↑

Properties

Label 18.12.431...624.1
Degree $18$
Signature $(12, 3)$
Discriminant $-4.311\times 10^{38}$
Root discriminant \(140.08\)
Ramified primes $2,7,43,4129,242689$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $S_3^3.S_3\wr C_3$ (as 18T807)

Related objects

Downloads

Learn more

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 - 131*x^16 - 172*x^15 + 7028*x^14 + 18636*x^13 - 185680*x^12 - 792908*x^11 + 2039902*x^10 + 16130092*x^9 + 7737010*x^8 - 141852516*x^7 - 361190300*x^6 + 92662532*x^5 + 1932933080*x^4 + 3989871708*x^3 + 3965722697*x^2 + 1987694596*x + 394525193)
 
Copy content gp:K = bnfinit(y^18 - 131*y^16 - 172*y^15 + 7028*y^14 + 18636*y^13 - 185680*y^12 - 792908*y^11 + 2039902*y^10 + 16130092*y^9 + 7737010*y^8 - 141852516*y^7 - 361190300*y^6 + 92662532*y^5 + 1932933080*y^4 + 3989871708*y^3 + 3965722697*y^2 + 1987694596*y + 394525193, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 131*x^16 - 172*x^15 + 7028*x^14 + 18636*x^13 - 185680*x^12 - 792908*x^11 + 2039902*x^10 + 16130092*x^9 + 7737010*x^8 - 141852516*x^7 - 361190300*x^6 + 92662532*x^5 + 1932933080*x^4 + 3989871708*x^3 + 3965722697*x^2 + 1987694596*x + 394525193);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 131*x^16 - 172*x^15 + 7028*x^14 + 18636*x^13 - 185680*x^12 - 792908*x^11 + 2039902*x^10 + 16130092*x^9 + 7737010*x^8 - 141852516*x^7 - 361190300*x^6 + 92662532*x^5 + 1932933080*x^4 + 3989871708*x^3 + 3965722697*x^2 + 1987694596*x + 394525193)
 

\( x^{18} - 131 x^{16} - 172 x^{15} + 7028 x^{14} + 18636 x^{13} - 185680 x^{12} - 792908 x^{11} + \cdots + 394525193 \) 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:  $(12, 3)$
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:   \(-431144884247498060798327803279650586624\) \(\medspace = -\,2^{24}\cdot 7^{12}\cdot 43^{2}\cdot 4129^{2}\cdot 242689^{2}\) 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:  \(140.08\)
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:   \(2\), \(7\), \(43\), \(4129\), \(242689\) 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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a+\frac{3}{8}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{2}+\frac{1}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{16}a^{12}+\frac{1}{16}a^{8}+\frac{3}{16}a^{4}-\frac{5}{16}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{9}-\frac{1}{8}a^{8}-\frac{1}{16}a^{5}-\frac{1}{4}a^{4}+\frac{1}{16}a+\frac{3}{8}$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{10}-\frac{1}{8}a^{8}-\frac{1}{16}a^{6}-\frac{1}{4}a^{4}+\frac{1}{16}a^{2}+\frac{3}{8}$, $\frac{1}{32}a^{15}-\frac{1}{32}a^{14}-\frac{1}{32}a^{13}-\frac{1}{32}a^{12}+\frac{1}{32}a^{11}-\frac{1}{32}a^{10}-\frac{1}{32}a^{9}-\frac{1}{32}a^{8}+\frac{3}{32}a^{7}-\frac{3}{32}a^{6}-\frac{3}{32}a^{5}-\frac{3}{32}a^{4}-\frac{5}{32}a^{3}+\frac{5}{32}a^{2}+\frac{5}{32}a+\frac{5}{32}$, $\frac{1}{32}a^{16}+\frac{1}{16}a^{8}-\frac{1}{4}a^{4}+\frac{5}{32}$, $\frac{1}{32\cdots 04}a^{17}-\frac{21\cdots 49}{16\cdots 52}a^{16}-\frac{85\cdots 40}{10\cdots 97}a^{15}+\frac{53\cdots 79}{40\cdots 88}a^{14}-\frac{31\cdots 95}{16\cdots 52}a^{13}-\frac{56\cdots 99}{37\cdots 64}a^{12}-\frac{36\cdots 21}{81\cdots 76}a^{11}-\frac{34\cdots 70}{10\cdots 97}a^{10}-\frac{10\cdots 57}{31\cdots 76}a^{9}+\frac{84\cdots 45}{16\cdots 52}a^{8}+\frac{10\cdots 90}{10\cdots 97}a^{7}-\frac{66\cdots 69}{20\cdots 94}a^{6}+\frac{74\cdots 05}{37\cdots 64}a^{5}-\frac{39\cdots 11}{16\cdots 52}a^{4}+\frac{19\cdots 85}{81\cdots 76}a^{3}+\frac{14\cdots 53}{40\cdots 88}a^{2}-\frac{28\cdots 45}{32\cdots 04}a+\frac{11\cdots 77}{40\cdots 88}$ 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:  $C_{3}$, which has order $3$ (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:  $C_{3}$, which has order $3$ (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:   $\frac{13\cdots 39}{81\cdots 76}a^{17}-\frac{58\cdots 37}{32\cdots 04}a^{16}-\frac{17\cdots 23}{81\cdots 76}a^{15}-\frac{20\cdots 43}{81\cdots 76}a^{14}+\frac{19\cdots 81}{16\cdots 52}a^{13}+\frac{29\cdots 95}{18\cdots 32}a^{12}-\frac{27\cdots 97}{81\cdots 76}a^{11}-\frac{71\cdots 21}{81\cdots 76}a^{10}+\frac{58\cdots 25}{12\cdots 04}a^{9}+\frac{33\cdots 53}{16\cdots 52}a^{8}-\frac{14\cdots 29}{81\cdots 76}a^{7}-\frac{17\cdots 57}{81\cdots 76}a^{6}-\frac{10\cdots 31}{37\cdots 64}a^{5}+\frac{48\cdots 73}{81\cdots 76}a^{4}+\frac{19\cdots 97}{81\cdots 76}a^{3}+\frac{25\cdots 45}{81\cdots 76}a^{2}+\frac{31\cdots 93}{16\cdots 52}a+\frac{14\cdots 31}{32\cdots 04}$, $\frac{45\cdots 87}{20\cdots 94}a^{17}-\frac{13\cdots 61}{40\cdots 88}a^{16}-\frac{23\cdots 89}{81\cdots 76}a^{15}+\frac{68\cdots 41}{16\cdots 52}a^{14}+\frac{25\cdots 67}{16\cdots 52}a^{13}+\frac{70\cdots 05}{37\cdots 64}a^{12}-\frac{17\cdots 97}{40\cdots 88}a^{11}-\frac{18\cdots 85}{16\cdots 52}a^{10}+\frac{76\cdots 49}{12\cdots 04}a^{9}+\frac{43\cdots 65}{16\cdots 52}a^{8}-\frac{17\cdots 91}{81\cdots 76}a^{7}-\frac{46\cdots 05}{16\cdots 52}a^{6}-\frac{14\cdots 13}{37\cdots 64}a^{5}+\frac{12\cdots 41}{16\cdots 52}a^{4}+\frac{12\cdots 97}{40\cdots 88}a^{3}+\frac{69\cdots 69}{16\cdots 52}a^{2}+\frac{42\cdots 83}{16\cdots 52}a+\frac{98\cdots 79}{16\cdots 52}$, $\frac{89\cdots 21}{40\cdots 88}a^{17}-\frac{90\cdots 13}{32\cdots 04}a^{16}-\frac{46\cdots 17}{16\cdots 52}a^{15}-\frac{17\cdots 59}{16\cdots 52}a^{14}+\frac{63\cdots 89}{40\cdots 88}a^{13}+\frac{78\cdots 51}{37\cdots 64}a^{12}-\frac{72\cdots 81}{16\cdots 52}a^{11}-\frac{19\cdots 55}{16\cdots 52}a^{10}+\frac{48\cdots 96}{78\cdots 69}a^{9}+\frac{28\cdots 16}{10\cdots 97}a^{8}-\frac{34\cdots 11}{16\cdots 52}a^{7}-\frac{47\cdots 45}{16\cdots 52}a^{6}-\frac{19\cdots 01}{47\cdots 58}a^{5}+\frac{12\cdots 91}{16\cdots 52}a^{4}+\frac{53\cdots 69}{16\cdots 52}a^{3}+\frac{71\cdots 19}{16\cdots 52}a^{2}+\frac{27\cdots 30}{10\cdots 97}a+\frac{20\cdots 89}{32\cdots 04}$, $\frac{65\cdots 95}{40\cdots 88}a^{17}-\frac{23\cdots 61}{32\cdots 04}a^{16}-\frac{14\cdots 63}{81\cdots 76}a^{15}+\frac{89\cdots 39}{16\cdots 52}a^{14}+\frac{15\cdots 67}{16\cdots 52}a^{13}-\frac{24\cdots 79}{18\cdots 32}a^{12}-\frac{21\cdots 19}{81\cdots 76}a^{11}-\frac{92\cdots 63}{16\cdots 52}a^{10}+\frac{52\cdots 75}{12\cdots 04}a^{9}+\frac{11\cdots 95}{16\cdots 52}a^{8}-\frac{24\cdots 29}{81\cdots 76}a^{7}-\frac{17\cdots 59}{16\cdots 52}a^{6}-\frac{46\cdots 33}{37\cdots 64}a^{5}+\frac{38\cdots 41}{81\cdots 76}a^{4}+\frac{77\cdots 59}{81\cdots 76}a^{3}+\frac{13\cdots 83}{16\cdots 52}a^{2}+\frac{51\cdots 33}{16\cdots 52}a+\frac{11\cdots 75}{32\cdots 04}$, $\frac{17\cdots 01}{32\cdots 04}a^{17}-\frac{65\cdots 47}{32\cdots 04}a^{16}-\frac{10\cdots 73}{16\cdots 52}a^{15}+\frac{11\cdots 67}{81\cdots 76}a^{14}+\frac{54\cdots 07}{16\cdots 52}a^{13}-\frac{48\cdots 36}{23\cdots 79}a^{12}-\frac{15\cdots 39}{16\cdots 52}a^{11}-\frac{17\cdots 19}{20\cdots 94}a^{10}+\frac{90\cdots 15}{62\cdots 52}a^{9}+\frac{57\cdots 63}{16\cdots 52}a^{8}-\frac{14\cdots 63}{16\cdots 52}a^{7}-\frac{37\cdots 61}{81\cdots 76}a^{6}-\frac{11\cdots 21}{37\cdots 64}a^{5}+\frac{16\cdots 50}{10\cdots 97}a^{4}+\frac{74\cdots 39}{16\cdots 52}a^{3}+\frac{21\cdots 11}{40\cdots 88}a^{2}+\frac{92\cdots 15}{32\cdots 04}a+\frac{19\cdots 97}{32\cdots 04}$, $\frac{27\cdots 61}{32\cdots 04}a^{17}-\frac{14\cdots 91}{32\cdots 04}a^{16}-\frac{30\cdots 81}{32\cdots 04}a^{15}+\frac{11\cdots 07}{32\cdots 04}a^{14}+\frac{15\cdots 97}{32\cdots 04}a^{13}-\frac{70\cdots 15}{75\cdots 28}a^{12}-\frac{43\cdots 21}{32\cdots 04}a^{11}+\frac{12\cdots 55}{32\cdots 04}a^{10}+\frac{54\cdots 59}{24\cdots 08}a^{9}+\frac{86\cdots 09}{32\cdots 04}a^{8}-\frac{54\cdots 31}{32\cdots 04}a^{7}-\frac{15\cdots 59}{32\cdots 04}a^{6}+\frac{10\cdots 33}{75\cdots 28}a^{5}+\frac{75\cdots 45}{32\cdots 04}a^{4}+\frac{12\cdots 25}{32\cdots 04}a^{3}+\frac{88\cdots 81}{32\cdots 04}a^{2}+\frac{42\cdots 31}{81\cdots 76}a-\frac{20\cdots 01}{16\cdots 52}$, $\frac{47\cdots 43}{16\cdots 52}a^{17}-\frac{55\cdots 03}{81\cdots 76}a^{16}-\frac{59\cdots 13}{16\cdots 52}a^{15}+\frac{56\cdots 99}{16\cdots 52}a^{14}+\frac{39\cdots 85}{20\cdots 94}a^{13}+\frac{88\cdots 11}{94\cdots 16}a^{12}-\frac{90\cdots 67}{16\cdots 52}a^{11}-\frac{16\cdots 55}{16\cdots 52}a^{10}+\frac{12\cdots 55}{15\cdots 38}a^{9}+\frac{22\cdots 33}{81\cdots 76}a^{8}-\frac{63\cdots 59}{16\cdots 52}a^{7}-\frac{51\cdots 23}{16\cdots 52}a^{6}-\frac{81\cdots 29}{23\cdots 79}a^{5}+\frac{40\cdots 47}{40\cdots 88}a^{4}+\frac{54\cdots 51}{16\cdots 52}a^{3}+\frac{68\cdots 63}{16\cdots 52}a^{2}+\frac{40\cdots 49}{16\cdots 52}a+\frac{22\cdots 31}{40\cdots 88}$, $\frac{43\cdots 93}{32\cdots 04}a^{17}-\frac{39\cdots 69}{16\cdots 52}a^{16}-\frac{27\cdots 83}{16\cdots 52}a^{15}+\frac{29\cdots 77}{40\cdots 88}a^{14}+\frac{15\cdots 69}{16\cdots 52}a^{13}+\frac{32\cdots 51}{37\cdots 64}a^{12}-\frac{42\cdots 65}{16\cdots 52}a^{11}-\frac{49\cdots 75}{81\cdots 76}a^{10}+\frac{59\cdots 81}{15\cdots 38}a^{9}+\frac{24\cdots 61}{16\cdots 52}a^{8}-\frac{25\cdots 13}{16\cdots 52}a^{7}-\frac{66\cdots 77}{40\cdots 88}a^{6}-\frac{77\cdots 43}{37\cdots 64}a^{5}+\frac{76\cdots 79}{16\cdots 52}a^{4}+\frac{28\cdots 45}{16\cdots 52}a^{3}+\frac{18\cdots 43}{81\cdots 76}a^{2}+\frac{45\cdots 27}{32\cdots 04}a+\frac{12\cdots 73}{40\cdots 88}$, $\frac{10\cdots 65}{32\cdots 04}a^{17}-\frac{18\cdots 47}{16\cdots 52}a^{16}-\frac{32\cdots 77}{81\cdots 76}a^{15}+\frac{17\cdots 69}{20\cdots 94}a^{14}+\frac{16\cdots 61}{81\cdots 76}a^{13}-\frac{25\cdots 35}{18\cdots 32}a^{12}-\frac{24\cdots 89}{40\cdots 88}a^{11}-\frac{37\cdots 65}{81\cdots 76}a^{10}+\frac{11\cdots 09}{12\cdots 04}a^{9}+\frac{16\cdots 13}{81\cdots 76}a^{8}-\frac{54\cdots 73}{81\cdots 76}a^{7}-\frac{11\cdots 13}{40\cdots 88}a^{6}-\frac{12\cdots 15}{18\cdots 32}a^{5}+\frac{93\cdots 83}{81\cdots 76}a^{4}+\frac{24\cdots 41}{10\cdots 97}a^{3}+\frac{16\cdots 59}{81\cdots 76}a^{2}+\frac{25\cdots 49}{32\cdots 04}a+\frac{17\cdots 37}{16\cdots 52}$, $\frac{44\cdots 15}{81\cdots 76}a^{17}-\frac{35\cdots 99}{16\cdots 52}a^{16}-\frac{20\cdots 95}{32\cdots 04}a^{15}+\frac{51\cdots 63}{32\cdots 04}a^{14}+\frac{10\cdots 23}{32\cdots 04}a^{13}-\frac{21\cdots 89}{75\cdots 28}a^{12}-\frac{29\cdots 03}{32\cdots 04}a^{11}-\frac{19\cdots 77}{32\cdots 04}a^{10}+\frac{34\cdots 67}{24\cdots 08}a^{9}+\frac{98\cdots 61}{32\cdots 04}a^{8}-\frac{29\cdots 41}{32\cdots 04}a^{7}-\frac{13\cdots 91}{32\cdots 04}a^{6}-\frac{16\cdots 29}{75\cdots 28}a^{5}+\frac{50\cdots 75}{32\cdots 04}a^{4}+\frac{12\cdots 35}{32\cdots 04}a^{3}+\frac{14\cdots 53}{32\cdots 04}a^{2}+\frac{73\cdots 17}{32\cdots 04}a+\frac{14\cdots 41}{32\cdots 04}$, $\frac{78\cdots 83}{32\cdots 04}a^{17}-\frac{15\cdots 53}{20\cdots 94}a^{16}-\frac{12\cdots 93}{40\cdots 88}a^{15}+\frac{39\cdots 59}{81\cdots 76}a^{14}+\frac{26\cdots 25}{16\cdots 52}a^{13}-\frac{58\cdots 49}{37\cdots 64}a^{12}-\frac{19\cdots 07}{40\cdots 88}a^{11}-\frac{14\cdots 09}{20\cdots 94}a^{10}+\frac{45\cdots 79}{62\cdots 52}a^{9}+\frac{37\cdots 09}{16\cdots 52}a^{8}-\frac{79\cdots 93}{20\cdots 94}a^{7}-\frac{22\cdots 77}{81\cdots 76}a^{6}-\frac{10\cdots 27}{37\cdots 64}a^{5}+\frac{15\cdots 63}{16\cdots 52}a^{4}+\frac{31\cdots 21}{10\cdots 97}a^{3}+\frac{15\cdots 59}{40\cdots 88}a^{2}+\frac{76\cdots 13}{32\cdots 04}a+\frac{85\cdots 07}{16\cdots 52}$, $\frac{90\cdots 01}{81\cdots 76}a^{17}-\frac{74\cdots 85}{32\cdots 04}a^{16}-\frac{22\cdots 63}{16\cdots 52}a^{15}+\frac{16\cdots 87}{16\cdots 52}a^{14}+\frac{62\cdots 09}{81\cdots 76}a^{13}+\frac{45\cdots 03}{94\cdots 16}a^{12}-\frac{35\cdots 13}{16\cdots 52}a^{11}-\frac{69\cdots 69}{16\cdots 52}a^{10}+\frac{19\cdots 53}{62\cdots 52}a^{9}+\frac{18\cdots 57}{16\cdots 52}a^{8}-\frac{25\cdots 93}{16\cdots 52}a^{7}-\frac{20\cdots 75}{16\cdots 52}a^{6}-\frac{25\cdots 35}{18\cdots 32}a^{5}+\frac{16\cdots 05}{40\cdots 88}a^{4}+\frac{21\cdots 29}{16\cdots 52}a^{3}+\frac{26\cdots 73}{16\cdots 52}a^{2}+\frac{38\cdots 79}{40\cdots 88}a+\frac{67\cdots 63}{32\cdots 04}$, $\frac{12\cdots 91}{16\cdots 52}a^{17}-\frac{43\cdots 17}{32\cdots 04}a^{16}-\frac{82\cdots 61}{81\cdots 76}a^{15}+\frac{36\cdots 95}{81\cdots 76}a^{14}+\frac{44\cdots 51}{81\cdots 76}a^{13}+\frac{83\cdots 15}{18\cdots 32}a^{12}-\frac{12\cdots 17}{81\cdots 76}a^{11}-\frac{32\cdots 64}{10\cdots 97}a^{10}+\frac{70\cdots 97}{31\cdots 76}a^{9}+\frac{13\cdots 39}{16\cdots 52}a^{8}-\frac{88\cdots 85}{81\cdots 76}a^{7}-\frac{71\cdots 01}{81\cdots 76}a^{6}-\frac{17\cdots 07}{18\cdots 32}a^{5}+\frac{22\cdots 35}{81\cdots 76}a^{4}+\frac{72\cdots 43}{81\cdots 76}a^{3}+\frac{42\cdots 19}{40\cdots 88}a^{2}+\frac{92\cdots 09}{16\cdots 52}a+\frac{35\cdots 75}{32\cdots 04}$, $\frac{74\cdots 97}{81\cdots 76}a^{17}-\frac{23\cdots 51}{16\cdots 52}a^{16}-\frac{19\cdots 71}{16\cdots 52}a^{15}+\frac{20\cdots 69}{81\cdots 76}a^{14}+\frac{25\cdots 27}{40\cdots 88}a^{13}+\frac{26\cdots 69}{37\cdots 64}a^{12}-\frac{29\cdots 47}{16\cdots 52}a^{11}-\frac{35\cdots 87}{81\cdots 76}a^{10}+\frac{19\cdots 37}{78\cdots 69}a^{9}+\frac{17\cdots 49}{16\cdots 52}a^{8}-\frac{15\cdots 89}{16\cdots 52}a^{7}-\frac{92\cdots 93}{81\cdots 76}a^{6}-\frac{14\cdots 57}{94\cdots 16}a^{5}+\frac{51\cdots 09}{16\cdots 52}a^{4}+\frac{20\cdots 75}{16\cdots 52}a^{3}+\frac{13\cdots 87}{81\cdots 76}a^{2}+\frac{82\cdots 55}{81\cdots 76}a+\frac{18\cdots 37}{81\cdots 76}$ 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:  \( 2931643563430 \) (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)
 
Unit signature rank:  \( 12 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{3}\cdot 2931643563430 \cdot 3}{2\cdot\sqrt{431144884247498060798327803279650586624}}\cr\approx \mathstrut & 0.215174280771490 \end{aligned}\] (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 - 131*x^16 - 172*x^15 + 7028*x^14 + 18636*x^13 - 185680*x^12 - 792908*x^11 + 2039902*x^10 + 16130092*x^9 + 7737010*x^8 - 141852516*x^7 - 361190300*x^6 + 92662532*x^5 + 1932933080*x^4 + 3989871708*x^3 + 3965722697*x^2 + 1987694596*x + 394525193) 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 - 131*x^16 - 172*x^15 + 7028*x^14 + 18636*x^13 - 185680*x^12 - 792908*x^11 + 2039902*x^10 + 16130092*x^9 + 7737010*x^8 - 141852516*x^7 - 361190300*x^6 + 92662532*x^5 + 1932933080*x^4 + 3989871708*x^3 + 3965722697*x^2 + 1987694596*x + 394525193, 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 - 131*x^16 - 172*x^15 + 7028*x^14 + 18636*x^13 - 185680*x^12 - 792908*x^11 + 2039902*x^10 + 16130092*x^9 + 7737010*x^8 - 141852516*x^7 - 361190300*x^6 + 92662532*x^5 + 1932933080*x^4 + 3989871708*x^3 + 3965722697*x^2 + 1987694596*x + 394525193); 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 - 131*x^16 - 172*x^15 + 7028*x^14 + 18636*x^13 - 185680*x^12 - 792908*x^11 + 2039902*x^10 + 16130092*x^9 + 7737010*x^8 - 141852516*x^7 - 361190300*x^6 + 92662532*x^5 + 1932933080*x^4 + 3989871708*x^3 + 3965722697*x^2 + 1987694596*x + 394525193); 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

$S_3^3.S_3\wr C_3$ (as 18T807):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 139968
The 267 conjugacy class representatives for $S_3^3.S_3\wr C_3$
Character table for $S_3^3.S_3\wr C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.153664.1

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 27 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 R ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{3}$ $18$ $18$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $18$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ R $18$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ $18$

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 2.3.6.24a2.9$x^{18} + 6 x^{16} + 8 x^{15} + 15 x^{14} + 40 x^{13} + 45 x^{12} + 80 x^{11} + 117 x^{10} + 120 x^{9} + 162 x^{8} + 166 x^{7} + 142 x^{6} + 140 x^{5} + 103 x^{4} + 62 x^{3} + 41 x^{2} + 18 x + 5$$6$$3$$24$18T60$$[2, 2, 2]_{3}^{6}$$
\(7\) Copy content Toggle raw display 7.6.3.12a1.3$x^{18} + 3 x^{16} + 15 x^{15} + 15 x^{14} + 48 x^{13} + 109 x^{12} + 171 x^{11} + 333 x^{10} + 497 x^{9} + 717 x^{8} + 1032 x^{7} + 1216 x^{6} + 1296 x^{5} + 1143 x^{4} + 783 x^{3} + 432 x^{2} + 162 x + 34$$3$$6$$12$$C_6 \times C_3$$$[\ ]_{3}^{6}$$
\(43\) Copy content Toggle raw display $\Q_{43}$$x + 40$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{43}$$x + 40$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{43}$$x + 40$$1$$1$$0$Trivial$$[\ ]$$
43.2.1.0a1.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
43.3.1.0a1.1$x^{3} + x + 40$$1$$3$$0$$C_3$$$[\ ]^{3}$$
43.3.1.0a1.1$x^{3} + x + 40$$1$$3$$0$$C_3$$$[\ ]^{3}$$
43.3.1.0a1.1$x^{3} + x + 40$$1$$3$$0$$C_3$$$[\ ]^{3}$$
43.2.2.2a1.2$x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(4129\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$3$$1$$2$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(242689\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$3$$1$$2$
Deg $6$$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)