Properties

Label 18.10.105...408.1
Degree $18$
Signature $(10, 4)$
Discriminant $1.057\times 10^{29}$
Root discriminant \(40.97\)
Ramified primes $2,37,101$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^6:S_3^2$ (as 18T376)

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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 - 6*x^17 + x^16 + 52*x^15 - 77*x^14 - 110*x^13 + 167*x^12 + 628*x^11 - 124*x^10 - 3760*x^9 + 916*x^8 + 8144*x^7 + 508*x^6 - 12000*x^5 - 4344*x^4 + 7704*x^3 + 1372*x^2 - 1528*x + 148)
 
Copy content gp:K = bnfinit(y^18 - 6*y^17 + y^16 + 52*y^15 - 77*y^14 - 110*y^13 + 167*y^12 + 628*y^11 - 124*y^10 - 3760*y^9 + 916*y^8 + 8144*y^7 + 508*y^6 - 12000*y^5 - 4344*y^4 + 7704*y^3 + 1372*y^2 - 1528*y + 148, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + x^16 + 52*x^15 - 77*x^14 - 110*x^13 + 167*x^12 + 628*x^11 - 124*x^10 - 3760*x^9 + 916*x^8 + 8144*x^7 + 508*x^6 - 12000*x^5 - 4344*x^4 + 7704*x^3 + 1372*x^2 - 1528*x + 148);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + x^16 + 52*x^15 - 77*x^14 - 110*x^13 + 167*x^12 + 628*x^11 - 124*x^10 - 3760*x^9 + 916*x^8 + 8144*x^7 + 508*x^6 - 12000*x^5 - 4344*x^4 + 7704*x^3 + 1372*x^2 - 1528*x + 148)
 

\( x^{18} - 6 x^{17} + x^{16} + 52 x^{15} - 77 x^{14} - 110 x^{13} + 167 x^{12} + 628 x^{11} - 124 x^{10} + \cdots + 148 \) 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:  $(10, 4)$
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:   \(105667206065482931222015377408\) \(\medspace = 2^{20}\cdot 37^{7}\cdot 101^{6}\) 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:  \(40.97\)
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:  $2^{19/12}37^{1/2}101^{1/2}\approx 183.1860403967972$
Ramified primes:   \(2\), \(37\), \(101\) 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:  \(\Q(\sqrt{37}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{81\cdots 32}a^{17}+\frac{14\cdots 71}{20\cdots 33}a^{16}-\frac{58\cdots 97}{40\cdots 66}a^{15}-\frac{44\cdots 47}{81\cdots 32}a^{14}+\frac{79\cdots 65}{81\cdots 32}a^{13}+\frac{50\cdots 91}{81\cdots 32}a^{12}+\frac{88\cdots 53}{81\cdots 32}a^{11}+\frac{15\cdots 97}{81\cdots 32}a^{10}+\frac{23\cdots 29}{40\cdots 66}a^{9}+\frac{26\cdots 19}{81\cdots 32}a^{8}-\frac{11\cdots 93}{81\cdots 32}a^{7}+\frac{73\cdots 56}{20\cdots 33}a^{6}-\frac{34\cdots 51}{40\cdots 66}a^{5}-\frac{59\cdots 43}{40\cdots 66}a^{4}-\frac{76\cdots 97}{40\cdots 66}a^{3}-\frac{55\cdots 15}{40\cdots 66}a^{2}-\frac{55\cdots 65}{40\cdots 66}a+\frac{10\cdots 94}{20\cdots 33}$ 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:  $C_{2}$, which has order $2$ (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:  $13$
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:   $\frac{62\cdots 52}{20\cdots 33}a^{17}-\frac{36\cdots 01}{20\cdots 33}a^{16}-\frac{62\cdots 69}{20\cdots 33}a^{15}+\frac{32\cdots 42}{20\cdots 33}a^{14}-\frac{41\cdots 15}{20\cdots 33}a^{13}-\frac{14\cdots 33}{40\cdots 66}a^{12}+\frac{87\cdots 42}{20\cdots 33}a^{11}+\frac{40\cdots 37}{20\cdots 33}a^{10}+\frac{79\cdots 74}{20\cdots 33}a^{9}-\frac{46\cdots 73}{40\cdots 66}a^{8}+\frac{12\cdots 38}{20\cdots 33}a^{7}+\frac{49\cdots 76}{20\cdots 33}a^{6}+\frac{12\cdots 52}{20\cdots 33}a^{5}-\frac{68\cdots 71}{20\cdots 33}a^{4}-\frac{37\cdots 64}{20\cdots 33}a^{3}+\frac{35\cdots 84}{20\cdots 33}a^{2}+\frac{85\cdots 20}{20\cdots 33}a-\frac{44\cdots 59}{20\cdots 33}$, $\frac{13\cdots 89}{81\cdots 32}a^{17}-\frac{37\cdots 73}{40\cdots 66}a^{16}-\frac{92\cdots 38}{20\cdots 33}a^{15}+\frac{16\cdots 95}{20\cdots 33}a^{14}-\frac{42\cdots 63}{40\cdots 66}a^{13}-\frac{15\cdots 03}{81\cdots 32}a^{12}+\frac{42\cdots 41}{20\cdots 33}a^{11}+\frac{84\cdots 71}{81\cdots 32}a^{10}+\frac{63\cdots 19}{81\cdots 32}a^{9}-\frac{47\cdots 15}{81\cdots 32}a^{8}+\frac{13\cdots 06}{20\cdots 33}a^{7}+\frac{10\cdots 09}{81\cdots 32}a^{6}+\frac{14\cdots 43}{40\cdots 66}a^{5}-\frac{68\cdots 25}{40\cdots 66}a^{4}-\frac{21\cdots 94}{20\cdots 33}a^{3}+\frac{16\cdots 79}{20\cdots 33}a^{2}+\frac{60\cdots 24}{20\cdots 33}a-\frac{35\cdots 81}{40\cdots 66}$, $\frac{15\cdots 19}{81\cdots 32}a^{17}-\frac{85\cdots 59}{81\cdots 32}a^{16}-\frac{47\cdots 93}{40\cdots 66}a^{15}+\frac{38\cdots 59}{40\cdots 66}a^{14}-\frac{46\cdots 89}{40\cdots 66}a^{13}-\frac{18\cdots 61}{81\cdots 32}a^{12}+\frac{91\cdots 47}{40\cdots 66}a^{11}+\frac{98\cdots 29}{81\cdots 32}a^{10}+\frac{14\cdots 11}{81\cdots 32}a^{9}-\frac{27\cdots 67}{40\cdots 66}a^{8}-\frac{81\cdots 34}{20\cdots 33}a^{7}+\frac{11\cdots 15}{81\cdots 32}a^{6}+\frac{11\cdots 67}{20\cdots 33}a^{5}-\frac{39\cdots 64}{20\cdots 33}a^{4}-\frac{29\cdots 54}{20\cdots 33}a^{3}+\frac{34\cdots 05}{40\cdots 66}a^{2}+\frac{11\cdots 83}{20\cdots 33}a-\frac{62\cdots 83}{40\cdots 66}$, $\frac{44\cdots 83}{81\cdots 32}a^{17}-\frac{33\cdots 43}{81\cdots 32}a^{16}+\frac{22\cdots 19}{40\cdots 66}a^{15}+\frac{56\cdots 23}{20\cdots 33}a^{14}-\frac{35\cdots 43}{40\cdots 66}a^{13}+\frac{76\cdots 81}{81\cdots 32}a^{12}+\frac{72\cdots 81}{40\cdots 66}a^{11}+\frac{14\cdots 87}{81\cdots 32}a^{10}-\frac{44\cdots 49}{81\cdots 32}a^{9}-\frac{39\cdots 62}{20\cdots 33}a^{8}+\frac{73\cdots 42}{20\cdots 33}a^{7}+\frac{27\cdots 63}{81\cdots 32}a^{6}-\frac{12\cdots 04}{20\cdots 33}a^{5}-\frac{10\cdots 97}{20\cdots 33}a^{4}+\frac{12\cdots 90}{20\cdots 33}a^{3}+\frac{24\cdots 07}{40\cdots 66}a^{2}-\frac{11\cdots 57}{20\cdots 33}a+\frac{27\cdots 43}{40\cdots 66}$, $\frac{13\cdots 15}{81\cdots 32}a^{17}-\frac{78\cdots 91}{81\cdots 32}a^{16}+\frac{18\cdots 45}{40\cdots 66}a^{15}+\frac{34\cdots 71}{40\cdots 66}a^{14}-\frac{45\cdots 57}{40\cdots 66}a^{13}-\frac{15\cdots 93}{81\cdots 32}a^{12}+\frac{88\cdots 33}{40\cdots 66}a^{11}+\frac{85\cdots 73}{81\cdots 32}a^{10}+\frac{59\cdots 63}{81\cdots 32}a^{9}-\frac{24\cdots 33}{40\cdots 66}a^{8}+\frac{61\cdots 74}{20\cdots 33}a^{7}+\frac{10\cdots 55}{81\cdots 32}a^{6}+\frac{70\cdots 62}{20\cdots 33}a^{5}-\frac{33\cdots 77}{20\cdots 33}a^{4}-\frac{22\cdots 46}{20\cdots 33}a^{3}+\frac{31\cdots 63}{40\cdots 66}a^{2}+\frac{74\cdots 01}{20\cdots 33}a-\frac{46\cdots 77}{40\cdots 66}$, $\frac{34\cdots 95}{81\cdots 32}a^{17}-\frac{45\cdots 92}{20\cdots 33}a^{16}-\frac{29\cdots 45}{20\cdots 33}a^{15}+\frac{88\cdots 27}{40\cdots 66}a^{14}-\frac{60\cdots 21}{40\cdots 66}a^{13}-\frac{54\cdots 53}{81\cdots 32}a^{12}+\frac{38\cdots 20}{20\cdots 33}a^{11}+\frac{26\cdots 15}{81\cdots 32}a^{10}+\frac{16\cdots 01}{81\cdots 32}a^{9}-\frac{13\cdots 51}{81\cdots 32}a^{8}-\frac{18\cdots 50}{20\cdots 33}a^{7}+\frac{29\cdots 47}{81\cdots 32}a^{6}+\frac{12\cdots 91}{40\cdots 66}a^{5}-\frac{18\cdots 69}{40\cdots 66}a^{4}-\frac{12\cdots 06}{20\cdots 33}a^{3}+\frac{20\cdots 26}{20\cdots 33}a^{2}+\frac{64\cdots 48}{20\cdots 33}a+\frac{24\cdots 41}{40\cdots 66}$, $\frac{51\cdots 59}{81\cdots 32}a^{17}-\frac{29\cdots 27}{81\cdots 32}a^{16}-\frac{15\cdots 29}{20\cdots 33}a^{15}+\frac{26\cdots 01}{81\cdots 32}a^{14}-\frac{85\cdots 15}{20\cdots 33}a^{13}-\frac{15\cdots 20}{20\cdots 33}a^{12}+\frac{17\cdots 90}{20\cdots 33}a^{11}+\frac{82\cdots 50}{20\cdots 33}a^{10}+\frac{11\cdots 07}{81\cdots 32}a^{9}-\frac{18\cdots 87}{81\cdots 32}a^{8}+\frac{21\cdots 73}{20\cdots 33}a^{7}+\frac{39\cdots 87}{81\cdots 32}a^{6}+\frac{51\cdots 31}{40\cdots 66}a^{5}-\frac{13\cdots 78}{20\cdots 33}a^{4}-\frac{79\cdots 58}{20\cdots 33}a^{3}+\frac{70\cdots 87}{20\cdots 33}a^{2}+\frac{20\cdots 95}{20\cdots 33}a-\frac{21\cdots 37}{40\cdots 66}$, $\frac{29\cdots 89}{81\cdots 32}a^{17}-\frac{16\cdots 97}{81\cdots 32}a^{16}-\frac{67\cdots 27}{20\cdots 33}a^{15}+\frac{15\cdots 01}{81\cdots 32}a^{14}-\frac{44\cdots 05}{20\cdots 33}a^{13}-\frac{94\cdots 39}{20\cdots 33}a^{12}+\frac{90\cdots 98}{20\cdots 33}a^{11}+\frac{97\cdots 59}{40\cdots 66}a^{10}+\frac{30\cdots 97}{81\cdots 32}a^{9}-\frac{10\cdots 33}{81\cdots 32}a^{8}-\frac{22\cdots 11}{20\cdots 33}a^{7}+\frac{23\cdots 97}{81\cdots 32}a^{6}+\frac{42\cdots 93}{40\cdots 66}a^{5}-\frac{78\cdots 41}{20\cdots 33}a^{4}-\frac{53\cdots 54}{20\cdots 33}a^{3}+\frac{37\cdots 01}{20\cdots 33}a^{2}+\frac{16\cdots 59}{20\cdots 33}a-\frac{13\cdots 91}{40\cdots 66}$, $\frac{33\cdots 17}{81\cdots 32}a^{17}-\frac{19\cdots 81}{81\cdots 32}a^{16}+\frac{32\cdots 49}{40\cdots 66}a^{15}+\frac{17\cdots 65}{81\cdots 32}a^{14}-\frac{23\cdots 71}{81\cdots 32}a^{13}-\frac{40\cdots 45}{81\cdots 32}a^{12}+\frac{49\cdots 85}{81\cdots 32}a^{11}+\frac{22\cdots 99}{81\cdots 32}a^{10}-\frac{48\cdots 19}{40\cdots 66}a^{9}-\frac{31\cdots 77}{20\cdots 33}a^{8}+\frac{12\cdots 97}{81\cdots 32}a^{7}+\frac{13\cdots 95}{40\cdots 66}a^{6}+\frac{15\cdots 44}{20\cdots 33}a^{5}-\frac{99\cdots 02}{20\cdots 33}a^{4}-\frac{10\cdots 41}{40\cdots 66}a^{3}+\frac{55\cdots 98}{20\cdots 33}a^{2}+\frac{34\cdots 21}{40\cdots 66}a-\frac{77\cdots 09}{20\cdots 33}$, $\frac{69\cdots 77}{81\cdots 32}a^{17}-\frac{19\cdots 17}{40\cdots 66}a^{16}-\frac{43\cdots 73}{40\cdots 66}a^{15}+\frac{36\cdots 85}{81\cdots 32}a^{14}-\frac{20\cdots 49}{40\cdots 66}a^{13}-\frac{49\cdots 15}{40\cdots 66}a^{12}+\frac{23\cdots 63}{20\cdots 33}a^{11}+\frac{11\cdots 12}{20\cdots 33}a^{10}+\frac{58\cdots 85}{81\cdots 32}a^{9}-\frac{66\cdots 85}{20\cdots 33}a^{8}-\frac{13\cdots 45}{40\cdots 66}a^{7}+\frac{60\cdots 05}{81\cdots 32}a^{6}+\frac{54\cdots 58}{20\cdots 33}a^{5}-\frac{42\cdots 93}{40\cdots 66}a^{4}-\frac{14\cdots 44}{20\cdots 33}a^{3}+\frac{24\cdots 55}{40\cdots 66}a^{2}+\frac{63\cdots 83}{20\cdots 33}a-\frac{58\cdots 65}{40\cdots 66}$, $\frac{23\cdots 09}{81\cdots 32}a^{17}-\frac{11\cdots 31}{81\cdots 32}a^{16}-\frac{10\cdots 65}{81\cdots 32}a^{15}+\frac{33\cdots 28}{20\cdots 33}a^{14}-\frac{46\cdots 67}{40\cdots 66}a^{13}-\frac{23\cdots 81}{40\cdots 66}a^{12}+\frac{96\cdots 69}{20\cdots 33}a^{11}+\frac{43\cdots 12}{20\cdots 33}a^{10}+\frac{66\cdots 53}{81\cdots 32}a^{9}-\frac{97\cdots 51}{81\cdots 32}a^{8}-\frac{39\cdots 23}{81\cdots 32}a^{7}+\frac{12\cdots 01}{40\cdots 66}a^{6}+\frac{48\cdots 87}{40\cdots 66}a^{5}-\frac{82\cdots 88}{20\cdots 33}a^{4}-\frac{68\cdots 47}{20\cdots 33}a^{3}+\frac{11\cdots 17}{40\cdots 66}a^{2}+\frac{51\cdots 87}{40\cdots 66}a-\frac{13\cdots 41}{20\cdots 33}$, $\frac{15\cdots 75}{81\cdots 32}a^{17}-\frac{80\cdots 99}{81\cdots 32}a^{16}-\frac{40\cdots 89}{81\cdots 32}a^{15}+\frac{37\cdots 17}{40\cdots 66}a^{14}-\frac{15\cdots 83}{20\cdots 33}a^{13}-\frac{50\cdots 71}{20\cdots 33}a^{12}+\frac{19\cdots 53}{20\cdots 33}a^{11}+\frac{51\cdots 05}{40\cdots 66}a^{10}+\frac{61\cdots 17}{81\cdots 32}a^{9}-\frac{52\cdots 73}{81\cdots 32}a^{8}-\frac{26\cdots 67}{81\cdots 32}a^{7}+\frac{50\cdots 51}{40\cdots 66}a^{6}+\frac{47\cdots 25}{40\cdots 66}a^{5}-\frac{27\cdots 74}{20\cdots 33}a^{4}-\frac{44\cdots 35}{20\cdots 33}a^{3}-\frac{46\cdots 75}{40\cdots 66}a^{2}+\frac{18\cdots 77}{40\cdots 66}a-\frac{74\cdots 65}{20\cdots 33}$, $\frac{12\cdots 34}{20\cdots 33}a^{17}-\frac{91\cdots 84}{20\cdots 33}a^{16}+\frac{84\cdots 58}{20\cdots 33}a^{15}+\frac{13\cdots 41}{40\cdots 66}a^{14}-\frac{32\cdots 39}{40\cdots 66}a^{13}-\frac{28\cdots 23}{81\cdots 32}a^{12}+\frac{68\cdots 35}{40\cdots 66}a^{11}+\frac{29\cdots 79}{81\cdots 32}a^{10}-\frac{18\cdots 25}{40\cdots 66}a^{9}-\frac{20\cdots 41}{81\cdots 32}a^{8}+\frac{53\cdots 75}{20\cdots 33}a^{7}+\frac{44\cdots 71}{81\cdots 32}a^{6}-\frac{58\cdots 65}{20\cdots 33}a^{5}-\frac{41\cdots 27}{40\cdots 66}a^{4}+\frac{46\cdots 31}{40\cdots 66}a^{3}+\frac{17\cdots 72}{20\cdots 33}a^{2}+\frac{35\cdots 61}{20\cdots 33}a-\frac{74\cdots 45}{40\cdots 66}$ 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:  \( 347353162.895 \) (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:  \( 9 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 347353162.895 \cdot 1}{2\cdot\sqrt{105667206065482931222015377408}}\cr\approx \mathstrut & 0.85268884197 \end{aligned}\] (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 + x^16 + 52*x^15 - 77*x^14 - 110*x^13 + 167*x^12 + 628*x^11 - 124*x^10 - 3760*x^9 + 916*x^8 + 8144*x^7 + 508*x^6 - 12000*x^5 - 4344*x^4 + 7704*x^3 + 1372*x^2 - 1528*x + 148) 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 + x^16 + 52*x^15 - 77*x^14 - 110*x^13 + 167*x^12 + 628*x^11 - 124*x^10 - 3760*x^9 + 916*x^8 + 8144*x^7 + 508*x^6 - 12000*x^5 - 4344*x^4 + 7704*x^3 + 1372*x^2 - 1528*x + 148, 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 + x^16 + 52*x^15 - 77*x^14 - 110*x^13 + 167*x^12 + 628*x^11 - 124*x^10 - 3760*x^9 + 916*x^8 + 8144*x^7 + 508*x^6 - 12000*x^5 - 4344*x^4 + 7704*x^3 + 1372*x^2 - 1528*x + 148); 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 + x^16 + 52*x^15 - 77*x^14 - 110*x^13 + 167*x^12 + 628*x^11 - 124*x^10 - 3760*x^9 + 916*x^8 + 8144*x^7 + 508*x^6 - 12000*x^5 - 4344*x^4 + 7704*x^3 + 1372*x^2 - 1528*x + 148); 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

$C_2^6:S_3^2$ (as 18T376):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 2304
The 30 conjugacy class representatives for $C_2^6:S_3^2$
Character table for $C_2^6:S_3^2$

Intermediate fields

3.3.148.1, 3.3.404.1, 9.9.3340021539392.1

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 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.4.173242481699658727424.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ R ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(2\) Copy content Toggle raw display 2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.2.6.16a1.8$x^{12} + 6 x^{11} + 23 x^{10} + 60 x^{9} + 120 x^{8} + 186 x^{7} + 233 x^{6} + 234 x^{5} + 192 x^{4} + 124 x^{3} + 63 x^{2} + 26 x + 7$$6$$2$$16$12T30$$[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
37.1.2.1a1.2$x^{2} + 74$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
37.1.2.1a1.1$x^{2} + 37$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.1.2.1a1.1$x^{2} + 37$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.2.2.2a1.2$x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
37.2.2.2a1.2$x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(101\) Copy content Toggle raw display 101.3.1.0a1.1$x^{3} + 3 x + 99$$1$$3$$0$$C_3$$$[\ ]^{3}$$
101.3.1.0a1.1$x^{3} + 3 x + 99$$1$$3$$0$$C_3$$$[\ ]^{3}$$
101.3.2.3a1.2$x^{6} + 6 x^{4} + 198 x^{3} + 9 x^{2} + 594 x + 9902$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
101.3.2.3a1.2$x^{6} + 6 x^{4} + 198 x^{3} + 9 x^{2} + 594 x + 9902$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$

Spectrum of ring of integers

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