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Properties

Label 18.8.569...707.1
Degree $18$
Signature $[8, 5]$
Discriminant $-5.699\times 10^{26}$
Root discriminant \(30.65\)
Ramified primes $3,73,577$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_4^3.C_6$ (as 18T765)

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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^18 - 6*x^17 + 3*x^16 + 48*x^15 - 168*x^14 + 300*x^13 - 138*x^12 - 1041*x^11 + 2739*x^10 - 1109*x^9 - 4908*x^8 + 6909*x^7 + 384*x^6 - 7284*x^5 + 4857*x^4 + 948*x^3 - 2631*x^2 + 1230*x - 107)
 
Copy content gp:K = bnfinit(y^18 - 6*y^17 + 3*y^16 + 48*y^15 - 168*y^14 + 300*y^13 - 138*y^12 - 1041*y^11 + 2739*y^10 - 1109*y^9 - 4908*y^8 + 6909*y^7 + 384*y^6 - 7284*y^5 + 4857*y^4 + 948*y^3 - 2631*y^2 + 1230*y - 107, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 3*x^16 + 48*x^15 - 168*x^14 + 300*x^13 - 138*x^12 - 1041*x^11 + 2739*x^10 - 1109*x^9 - 4908*x^8 + 6909*x^7 + 384*x^6 - 7284*x^5 + 4857*x^4 + 948*x^3 - 2631*x^2 + 1230*x - 107);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 3*x^16 + 48*x^15 - 168*x^14 + 300*x^13 - 138*x^12 - 1041*x^11 + 2739*x^10 - 1109*x^9 - 4908*x^8 + 6909*x^7 + 384*x^6 - 7284*x^5 + 4857*x^4 + 948*x^3 - 2631*x^2 + 1230*x - 107)
 

\( x^{18} - 6 x^{17} + 3 x^{16} + 48 x^{15} - 168 x^{14} + 300 x^{13} - 138 x^{12} - 1041 x^{11} + \cdots - 107 \) 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:  $18$
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:  $[8, 5]$
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:   \(-569862261508654647313096707\) \(\medspace = -\,3^{27}\cdot 73^{3}\cdot 577^{3}\) 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:  \(30.65\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(3\), \(73\), \(577\) 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:  \(\Q(\sqrt{-126363}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{9}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{9}-\frac{1}{9}a^{3}+\frac{1}{9}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{10}-\frac{1}{9}a^{4}+\frac{1}{9}a$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{11}-\frac{1}{9}a^{5}+\frac{1}{9}a^{2}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{9}-\frac{1}{9}a^{6}+\frac{1}{9}$, $\frac{1}{9}a^{16}-\frac{1}{9}a^{10}-\frac{1}{9}a^{7}+\frac{1}{9}a$, $\frac{1}{81\cdots 63}a^{17}+\frac{33\cdots 02}{81\cdots 63}a^{16}+\frac{13\cdots 51}{27\cdots 21}a^{15}-\frac{41\cdots 43}{90\cdots 07}a^{14}-\frac{41\cdots 65}{81\cdots 63}a^{13}-\frac{19\cdots 39}{81\cdots 63}a^{12}-\frac{73\cdots 91}{81\cdots 63}a^{11}+\frac{37\cdots 75}{27\cdots 21}a^{10}-\frac{27\cdots 43}{81\cdots 63}a^{9}+\frac{62\cdots 72}{81\cdots 63}a^{8}-\frac{11\cdots 28}{81\cdots 63}a^{7}+\frac{45\cdots 27}{27\cdots 21}a^{6}+\frac{17\cdots 99}{27\cdots 21}a^{5}-\frac{25\cdots 96}{81\cdots 63}a^{4}-\frac{23\cdots 35}{81\cdots 63}a^{3}-\frac{23\cdots 42}{81\cdots 63}a^{2}-\frac{75\cdots 08}{27\cdots 21}a-\frac{25\cdots 46}{81\cdots 63}$ 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:  $12$
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:   $\frac{37\cdots 92}{81\cdots 63}a^{17}-\frac{21\cdots 68}{81\cdots 63}a^{16}+\frac{23\cdots 93}{27\cdots 21}a^{15}+\frac{54\cdots 85}{27\cdots 21}a^{14}-\frac{56\cdots 37}{81\cdots 63}a^{13}+\frac{10\cdots 05}{81\cdots 63}a^{12}-\frac{66\cdots 83}{81\cdots 63}a^{11}-\frac{10\cdots 92}{27\cdots 21}a^{10}+\frac{83\cdots 89}{81\cdots 63}a^{9}-\frac{36\cdots 05}{81\cdots 63}a^{8}-\frac{12\cdots 89}{81\cdots 63}a^{7}+\frac{58\cdots 08}{27\cdots 21}a^{6}-\frac{14\cdots 13}{90\cdots 07}a^{5}-\frac{14\cdots 97}{81\cdots 63}a^{4}+\frac{10\cdots 23}{81\cdots 63}a^{3}+\frac{29\cdots 92}{81\cdots 63}a^{2}-\frac{12\cdots 51}{27\cdots 21}a+\frac{20\cdots 36}{81\cdots 63}$, $\frac{29\cdots 80}{41\cdots 37}a^{17}-\frac{50\cdots 21}{13\cdots 79}a^{16}-\frac{47\cdots 92}{45\cdots 93}a^{15}+\frac{45\cdots 56}{13\cdots 79}a^{14}-\frac{37\cdots 49}{41\cdots 37}a^{13}+\frac{56\cdots 20}{41\cdots 37}a^{12}+\frac{30\cdots 14}{41\cdots 37}a^{11}-\frac{28\cdots 27}{41\cdots 37}a^{10}+\frac{52\cdots 20}{41\cdots 37}a^{9}+\frac{15\cdots 60}{41\cdots 37}a^{8}-\frac{13\cdots 91}{45\cdots 93}a^{7}+\frac{23\cdots 82}{13\cdots 79}a^{6}+\frac{29\cdots 89}{13\cdots 79}a^{5}-\frac{10\cdots 04}{41\cdots 37}a^{4}-\frac{21\cdots 67}{41\cdots 37}a^{3}+\frac{32\cdots 66}{41\cdots 37}a^{2}-\frac{12\cdots 98}{41\cdots 37}a-\frac{37\cdots 13}{41\cdots 37}$, $\frac{29\cdots 80}{41\cdots 37}a^{17}-\frac{50\cdots 21}{13\cdots 79}a^{16}-\frac{47\cdots 92}{45\cdots 93}a^{15}+\frac{45\cdots 56}{13\cdots 79}a^{14}-\frac{37\cdots 49}{41\cdots 37}a^{13}+\frac{56\cdots 20}{41\cdots 37}a^{12}+\frac{30\cdots 14}{41\cdots 37}a^{11}-\frac{28\cdots 27}{41\cdots 37}a^{10}+\frac{52\cdots 20}{41\cdots 37}a^{9}+\frac{15\cdots 60}{41\cdots 37}a^{8}-\frac{13\cdots 91}{45\cdots 93}a^{7}+\frac{23\cdots 82}{13\cdots 79}a^{6}+\frac{29\cdots 89}{13\cdots 79}a^{5}-\frac{10\cdots 04}{41\cdots 37}a^{4}-\frac{21\cdots 67}{41\cdots 37}a^{3}+\frac{32\cdots 66}{41\cdots 37}a^{2}-\frac{12\cdots 98}{41\cdots 37}a+\frac{37\cdots 24}{41\cdots 37}$, $\frac{29\cdots 10}{27\cdots 21}a^{17}-\frac{41\cdots 39}{81\cdots 63}a^{16}-\frac{28\cdots 61}{81\cdots 63}a^{15}+\frac{38\cdots 99}{81\cdots 63}a^{14}-\frac{31\cdots 49}{27\cdots 21}a^{13}+\frac{13\cdots 55}{81\cdots 63}a^{12}+\frac{50\cdots 25}{81\cdots 63}a^{11}-\frac{82\cdots 40}{81\cdots 63}a^{10}+\frac{41\cdots 30}{27\cdots 21}a^{9}+\frac{26\cdots 72}{27\cdots 21}a^{8}-\frac{31\cdots 25}{81\cdots 63}a^{7}+\frac{13\cdots 57}{81\cdots 63}a^{6}+\frac{24\cdots 76}{81\cdots 63}a^{5}-\frac{85\cdots 81}{27\cdots 21}a^{4}-\frac{81\cdots 97}{81\cdots 63}a^{3}+\frac{98\cdots 57}{81\cdots 63}a^{2}-\frac{49\cdots 07}{81\cdots 63}a+\frac{11\cdots 75}{90\cdots 07}$, $\frac{24\cdots 42}{27\cdots 21}a^{17}-\frac{69\cdots 09}{81\cdots 63}a^{16}+\frac{19\cdots 73}{81\cdots 63}a^{15}+\frac{17\cdots 11}{81\cdots 63}a^{14}-\frac{23\cdots 83}{81\cdots 63}a^{13}+\frac{69\cdots 42}{81\cdots 63}a^{12}-\frac{11\cdots 82}{81\cdots 63}a^{11}+\frac{92\cdots 27}{27\cdots 21}a^{10}+\frac{12\cdots 79}{27\cdots 21}a^{9}-\frac{26\cdots 97}{27\cdots 21}a^{8}+\frac{21\cdots 98}{81\cdots 63}a^{7}+\frac{12\cdots 81}{81\cdots 63}a^{6}-\frac{14\cdots 65}{81\cdots 63}a^{5}-\frac{19\cdots 00}{81\cdots 63}a^{4}+\frac{12\cdots 84}{81\cdots 63}a^{3}-\frac{50\cdots 32}{81\cdots 63}a^{2}-\frac{14\cdots 02}{90\cdots 07}a+\frac{18\cdots 51}{27\cdots 21}$, $\frac{58\cdots 64}{81\cdots 63}a^{17}-\frac{27\cdots 80}{81\cdots 63}a^{16}-\frac{16\cdots 67}{81\cdots 63}a^{15}+\frac{24\cdots 55}{81\cdots 63}a^{14}-\frac{21\cdots 31}{27\cdots 21}a^{13}+\frac{98\cdots 92}{81\cdots 63}a^{12}+\frac{15\cdots 42}{27\cdots 21}a^{11}-\frac{48\cdots 83}{81\cdots 63}a^{10}+\frac{79\cdots 38}{81\cdots 63}a^{9}+\frac{35\cdots 30}{81\cdots 63}a^{8}-\frac{17\cdots 38}{81\cdots 63}a^{7}+\frac{84\cdots 14}{81\cdots 63}a^{6}+\frac{11\cdots 28}{81\cdots 63}a^{5}-\frac{43\cdots 82}{27\cdots 21}a^{4}+\frac{70\cdots 64}{81\cdots 63}a^{3}+\frac{14\cdots 11}{27\cdots 21}a^{2}-\frac{25\cdots 72}{81\cdots 63}a-\frac{49\cdots 08}{81\cdots 63}$, $\frac{89\cdots 86}{81\cdots 63}a^{17}-\frac{14\cdots 32}{27\cdots 21}a^{16}-\frac{20\cdots 66}{81\cdots 63}a^{15}+\frac{41\cdots 76}{81\cdots 63}a^{14}-\frac{10\cdots 05}{81\cdots 63}a^{13}+\frac{14\cdots 43}{81\cdots 63}a^{12}+\frac{57\cdots 65}{90\cdots 07}a^{11}-\frac{90\cdots 30}{81\cdots 63}a^{10}+\frac{15\cdots 32}{81\cdots 63}a^{9}+\frac{72\cdots 26}{81\cdots 63}a^{8}-\frac{12\cdots 56}{27\cdots 21}a^{7}+\frac{19\cdots 97}{81\cdots 63}a^{6}+\frac{31\cdots 81}{81\cdots 63}a^{5}-\frac{33\cdots 93}{81\cdots 63}a^{4}-\frac{19\cdots 94}{81\cdots 63}a^{3}+\frac{39\cdots 86}{27\cdots 21}a^{2}-\frac{49\cdots 37}{81\cdots 63}a+\frac{14\cdots 96}{81\cdots 63}$, $\frac{16\cdots 49}{81\cdots 63}a^{17}-\frac{88\cdots 51}{90\cdots 07}a^{16}-\frac{12\cdots 44}{27\cdots 21}a^{15}+\frac{72\cdots 50}{81\cdots 63}a^{14}-\frac{19\cdots 28}{81\cdots 63}a^{13}+\frac{31\cdots 85}{90\cdots 07}a^{12}+\frac{17\cdots 71}{27\cdots 21}a^{11}-\frac{15\cdots 52}{81\cdots 63}a^{10}+\frac{89\cdots 59}{27\cdots 21}a^{9}+\frac{10\cdots 54}{81\cdots 63}a^{8}-\frac{21\cdots 97}{27\cdots 21}a^{7}+\frac{12\cdots 55}{27\cdots 21}a^{6}+\frac{44\cdots 51}{81\cdots 63}a^{5}-\frac{57\cdots 80}{81\cdots 63}a^{4}+\frac{16\cdots 69}{27\cdots 21}a^{3}+\frac{21\cdots 23}{90\cdots 07}a^{2}-\frac{12\cdots 78}{81\cdots 63}a+\frac{16\cdots 90}{27\cdots 21}$, $\frac{51\cdots 09}{27\cdots 21}a^{17}-\frac{26\cdots 92}{27\cdots 21}a^{16}-\frac{11\cdots 50}{90\cdots 07}a^{15}+\frac{65\cdots 07}{81\cdots 63}a^{14}-\frac{19\cdots 56}{81\cdots 63}a^{13}+\frac{34\cdots 14}{81\cdots 63}a^{12}-\frac{11\cdots 54}{81\cdots 63}a^{11}-\frac{12\cdots 65}{81\cdots 63}a^{10}+\frac{27\cdots 61}{81\cdots 63}a^{9}-\frac{23\cdots 10}{90\cdots 07}a^{8}-\frac{15\cdots 25}{27\cdots 21}a^{7}+\frac{14\cdots 14}{27\cdots 21}a^{6}+\frac{12\cdots 12}{81\cdots 63}a^{5}-\frac{37\cdots 37}{81\cdots 63}a^{4}+\frac{16\cdots 54}{81\cdots 63}a^{3}+\frac{11\cdots 18}{81\cdots 63}a^{2}-\frac{35\cdots 45}{81\cdots 63}a-\frac{64\cdots 27}{81\cdots 63}$, $\frac{87\cdots 89}{27\cdots 21}a^{17}-\frac{68\cdots 02}{27\cdots 21}a^{16}+\frac{11\cdots 58}{27\cdots 21}a^{15}+\frac{11\cdots 82}{81\cdots 63}a^{14}-\frac{64\cdots 04}{81\cdots 63}a^{13}+\frac{14\cdots 83}{81\cdots 63}a^{12}-\frac{17\cdots 94}{81\cdots 63}a^{11}-\frac{16\cdots 51}{81\cdots 63}a^{10}+\frac{10\cdots 47}{81\cdots 63}a^{9}-\frac{41\cdots 40}{27\cdots 21}a^{8}-\frac{90\cdots 52}{90\cdots 07}a^{7}+\frac{95\cdots 73}{27\cdots 21}a^{6}-\frac{13\cdots 01}{81\cdots 63}a^{5}-\frac{14\cdots 06}{81\cdots 63}a^{4}+\frac{15\cdots 32}{81\cdots 63}a^{3}-\frac{29\cdots 94}{81\cdots 63}a^{2}-\frac{27\cdots 23}{81\cdots 63}a+\frac{33\cdots 96}{81\cdots 63}$, $\frac{19\cdots 94}{81\cdots 63}a^{17}-\frac{13\cdots 82}{81\cdots 63}a^{16}+\frac{22\cdots 04}{81\cdots 63}a^{15}+\frac{67\cdots 66}{81\cdots 63}a^{14}-\frac{14\cdots 73}{27\cdots 21}a^{13}+\frac{36\cdots 43}{27\cdots 21}a^{12}-\frac{46\cdots 58}{27\cdots 21}a^{11}-\frac{74\cdots 02}{81\cdots 63}a^{10}+\frac{68\cdots 11}{81\cdots 63}a^{9}-\frac{10\cdots 04}{81\cdots 63}a^{8}-\frac{79\cdots 23}{81\cdots 63}a^{7}+\frac{20\cdots 53}{81\cdots 63}a^{6}-\frac{19\cdots 00}{81\cdots 63}a^{5}-\frac{15\cdots 45}{27\cdots 21}a^{4}+\frac{19\cdots 22}{90\cdots 07}a^{3}-\frac{90\cdots 61}{90\cdots 07}a^{2}-\frac{92\cdots 97}{81\cdots 63}a-\frac{11\cdots 83}{81\cdots 63}$, $\frac{34\cdots 24}{81\cdots 63}a^{17}-\frac{15\cdots 69}{81\cdots 63}a^{16}-\frac{16\cdots 83}{81\cdots 63}a^{15}+\frac{18\cdots 15}{90\cdots 07}a^{14}-\frac{35\cdots 83}{81\cdots 63}a^{13}+\frac{38\cdots 17}{81\cdots 63}a^{12}+\frac{60\cdots 59}{81\cdots 63}a^{11}-\frac{42\cdots 13}{90\cdots 07}a^{10}+\frac{54\cdots 86}{90\cdots 07}a^{9}+\frac{55\cdots 88}{81\cdots 63}a^{8}-\frac{16\cdots 91}{81\cdots 63}a^{7}+\frac{50\cdots 94}{81\cdots 63}a^{6}+\frac{18\cdots 25}{90\cdots 07}a^{5}-\frac{15\cdots 22}{81\cdots 63}a^{4}-\frac{11\cdots 39}{81\cdots 63}a^{3}+\frac{77\cdots 89}{81\cdots 63}a^{2}-\frac{14\cdots 44}{27\cdots 21}a+\frac{69\cdots 49}{27\cdots 21}$ 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:  \( 3198073.57437 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{5}\cdot 3198073.57437 \cdot 1}{2\cdot\sqrt{569862261508654647313096707}}\cr\approx \mathstrut & 0.167924016596 \end{aligned}\] (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^18 - 6*x^17 + 3*x^16 + 48*x^15 - 168*x^14 + 300*x^13 - 138*x^12 - 1041*x^11 + 2739*x^10 - 1109*x^9 - 4908*x^8 + 6909*x^7 + 384*x^6 - 7284*x^5 + 4857*x^4 + 948*x^3 - 2631*x^2 + 1230*x - 107) 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^18 - 6*x^17 + 3*x^16 + 48*x^15 - 168*x^14 + 300*x^13 - 138*x^12 - 1041*x^11 + 2739*x^10 - 1109*x^9 - 4908*x^8 + 6909*x^7 + 384*x^6 - 7284*x^5 + 4857*x^4 + 948*x^3 - 2631*x^2 + 1230*x - 107, 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^18 - 6*x^17 + 3*x^16 + 48*x^15 - 168*x^14 + 300*x^13 - 138*x^12 - 1041*x^11 + 2739*x^10 - 1109*x^9 - 4908*x^8 + 6909*x^7 + 384*x^6 - 7284*x^5 + 4857*x^4 + 948*x^3 - 2631*x^2 + 1230*x - 107); 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 = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 3*x^16 + 48*x^15 - 168*x^14 + 300*x^13 - 138*x^12 - 1041*x^11 + 2739*x^10 - 1109*x^9 - 4908*x^8 + 6909*x^7 + 384*x^6 - 7284*x^5 + 4857*x^4 + 948*x^3 - 2631*x^2 + 1230*x - 107); 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

$S_4^3.C_6$ (as 18T765):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 82944
The 110 conjugacy class representatives for $S_4^3.C_6$
Character table for $S_4^3.C_6$

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $18$ R $18$ ${\href{/padicField/7.9.0.1}{9} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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 3.3.6.27a1.43$x^{18} + 12 x^{16} + 6 x^{15} + 60 x^{14} + 60 x^{13} + 181 x^{12} + 240 x^{11} + 408 x^{10} + 524 x^{9} + 696 x^{8} + 744 x^{7} + 787 x^{6} + 720 x^{5} + 540 x^{4} + 382 x^{3} + 204 x^{2} + 60 x + 10$$6$$3$$27$not computednot computed
\(73\) Copy content Toggle raw display $\Q_{73}$$x + 68$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{73}$$x + 68$$1$$1$$0$Trivial$$[\ ]$$
73.2.1.0a1.1$x^{2} + 70 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
73.2.1.0a1.1$x^{2} + 70 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
73.2.1.0a1.1$x^{2} + 70 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
73.3.1.0a1.1$x^{3} + 2 x + 68$$1$$3$$0$$C_3$$$[\ ]^{3}$$
73.3.1.0a1.1$x^{3} + 2 x + 68$$1$$3$$0$$C_3$$$[\ ]^{3}$$
73.1.4.3a1.3$x^{4} + 1825$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(577\) Copy content Toggle raw display $\Q_{577}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{577}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{577}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{577}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$

Spectrum of ring of integers

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