}a^{14}-\frac{701901406216755}{56\!\cdots\!22}a^{13}-\frac{49\!\cdots\!97}{12\!\cdots\!06}a^{12}+\frac{36\!\cdots\!31}{12\!\cdots\!06}a^{11}+\frac{10\!\cdots\!47}{12\!\cdots\!06}a^{10}+\frac{32\!\cdots\!45}{12\!\cdots\!06}a^{9}-\frac{58\!\cdots\!67}{64\!\cdots\!03}a^{8}-\frac{69\!\cdots\!31}{64\!\cdots\!03}a^{7}+\frac{53\!\cdots\!05}{12\!\cdots\!06}a^{6}+\frac{48\!\cdots\!03}{56\!\cdots\!22}a^{5}-\frac{17\!\cdots\!35}{56\!\cdots\!22}a^{4}-\frac{58\!\cdots\!60}{28\!\cdots\!61}a^{3}-\frac{21\!\cdots\!70}{122541950361907}a^{2}-\frac{49\!\cdots\!66}{122541950361907}a+\frac{10\!\cdots\!09}{245083900723814}$, $\frac{71\!\cdots\!29}{64\!\cdots\!03}a^{14}-\frac{12\!\cdots\!57}{56\!\cdots\!22}a^{13}-\frac{80\!\cdots\!71}{12\!\cdots\!06}a^{12}+\frac{36\!\cdots\!62}{64\!\cdots\!03}a^{11}+\frac{89\!\cdots\!18}{64\!\cdots\!03}a^{10}+\frac{14\!\cdots\!43}{64\!\cdots\!03}a^{9}-\frac{94\!\cdots\!62}{64\!\cdots\!03}a^{8}-\frac{20\!\cdots\!39}{12\!\cdots\!06}a^{7}+\frac{43\!\cdots\!82}{64\!\cdots\!03}a^{6}+\frac{37\!\cdots\!00}{28\!\cdots\!61}a^{5}-\frac{15\!\cdots\!90}{28\!\cdots\!61}a^{4}-\frac{91\!\cdots\!66}{28\!\cdots\!61}a^{3}-\frac{67\!\cdots\!23}{245083900723814}a^{2}-\frac{15\!\cdots\!51}{245083900723814}a+\frac{79\!\cdots\!98}{122541950361907}$, $\frac{58\!\cdots\!83}{12\!\cdots\!06}a^{14}-\frac{217268964363708}{28\!\cdots\!61}a^{13}-\frac{33\!\cdots\!45}{12\!\cdots\!06}a^{12}+\frac{20\!\cdots\!99}{12\!\cdots\!06}a^{11}+\frac{37\!\cdots\!80}{64\!\cdots\!03}a^{10}+\frac{31\!\cdots\!59}{12\!\cdots\!06}a^{9}-\frac{39\!\cdots\!61}{64\!\cdots\!03}a^{8}-\frac{10\!\cdots\!01}{12\!\cdots\!06}a^{7}+\frac{36\!\cdots\!79}{12\!\cdots\!06}a^{6}+\frac{17\!\cdots\!20}{28\!\cdots\!61}a^{5}-\frac{44\!\cdots\!31}{245083900723814}a^{4}-\frac{41\!\cdots\!54}{28\!\cdots\!61}a^{3}-\frac{16\!\cdots\!13}{122541950361907}a^{2}-\frac{38\!\cdots\!53}{122541950361907}a+\frac{38\!\cdots\!39}{122541950361907}$, $\frac{10\!\cdots\!27}{12\!\cdots\!06}a^{14}-\frac{926761243064371}{56\!\cdots\!22}a^{13}-\frac{55\!\cdots\!47}{12\!\cdots\!06}a^{12}+\frac{54\!\cdots\!13}{12\!\cdots\!06}a^{11}+\frac{61\!\cdots\!61}{64\!\cdots\!03}a^{10}+\frac{68\!\cdots\!15}{64\!\cdots\!03}a^{9}-\frac{13\!\cdots\!27}{12\!\cdots\!06}a^{8}-\frac{67\!\cdots\!19}{64\!\cdots\!03}a^{7}+\frac{60\!\cdots\!21}{12\!\cdots\!06}a^{6}+\frac{25\!\cdots\!72}{28\!\cdots\!61}a^{5}-\frac{11\!\cdots\!86}{28\!\cdots\!61}a^{4}-\frac{12\!\cdots\!21}{56\!\cdots\!22}a^{3}-\frac{45\!\cdots\!93}{245083900723814}a^{2}-\frac{10\!\cdots\!35}{245083900723814}a+\frac{10\!\cdots\!91}{245083900723814}$, $\frac{10\!\cdots\!09}{12\!\cdots\!06}a^{14}-\frac{872665231557377}{56\!\cdots\!22}a^{13}-\frac{56\!\cdots\!89}{12\!\cdots\!06}a^{12}+\frac{49\!\cdots\!49}{12\!\cdots\!06}a^{11}+\frac{12\!\cdots\!69}{12\!\cdots\!06}a^{10}+\frac{11\!\cdots\!00}{64\!\cdots\!03}a^{9}-\frac{66\!\cdots\!30}{64\!\cdots\!03}a^{8}-\frac{14\!\cdots\!85}{12\!\cdots\!06}a^{7}+\frac{61\!\cdots\!67}{12\!\cdots\!06}a^{6}+\frac{52\!\cdots\!23}{56\!\cdots\!22}a^{5}-\frac{11\!\cdots\!07}{28\!\cdots\!61}a^{4}-\frac{64\!\cdots\!80}{28\!\cdots\!61}a^{3}-\frac{23\!\cdots\!93}{122541950361907}a^{2}-\frac{10\!\cdots\!83}{245083900723814}a+\frac{54\!\cdots\!70}{122541950361907}$ Copy content Toggle raw display (assuming GRH)
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:  \( 181222525.173 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 181222525.173 \cdot 1}{2\cdot\sqrt{138622929316925604547946697}}\cr\approx \mathstrut & 0.252182288159 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 60*x^13 - 63*x^12 + 1341*x^11 + 2727*x^10 - 12674*x^9 - 40869*x^8 + 31254*x^7 + 243271*x^6 + 196857*x^5 - 393162*x^4 - 847458*x^3 - 566559*x^2 - 111090*x + 12167)
 
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^15 - 60*x^13 - 63*x^12 + 1341*x^11 + 2727*x^10 - 12674*x^9 - 40869*x^8 + 31254*x^7 + 243271*x^6 + 196857*x^5 - 393162*x^4 - 847458*x^3 - 566559*x^2 - 111090*x + 12167, 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^15 - 60*x^13 - 63*x^12 + 1341*x^11 + 2727*x^10 - 12674*x^9 - 40869*x^8 + 31254*x^7 + 243271*x^6 + 196857*x^5 - 393162*x^4 - 847458*x^3 - 566559*x^2 - 111090*x + 12167);
 
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^15 - 60*x^13 - 63*x^12 + 1341*x^11 + 2727*x^10 - 12674*x^9 - 40869*x^8 + 31254*x^7 + 243271*x^6 + 196857*x^5 - 393162*x^4 - 847458*x^3 - 566559*x^2 - 111090*x + 12167);
 
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

$C_3^4:C_{10}$ (as 15T33):

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 solvable group of order 810
The 18 conjugacy class representatives for $C_3^4:C_{10}$
Character table for $C_3^4:C_{10}$

Intermediate fields

\(\Q(\zeta_{11})^+\)

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]
 

Sibling fields

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 siblings: data not computed
Minimal sibling: 15.15.138622929316925604547946697.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{3}$ R ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ R ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ ${\href{/padicField/17.5.0.1}{5} }^{3}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ R ${\href{/padicField/29.5.0.1}{5} }^{3}$ ${\href{/padicField/31.5.0.1}{5} }^{3}$ ${\href{/padicField/37.5.0.1}{5} }^{3}$ ${\href{/padicField/41.5.0.1}{5} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$

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 3.15.15.31$x^{15} - 9 x^{14} + 111 x^{13} + 4596 x^{12} + 29925 x^{11} + 55764 x^{10} + 33786 x^{9} + 2916 x^{8} - 2025 x^{7} + 40851 x^{6} + 21384 x^{5} - 20331 x^{4} - 3483 x^{3} + 10449 x^{2} - 2673 x + 243$$3$$5$$15$15T33$[3/2, 3/2, 3/2, 3/2]_{2}^{5}$
\(11\) Copy content Toggle raw display 11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} + 110$$10$$1$$9$$C_{10}$$[\ ]_{10}$
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.3.2.1$x^{3} + 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.3.2.1$x^{3} + 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$