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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 / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content 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)
 
Copy content 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)
 
Copy content 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);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(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)
 

\( x^{15} - 60 x^{13} - 63 x^{12} + 1341 x^{11} + 2727 x^{10} - 12674 x^{9} - 40869 x^{8} + 31254 x^{7} + \cdots + 12167 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $15$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[15, 0]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(138622929316925604547946697\) \(\medspace = 3^{15}\cdot 11^{13}\cdot 23^{4}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(55.31\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content 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
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{33}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $14$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content 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)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 181222525.173 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content 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)

Copy content comment:Analytic class number formula
 
Copy content sage:# 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))))
 
Copy content gp:\\ 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<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); 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)));
 
Copy content oscar:# 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):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content 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.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content 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.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 3.5.3.15a27.1$x^{15} + 6 x^{11} + 3 x^{10} + 6 x^{9} + 6 x^{8} + 12 x^{7} + 18 x^{6} + 18 x^{5} + 18 x^{4} + 14 x^{3} + 24 x^{2} + 18 x + 7$$3$$5$$15$15T33$$[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{5}$$
\(11\) Copy content Toggle raw display 11.1.5.4a1.1$x^{5} + 11$$5$$1$$4$$C_5$$$[\ ]_{5}$$
11.1.10.9a1.10$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$$[\ ]$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.1.3.2a1.1$x^{3} + 23$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
23.1.3.2a1.1$x^{3} + 23$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)