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Properties

Label 18.18.176...000.1
Degree $18$
Signature $[18, 0]$
Discriminant $1.769\times 10^{32}$
Root discriminant \(61.88\)
Ramified primes $2,3,5,23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 57*x^16 + 114*x^15 + 1281*x^14 - 1467*x^13 - 14285*x^12 + 6657*x^11 + 82890*x^10 + 2168*x^9 - 247929*x^8 - 86271*x^7 + 358513*x^6 + 187347*x^5 - 215109*x^4 - 108905*x^3 + 56235*x^2 + 16050*x - 5725)
 
Copy content gp:K = bnfinit(y^18 - 3*y^17 - 57*y^16 + 114*y^15 + 1281*y^14 - 1467*y^13 - 14285*y^12 + 6657*y^11 + 82890*y^10 + 2168*y^9 - 247929*y^8 - 86271*y^7 + 358513*y^6 + 187347*y^5 - 215109*y^4 - 108905*y^3 + 56235*y^2 + 16050*y - 5725, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 57*x^16 + 114*x^15 + 1281*x^14 - 1467*x^13 - 14285*x^12 + 6657*x^11 + 82890*x^10 + 2168*x^9 - 247929*x^8 - 86271*x^7 + 358513*x^6 + 187347*x^5 - 215109*x^4 - 108905*x^3 + 56235*x^2 + 16050*x - 5725);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 3*x^17 - 57*x^16 + 114*x^15 + 1281*x^14 - 1467*x^13 - 14285*x^12 + 6657*x^11 + 82890*x^10 + 2168*x^9 - 247929*x^8 - 86271*x^7 + 358513*x^6 + 187347*x^5 - 215109*x^4 - 108905*x^3 + 56235*x^2 + 16050*x - 5725)
 

\( x^{18} - 3 x^{17} - 57 x^{16} + 114 x^{15} + 1281 x^{14} - 1467 x^{13} - 14285 x^{12} + 6657 x^{11} + \cdots - 5725 \) 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:  $[18, 0]$
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:   \(176938682194265948289288000000000\) \(\medspace = 2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 23^{8}\) 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:  \(61.88\)
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:  $2^{2/3}3^{4/3}5^{1/2}23^{2/3}\approx 124.20866564536522$
Ramified primes:   \(2\), \(3\), \(5\), \(23\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{6}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{7}+\frac{1}{5}a^{3}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{8}+\frac{1}{5}a^{4}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{9}+\frac{1}{5}a^{5}$, $\frac{1}{25}a^{14}-\frac{1}{25}a^{13}-\frac{1}{25}a^{12}-\frac{1}{25}a^{11}+\frac{1}{25}a^{10}+\frac{12}{25}a^{9}-\frac{8}{25}a^{8}+\frac{12}{25}a^{7}+\frac{2}{5}a^{6}-\frac{11}{25}a^{5}-\frac{11}{25}a^{4}+\frac{9}{25}a^{3}+\frac{8}{25}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{425}a^{15}+\frac{1}{85}a^{14}-\frac{2}{425}a^{13}-\frac{37}{425}a^{12}-\frac{2}{85}a^{11}+\frac{18}{425}a^{10}-\frac{146}{425}a^{9}+\frac{124}{425}a^{8}-\frac{33}{425}a^{7}-\frac{1}{425}a^{6}+\frac{78}{425}a^{5}+\frac{63}{425}a^{4}+\frac{82}{425}a^{3}+\frac{18}{425}a^{2}-\frac{3}{85}a+\frac{33}{85}$, $\frac{1}{19975}a^{16}+\frac{2}{3995}a^{15}-\frac{62}{19975}a^{14}+\frac{1483}{19975}a^{13}-\frac{277}{3995}a^{12}+\frac{988}{19975}a^{11}-\frac{821}{19975}a^{10}+\frac{3134}{19975}a^{9}-\frac{1708}{19975}a^{8}+\frac{5019}{19975}a^{7}-\frac{2392}{19975}a^{6}-\frac{4392}{19975}a^{5}+\frac{5582}{19975}a^{4}-\frac{2802}{19975}a^{3}-\frac{2}{3995}a^{2}-\frac{1002}{3995}a-\frac{1}{799}$, $\frac{1}{60\cdots 75}a^{17}-\frac{10\cdots 42}{12\cdots 15}a^{16}+\frac{81\cdots 28}{70\cdots 95}a^{15}+\frac{13\cdots 74}{60\cdots 75}a^{14}+\frac{13\cdots 67}{24\cdots 43}a^{13}+\frac{13\cdots 08}{60\cdots 75}a^{12}-\frac{36\cdots 77}{60\cdots 75}a^{11}+\frac{54\cdots 66}{60\cdots 75}a^{10}-\frac{62\cdots 68}{60\cdots 75}a^{9}+\frac{22\cdots 54}{60\cdots 75}a^{8}-\frac{74\cdots 71}{60\cdots 75}a^{7}-\frac{19\cdots 14}{60\cdots 75}a^{6}-\frac{15\cdots 74}{35\cdots 75}a^{5}-\frac{19\cdots 82}{60\cdots 75}a^{4}-\frac{11\cdots 17}{60\cdots 75}a^{3}-\frac{16\cdots 61}{60\cdots 75}a^{2}+\frac{14\cdots 68}{12\cdots 15}a+\frac{28\cdots 04}{12\cdots 15}$ 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:  No
Index:  Not computed
Inessential primes:  $5$

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:  $17$
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:   $\frac{19\cdots 38}{75\cdots 25}a^{17}-\frac{15\cdots 99}{12\cdots 25}a^{16}-\frac{17\cdots 08}{12\cdots 25}a^{15}+\frac{63\cdots 42}{12\cdots 25}a^{14}+\frac{34\cdots 14}{12\cdots 25}a^{13}-\frac{98\cdots 96}{12\cdots 25}a^{12}-\frac{33\cdots 54}{12\cdots 25}a^{11}+\frac{70\cdots 86}{12\cdots 25}a^{10}+\frac{17\cdots 44}{12\cdots 25}a^{9}-\frac{23\cdots 71}{12\cdots 25}a^{8}-\frac{47\cdots 78}{12\cdots 25}a^{7}+\frac{35\cdots 72}{12\cdots 25}a^{6}+\frac{62\cdots 06}{12\cdots 25}a^{5}-\frac{20\cdots 99}{12\cdots 25}a^{4}-\frac{12\cdots 46}{51\cdots 69}a^{3}+\frac{11\cdots 82}{25\cdots 45}a^{2}+\frac{92\cdots 10}{30\cdots 57}a-\frac{26\cdots 87}{51\cdots 69}$, $\frac{20\cdots 36}{12\cdots 15}a^{17}-\frac{39\cdots 97}{60\cdots 75}a^{16}-\frac{21\cdots 32}{24\cdots 43}a^{15}+\frac{16\cdots 68}{60\cdots 75}a^{14}+\frac{22\cdots 78}{12\cdots 15}a^{13}-\frac{25\cdots 09}{60\cdots 75}a^{12}-\frac{47\cdots 44}{24\cdots 43}a^{11}+\frac{18\cdots 21}{60\cdots 75}a^{10}+\frac{13\cdots 64}{12\cdots 15}a^{9}-\frac{61\cdots 06}{60\cdots 75}a^{8}-\frac{37\cdots 72}{12\cdots 15}a^{7}+\frac{98\cdots 29}{60\cdots 75}a^{6}+\frac{52\cdots 02}{12\cdots 15}a^{5}-\frac{41\cdots 94}{35\cdots 75}a^{4}-\frac{28\cdots 38}{12\cdots 15}a^{3}+\frac{33\cdots 32}{60\cdots 75}a^{2}+\frac{86\cdots 69}{24\cdots 43}a-\frac{11\cdots 63}{12\cdots 15}$, $\frac{20\cdots 36}{12\cdots 15}a^{17}-\frac{39\cdots 97}{60\cdots 75}a^{16}-\frac{21\cdots 32}{24\cdots 43}a^{15}+\frac{16\cdots 68}{60\cdots 75}a^{14}+\frac{22\cdots 78}{12\cdots 15}a^{13}-\frac{25\cdots 09}{60\cdots 75}a^{12}-\frac{47\cdots 44}{24\cdots 43}a^{11}+\frac{18\cdots 21}{60\cdots 75}a^{10}+\frac{13\cdots 64}{12\cdots 15}a^{9}-\frac{61\cdots 06}{60\cdots 75}a^{8}-\frac{37\cdots 72}{12\cdots 15}a^{7}+\frac{98\cdots 29}{60\cdots 75}a^{6}+\frac{52\cdots 02}{12\cdots 15}a^{5}-\frac{41\cdots 94}{35\cdots 75}a^{4}-\frac{28\cdots 38}{12\cdots 15}a^{3}+\frac{33\cdots 32}{60\cdots 75}a^{2}+\frac{86\cdots 69}{24\cdots 43}a-\frac{11\cdots 48}{12\cdots 15}$, $\frac{64\cdots 83}{60\cdots 75}a^{17}-\frac{27\cdots 93}{60\cdots 75}a^{16}-\frac{33\cdots 93}{60\cdots 75}a^{15}+\frac{22\cdots 53}{12\cdots 15}a^{14}+\frac{13\cdots 59}{12\cdots 15}a^{13}-\frac{17\cdots 37}{60\cdots 75}a^{12}-\frac{69\cdots 97}{60\cdots 75}a^{11}+\frac{12\cdots 14}{60\cdots 75}a^{10}+\frac{37\cdots 36}{60\cdots 75}a^{9}-\frac{42\cdots 97}{60\cdots 75}a^{8}-\frac{10\cdots 62}{60\cdots 75}a^{7}+\frac{66\cdots 12}{60\cdots 75}a^{6}+\frac{14\cdots 21}{60\cdots 75}a^{5}-\frac{47\cdots 23}{60\cdots 75}a^{4}-\frac{76\cdots 78}{60\cdots 75}a^{3}+\frac{19\cdots 54}{60\cdots 75}a^{2}+\frac{22\cdots 77}{12\cdots 15}a-\frac{59\cdots 01}{12\cdots 15}$, $\frac{18\cdots 89}{60\cdots 75}a^{17}-\frac{77\cdots 17}{60\cdots 75}a^{16}-\frac{19\cdots 77}{12\cdots 15}a^{15}+\frac{64\cdots 63}{12\cdots 15}a^{14}+\frac{20\cdots 64}{60\cdots 75}a^{13}-\frac{50\cdots 98}{60\cdots 75}a^{12}-\frac{21\cdots 59}{60\cdots 75}a^{11}+\frac{35\cdots 21}{60\cdots 75}a^{10}+\frac{23\cdots 29}{12\cdots 15}a^{9}-\frac{12\cdots 18}{60\cdots 75}a^{8}-\frac{32\cdots 67}{60\cdots 75}a^{7}+\frac{19\cdots 88}{60\cdots 75}a^{6}+\frac{45\cdots 77}{60\cdots 75}a^{5}-\frac{14\cdots 17}{60\cdots 75}a^{4}-\frac{24\cdots 09}{60\cdots 75}a^{3}+\frac{61\cdots 81}{60\cdots 75}a^{2}+\frac{72\cdots 11}{12\cdots 15}a-\frac{19\cdots 19}{12\cdots 15}$, $\frac{12\cdots 24}{60\cdots 75}a^{17}-\frac{50\cdots 59}{60\cdots 75}a^{16}-\frac{62\cdots 08}{60\cdots 75}a^{15}+\frac{21\cdots 39}{60\cdots 75}a^{14}+\frac{12\cdots 19}{60\cdots 75}a^{13}-\frac{32\cdots 27}{60\cdots 75}a^{12}-\frac{26\cdots 51}{12\cdots 15}a^{11}+\frac{23\cdots 89}{60\cdots 75}a^{10}+\frac{14\cdots 96}{12\cdots 15}a^{9}-\frac{81\cdots 84}{60\cdots 75}a^{8}-\frac{20\cdots 31}{60\cdots 75}a^{7}+\frac{26\cdots 24}{12\cdots 15}a^{6}+\frac{28\cdots 37}{60\cdots 75}a^{5}-\frac{19\cdots 77}{12\cdots 15}a^{4}-\frac{15\cdots 01}{60\cdots 75}a^{3}+\frac{43\cdots 32}{60\cdots 75}a^{2}+\frac{46\cdots 39}{12\cdots 15}a-\frac{13\cdots 28}{12\cdots 15}$, $\frac{30\cdots 84}{35\cdots 75}a^{17}-\frac{21\cdots 56}{60\cdots 75}a^{16}-\frac{55\cdots 57}{12\cdots 15}a^{15}+\frac{90\cdots 93}{60\cdots 75}a^{14}+\frac{57\cdots 33}{60\cdots 75}a^{13}-\frac{28\cdots 82}{12\cdots 15}a^{12}-\frac{59\cdots 78}{60\cdots 75}a^{11}+\frac{10\cdots 48}{60\cdots 75}a^{10}+\frac{64\cdots 84}{12\cdots 15}a^{9}-\frac{34\cdots 82}{60\cdots 75}a^{8}-\frac{53\cdots 87}{35\cdots 75}a^{7}+\frac{22\cdots 92}{24\cdots 43}a^{6}+\frac{12\cdots 44}{60\cdots 75}a^{5}-\frac{40\cdots 12}{60\cdots 75}a^{4}-\frac{67\cdots 73}{60\cdots 75}a^{3}+\frac{17\cdots 19}{60\cdots 75}a^{2}+\frac{20\cdots 12}{12\cdots 15}a-\frac{53\cdots 61}{12\cdots 15}$, $\frac{42\cdots 87}{35\cdots 75}a^{17}-\frac{27\cdots 46}{60\cdots 75}a^{16}-\frac{23\cdots 84}{35\cdots 75}a^{15}+\frac{11\cdots 54}{60\cdots 75}a^{14}+\frac{85\cdots 77}{60\cdots 75}a^{13}-\frac{17\cdots 73}{60\cdots 75}a^{12}-\frac{53\cdots 01}{35\cdots 75}a^{11}+\frac{12\cdots 67}{60\cdots 75}a^{10}+\frac{51\cdots 34}{60\cdots 75}a^{9}-\frac{44\cdots 16}{60\cdots 75}a^{8}-\frac{15\cdots 87}{60\cdots 75}a^{7}+\frac{74\cdots 74}{60\cdots 75}a^{6}+\frac{42\cdots 99}{12\cdots 15}a^{5}-\frac{12\cdots 59}{12\cdots 15}a^{4}-\frac{12\cdots 48}{60\cdots 75}a^{3}+\frac{12\cdots 45}{24\cdots 43}a^{2}+\frac{37\cdots 17}{12\cdots 15}a-\frac{20\cdots 96}{24\cdots 43}$, $\frac{22\cdots 57}{60\cdots 75}a^{17}-\frac{91\cdots 03}{60\cdots 75}a^{16}-\frac{11\cdots 01}{60\cdots 75}a^{15}+\frac{45\cdots 37}{70\cdots 95}a^{14}+\frac{24\cdots 02}{60\cdots 75}a^{13}-\frac{59\cdots 63}{60\cdots 75}a^{12}-\frac{25\cdots 34}{60\cdots 75}a^{11}+\frac{42\cdots 18}{60\cdots 75}a^{10}+\frac{27\cdots 56}{12\cdots 15}a^{9}-\frac{29\cdots 84}{12\cdots 15}a^{8}-\frac{38\cdots 89}{60\cdots 75}a^{7}+\frac{23\cdots 84}{60\cdots 75}a^{6}+\frac{52\cdots 06}{60\cdots 75}a^{5}-\frac{17\cdots 14}{60\cdots 75}a^{4}-\frac{28\cdots 36}{60\cdots 75}a^{3}+\frac{74\cdots 18}{60\cdots 75}a^{2}+\frac{84\cdots 44}{12\cdots 15}a-\frac{23\cdots 27}{12\cdots 15}$, $\frac{10\cdots 97}{60\cdots 75}a^{17}-\frac{42\cdots 71}{60\cdots 75}a^{16}-\frac{56\cdots 57}{60\cdots 75}a^{15}+\frac{35\cdots 66}{12\cdots 15}a^{14}+\frac{11\cdots 16}{60\cdots 75}a^{13}-\frac{11\cdots 34}{25\cdots 45}a^{12}-\frac{12\cdots 22}{60\cdots 75}a^{11}+\frac{19\cdots 17}{60\cdots 75}a^{10}+\frac{68\cdots 22}{60\cdots 75}a^{9}-\frac{68\cdots 02}{60\cdots 75}a^{8}-\frac{39\cdots 26}{12\cdots 15}a^{7}+\frac{10\cdots 76}{60\cdots 75}a^{6}+\frac{54\cdots 86}{12\cdots 15}a^{5}-\frac{78\cdots 12}{60\cdots 75}a^{4}-\frac{14\cdots 56}{60\cdots 75}a^{3}+\frac{35\cdots 12}{60\cdots 75}a^{2}+\frac{25\cdots 27}{70\cdots 95}a-\frac{11\cdots 93}{12\cdots 15}$, $\frac{24\cdots 61}{60\cdots 75}a^{17}-\frac{61\cdots 03}{35\cdots 75}a^{16}-\frac{24\cdots 16}{12\cdots 15}a^{15}+\frac{43\cdots 43}{60\cdots 75}a^{14}+\frac{50\cdots 17}{12\cdots 15}a^{13}-\frac{40\cdots 57}{35\cdots 75}a^{12}-\frac{25\cdots 02}{60\cdots 75}a^{11}+\frac{49\cdots 14}{60\cdots 75}a^{10}+\frac{13\cdots 52}{60\cdots 75}a^{9}-\frac{17\cdots 89}{60\cdots 75}a^{8}-\frac{37\cdots 96}{60\cdots 75}a^{7}+\frac{27\cdots 08}{60\cdots 75}a^{6}+\frac{50\cdots 77}{60\cdots 75}a^{5}-\frac{19\cdots 51}{60\cdots 75}a^{4}-\frac{26\cdots 07}{60\cdots 75}a^{3}+\frac{31\cdots 79}{24\cdots 43}a^{2}+\frac{79\cdots 83}{12\cdots 15}a-\frac{26\cdots 60}{14\cdots 79}$, $\frac{32\cdots 29}{60\cdots 75}a^{17}-\frac{13\cdots 66}{60\cdots 75}a^{16}-\frac{67\cdots 80}{24\cdots 43}a^{15}+\frac{11\cdots 02}{12\cdots 15}a^{14}+\frac{69\cdots 96}{12\cdots 15}a^{13}-\frac{38\cdots 48}{25\cdots 45}a^{12}-\frac{35\cdots 83}{60\cdots 75}a^{11}+\frac{64\cdots 82}{60\cdots 75}a^{10}+\frac{18\cdots 88}{60\cdots 75}a^{9}-\frac{22\cdots 02}{60\cdots 75}a^{8}-\frac{52\cdots 04}{60\cdots 75}a^{7}+\frac{35\cdots 16}{60\cdots 75}a^{6}+\frac{71\cdots 03}{60\cdots 75}a^{5}-\frac{25\cdots 77}{60\cdots 75}a^{4}-\frac{37\cdots 03}{60\cdots 75}a^{3}+\frac{10\cdots 77}{60\cdots 75}a^{2}+\frac{11\cdots 77}{12\cdots 15}a-\frac{30\cdots 03}{12\cdots 15}$, $\frac{34\cdots 94}{60\cdots 75}a^{17}-\frac{13\cdots 38}{60\cdots 75}a^{16}-\frac{18\cdots 59}{60\cdots 75}a^{15}+\frac{72\cdots 51}{75\cdots 25}a^{14}+\frac{38\cdots 07}{60\cdots 75}a^{13}-\frac{90\cdots 62}{60\cdots 75}a^{12}-\frac{40\cdots 89}{60\cdots 75}a^{11}+\frac{38\cdots 66}{35\cdots 75}a^{10}+\frac{22\cdots 89}{60\cdots 75}a^{9}-\frac{22\cdots 97}{60\cdots 75}a^{8}-\frac{12\cdots 48}{12\cdots 15}a^{7}+\frac{21\cdots 02}{35\cdots 75}a^{6}+\frac{17\cdots 48}{12\cdots 15}a^{5}-\frac{26\cdots 83}{60\cdots 75}a^{4}-\frac{48\cdots 92}{60\cdots 75}a^{3}+\frac{24\cdots 78}{12\cdots 15}a^{2}+\frac{14\cdots 33}{12\cdots 15}a-\frac{78\cdots 01}{24\cdots 43}$, $\frac{98\cdots 96}{60\cdots 75}a^{17}-\frac{40\cdots 37}{60\cdots 75}a^{16}-\frac{20\cdots 43}{24\cdots 43}a^{15}+\frac{17\cdots 57}{60\cdots 75}a^{14}+\frac{21\cdots 07}{12\cdots 15}a^{13}-\frac{26\cdots 66}{60\cdots 75}a^{12}-\frac{11\cdots 12}{60\cdots 75}a^{11}+\frac{19\cdots 22}{60\cdots 75}a^{10}+\frac{60\cdots 42}{60\cdots 75}a^{9}-\frac{66\cdots 82}{60\cdots 75}a^{8}-\frac{17\cdots 21}{60\cdots 75}a^{7}+\frac{43\cdots 48}{24\cdots 43}a^{6}+\frac{23\cdots 02}{60\cdots 75}a^{5}-\frac{16\cdots 01}{12\cdots 15}a^{4}-\frac{13\cdots 27}{60\cdots 75}a^{3}+\frac{36\cdots 11}{60\cdots 75}a^{2}+\frac{42\cdots 03}{12\cdots 15}a-\frac{12\cdots 44}{12\cdots 15}$, $\frac{83\cdots 41}{60\cdots 75}a^{17}-\frac{37\cdots 18}{60\cdots 75}a^{16}-\frac{41\cdots 81}{60\cdots 75}a^{15}+\frac{15\cdots 16}{60\cdots 75}a^{14}+\frac{81\cdots 77}{60\cdots 75}a^{13}-\frac{24\cdots 36}{60\cdots 75}a^{12}-\frac{79\cdots 32}{60\cdots 75}a^{11}+\frac{18\cdots 62}{60\cdots 75}a^{10}+\frac{40\cdots 93}{60\cdots 75}a^{9}-\frac{62\cdots 19}{60\cdots 75}a^{8}-\frac{10\cdots 68}{60\cdots 75}a^{7}+\frac{10\cdots 93}{60\cdots 75}a^{6}+\frac{14\cdots 09}{60\cdots 75}a^{5}-\frac{40\cdots 91}{35\cdots 75}a^{4}-\frac{72\cdots 94}{60\cdots 75}a^{3}+\frac{23\cdots 29}{60\cdots 75}a^{2}+\frac{20\cdots 16}{12\cdots 15}a-\frac{60\cdots 06}{12\cdots 15}$, $\frac{42\cdots 99}{35\cdots 75}a^{17}-\frac{27\cdots 74}{60\cdots 75}a^{16}-\frac{38\cdots 96}{60\cdots 75}a^{15}+\frac{11\cdots 51}{60\cdots 75}a^{14}+\frac{81\cdots 34}{60\cdots 75}a^{13}-\frac{34\cdots 67}{12\cdots 15}a^{12}-\frac{86\cdots 64}{60\cdots 75}a^{11}+\frac{24\cdots 44}{12\cdots 15}a^{10}+\frac{28\cdots 19}{35\cdots 75}a^{9}-\frac{39\cdots 23}{60\cdots 75}a^{8}-\frac{13\cdots 04}{60\cdots 75}a^{7}+\frac{56\cdots 72}{60\cdots 75}a^{6}+\frac{22\cdots 28}{70\cdots 95}a^{5}-\frac{31\cdots 13}{60\cdots 75}a^{4}-\frac{10\cdots 21}{60\cdots 75}a^{3}+\frac{14\cdots 22}{60\cdots 75}a^{2}+\frac{29\cdots 89}{12\cdots 15}a-\frac{64\cdots 78}{12\cdots 15}$, $\frac{85\cdots 54}{35\cdots 75}a^{17}-\frac{35\cdots 99}{35\cdots 75}a^{16}-\frac{75\cdots 72}{60\cdots 75}a^{15}+\frac{25\cdots 19}{60\cdots 75}a^{14}+\frac{15\cdots 09}{60\cdots 75}a^{13}-\frac{39\cdots 38}{60\cdots 75}a^{12}-\frac{16\cdots 41}{60\cdots 75}a^{11}+\frac{28\cdots 36}{60\cdots 75}a^{10}+\frac{87\cdots 68}{60\cdots 75}a^{9}-\frac{19\cdots 13}{12\cdots 15}a^{8}-\frac{24\cdots 97}{60\cdots 75}a^{7}+\frac{15\cdots 66}{60\cdots 75}a^{6}+\frac{67\cdots 34}{12\cdots 15}a^{5}-\frac{11\cdots 94}{60\cdots 75}a^{4}-\frac{72\cdots 39}{24\cdots 43}a^{3}+\frac{48\cdots 69}{60\cdots 75}a^{2}+\frac{10\cdots 36}{24\cdots 43}a-\frac{14\cdots 16}{12\cdots 15}$ 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:  \( 100698193013 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 100698193013 \cdot 1}{2\cdot\sqrt{176938682194265948289288000000000}}\cr\approx \mathstrut & 0.992247948069549 \end{aligned}\] (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 - 3*x^17 - 57*x^16 + 114*x^15 + 1281*x^14 - 1467*x^13 - 14285*x^12 + 6657*x^11 + 82890*x^10 + 2168*x^9 - 247929*x^8 - 86271*x^7 + 358513*x^6 + 187347*x^5 - 215109*x^4 - 108905*x^3 + 56235*x^2 + 16050*x - 5725) 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 - 3*x^17 - 57*x^16 + 114*x^15 + 1281*x^14 - 1467*x^13 - 14285*x^12 + 6657*x^11 + 82890*x^10 + 2168*x^9 - 247929*x^8 - 86271*x^7 + 358513*x^6 + 187347*x^5 - 215109*x^4 - 108905*x^3 + 56235*x^2 + 16050*x - 5725, 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 - 3*x^17 - 57*x^16 + 114*x^15 + 1281*x^14 - 1467*x^13 - 14285*x^12 + 6657*x^11 + 82890*x^10 + 2168*x^9 - 247929*x^8 - 86271*x^7 + 358513*x^6 + 187347*x^5 - 215109*x^4 - 108905*x^3 + 56235*x^2 + 16050*x - 5725); 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^18 - 3*x^17 - 57*x^16 + 114*x^15 + 1281*x^14 - 1467*x^13 - 14285*x^12 + 6657*x^11 + 82890*x^10 + 2168*x^9 - 247929*x^8 - 86271*x^7 + 358513*x^6 + 187347*x^5 - 215109*x^4 - 108905*x^3 + 56235*x^2 + 16050*x - 5725); 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

$C_3^2:S_3$ (as 18T24):

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 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.1620.1 x3, 6.6.13122000.1, 9.9.1189751847048000.1

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

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: data not computed
Degree 27 sibling: data not computed
Minimal sibling: 9.9.1189751847048000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ R ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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
\(2\) Copy content Toggle raw display 2.6.3.12a1.2$x^{18} + 3 x^{16} + 3 x^{15} + 3 x^{14} + 9 x^{13} + 7 x^{12} + 9 x^{11} + 15 x^{10} + 10 x^{9} + 12 x^{8} + 15 x^{7} + 9 x^{6} + 9 x^{5} + 9 x^{4} + 4 x^{3} + 3 x^{2} + 3 x + 3$$3$$6$$12$$S_3 \times C_3$$$[\ ]_{3}^{6}$$
\(3\) Copy content Toggle raw display 3.6.3.24a1.1$x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 29 x^{12} + 24 x^{11} + 51 x^{10} + 36 x^{9} + 72 x^{8} + 60 x^{7} + 85 x^{6} + 78 x^{5} + 69 x^{4} + 44 x^{3} + 60 x^{2} + 48 x + 23$$3$$6$$24$$S_3 \times C_3$not computed
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(23\) Copy content Toggle raw display 23.2.3.4a1.1$x^{6} + 63 x^{5} + 1338 x^{4} + 9891 x^{3} + 6690 x^{2} + 1598 x + 125$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
23.2.3.4a1.1$x^{6} + 63 x^{5} + 1338 x^{4} + 9891 x^{3} + 6690 x^{2} + 1598 x + 125$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
23.6.1.0a1.1$x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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