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Properties

Label 18.18.290...376.1
Degree $18$
Signature $[18, 0]$
Discriminant $2.909\times 10^{56}$
Root discriminant \(1370.48\)
Ramified primes $2,3,197,229$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $C_3^4:S_4$ (as 18T360)

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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 - 2694*x^16 - 3336*x^15 + 2597787*x^14 + 8641692*x^13 - 1119005570*x^12 - 7198543116*x^11 + 225827964465*x^10 + 2300154078516*x^9 - 16615938449916*x^8 - 292200954101640*x^7 - 420745662756108*x^6 + 10758471758781408*x^5 + 64627251132351120*x^4 + 123813367121310144*x^3 + 26830278605886912*x^2 - 138989639991151872*x - 89792650683453952)
 
Copy content gp:K = bnfinit(y^18 - 2694*y^16 - 3336*y^15 + 2597787*y^14 + 8641692*y^13 - 1119005570*y^12 - 7198543116*y^11 + 225827964465*y^10 + 2300154078516*y^9 - 16615938449916*y^8 - 292200954101640*y^7 - 420745662756108*y^6 + 10758471758781408*y^5 + 64627251132351120*y^4 + 123813367121310144*y^3 + 26830278605886912*y^2 - 138989639991151872*y - 89792650683453952, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2694*x^16 - 3336*x^15 + 2597787*x^14 + 8641692*x^13 - 1119005570*x^12 - 7198543116*x^11 + 225827964465*x^10 + 2300154078516*x^9 - 16615938449916*x^8 - 292200954101640*x^7 - 420745662756108*x^6 + 10758471758781408*x^5 + 64627251132351120*x^4 + 123813367121310144*x^3 + 26830278605886912*x^2 - 138989639991151872*x - 89792650683453952);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 2694*x^16 - 3336*x^15 + 2597787*x^14 + 8641692*x^13 - 1119005570*x^12 - 7198543116*x^11 + 225827964465*x^10 + 2300154078516*x^9 - 16615938449916*x^8 - 292200954101640*x^7 - 420745662756108*x^6 + 10758471758781408*x^5 + 64627251132351120*x^4 + 123813367121310144*x^3 + 26830278605886912*x^2 - 138989639991151872*x - 89792650683453952)
 

\( x^{18} - 2694 x^{16} - 3336 x^{15} + 2597787 x^{14} + 8641692 x^{13} - 1119005570 x^{12} + \cdots - 89\!\cdots\!52 \) 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:   \(290870170556598330269359394760370258478274982077485285376\) \(\medspace = 2^{18}\cdot 3^{24}\cdot 197^{8}\cdot 229^{9}\) 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:  \(1370.48\)
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:   \(2\), \(3\), \(197\), \(229\) 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:  \(\Q(\sqrt{229}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{788}a^{9}+\frac{16}{197}a^{7}-\frac{46}{197}a^{6}-\frac{13}{197}a^{5}+\frac{45}{394}a^{4}-\frac{63}{788}a^{3}-\frac{1}{2}a$, $\frac{1}{1576}a^{10}-\frac{1}{1576}a^{9}+\frac{8}{197}a^{8}-\frac{51}{1576}a^{7}-\frac{131}{788}a^{6}+\frac{339}{1576}a^{5}-\frac{153}{1576}a^{4}+\frac{65}{394}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{1576}a^{11}-\frac{1}{1576}a^{9}+\frac{13}{1576}a^{8}-\frac{75}{1576}a^{7}+\frac{33}{1576}a^{6}-\frac{4}{197}a^{5}+\frac{651}{1576}a^{4}+\frac{44}{197}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{310472}a^{12}+\frac{8}{38809}a^{10}-\frac{23}{38809}a^{9}-\frac{19949}{310472}a^{8}+\frac{15411}{155236}a^{7}+\frac{16483}{77618}a^{6}-\frac{223}{788}a^{5}+\frac{677}{1576}a^{4}-\frac{135}{788}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{3104720}a^{13}+\frac{1}{776180}a^{12}+\frac{213}{776180}a^{11}+\frac{43}{155236}a^{10}+\frac{7}{15760}a^{9}+\frac{2233}{77618}a^{8}+\frac{90251}{776180}a^{7}+\frac{86353}{388090}a^{6}+\frac{973}{15760}a^{5}+\frac{771}{1970}a^{4}-\frac{501}{1576}a^{3}-\frac{3}{20}a^{2}+\frac{1}{10}a-\frac{2}{5}$, $\frac{1}{6209440}a^{14}-\frac{1}{1552360}a^{12}+\frac{87}{388090}a^{11}-\frac{661}{6209440}a^{10}-\frac{247}{776180}a^{9}+\frac{5052}{194045}a^{8}+\frac{140507}{1552360}a^{7}-\frac{23847}{1241888}a^{6}+\frac{1709}{7880}a^{5}-\frac{4831}{15760}a^{4}-\frac{961}{7880}a^{3}+\frac{1}{10}a^{2}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{1223259680}a^{15}+\frac{2}{38226865}a^{13}+\frac{151}{305814920}a^{12}-\frac{35037}{244651936}a^{11}-\frac{76709}{305814920}a^{10}-\frac{175001}{305814920}a^{9}+\frac{92131}{1552360}a^{8}+\frac{428921}{6209440}a^{7}+\frac{95529}{776180}a^{6}+\frac{6993}{15760}a^{5}-\frac{57}{788}a^{4}+\frac{1767}{3940}a^{3}+\frac{1}{4}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{31804751680}a^{16}+\frac{607}{7951187920}a^{14}-\frac{417}{3975593960}a^{13}+\frac{16299}{31804751680}a^{12}-\frac{93257}{7951187920}a^{11}+\frac{308043}{7951187920}a^{10}-\frac{11677}{40361360}a^{9}+\frac{2759269}{161445440}a^{8}+\frac{26937}{40361360}a^{7}-\frac{417513}{6209440}a^{6}+\frac{42599}{102440}a^{5}+\frac{4929}{102440}a^{4}-\frac{16521}{102440}a^{3}-\frac{87}{260}a^{2}-\frac{63}{130}a-\frac{2}{5}$, $\frac{1}{14\cdots 00}a^{17}-\frac{77\cdots 83}{56\cdots 00}a^{16}-\frac{13\cdots 97}{14\cdots 80}a^{15}+\frac{23\cdots 31}{36\cdots 00}a^{14}+\frac{17\cdots 67}{29\cdots 60}a^{13}-\frac{96\cdots 39}{73\cdots 00}a^{12}+\frac{11\cdots 17}{73\cdots 00}a^{11}-\frac{48\cdots 07}{15\cdots 00}a^{10}+\frac{40\cdots 89}{14\cdots 00}a^{9}-\frac{31\cdots 69}{37\cdots 00}a^{8}-\frac{67\cdots 41}{71\cdots 00}a^{7}+\frac{27\cdots 43}{11\cdots 50}a^{6}-\frac{26\cdots 83}{18\cdots 60}a^{5}-\frac{85\cdots 41}{47\cdots 00}a^{4}+\frac{14\cdots 48}{29\cdots 75}a^{3}-\frac{26\cdots 46}{15\cdots 75}a^{2}-\frac{18\cdots 71}{50\cdots 25}a+\frac{66\cdots 51}{88\cdots 75}$ 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:  $2$

Class group and class number

Ideal class group:  $C_{6}$, which has order $6$ (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:  $C_{2}\times C_{6}$, which has order $12$ (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:   $\frac{12\cdots 79}{29\cdots 00}a^{17}-\frac{12\cdots 09}{58\cdots 00}a^{16}-\frac{34\cdots 31}{29\cdots 80}a^{15}+\frac{12\cdots 61}{29\cdots 00}a^{14}+\frac{65\cdots 29}{58\cdots 60}a^{13}-\frac{89\cdots 59}{58\cdots 00}a^{12}-\frac{43\cdots 52}{90\cdots 75}a^{11}-\frac{27\cdots 17}{29\cdots 00}a^{10}+\frac{29\cdots 01}{29\cdots 00}a^{9}+\frac{16\cdots 31}{29\cdots 00}a^{8}-\frac{67\cdots 53}{73\cdots 00}a^{7}-\frac{31\cdots 23}{36\cdots 00}a^{6}+\frac{12\cdots 01}{93\cdots 90}a^{5}+\frac{14\cdots 79}{37\cdots 00}a^{4}+\frac{31\cdots 73}{23\cdots 75}a^{3}+\frac{23\cdots 43}{23\cdots 50}a^{2}-\frac{17\cdots 14}{11\cdots 75}a-\frac{15\cdots 61}{11\cdots 75}$, $\frac{12\cdots 79}{29\cdots 00}a^{17}-\frac{12\cdots 09}{58\cdots 00}a^{16}-\frac{34\cdots 31}{29\cdots 80}a^{15}+\frac{12\cdots 61}{29\cdots 00}a^{14}+\frac{65\cdots 29}{58\cdots 60}a^{13}-\frac{89\cdots 59}{58\cdots 00}a^{12}-\frac{43\cdots 52}{90\cdots 75}a^{11}-\frac{27\cdots 17}{29\cdots 00}a^{10}+\frac{29\cdots 01}{29\cdots 00}a^{9}+\frac{16\cdots 31}{29\cdots 00}a^{8}-\frac{67\cdots 53}{73\cdots 00}a^{7}-\frac{31\cdots 23}{36\cdots 00}a^{6}+\frac{12\cdots 01}{93\cdots 90}a^{5}+\frac{14\cdots 79}{37\cdots 00}a^{4}+\frac{31\cdots 73}{23\cdots 75}a^{3}+\frac{23\cdots 43}{23\cdots 50}a^{2}-\frac{17\cdots 14}{11\cdots 75}a-\frac{15\cdots 11}{11\cdots 75}$, $\frac{44\cdots 79}{37\cdots 00}a^{17}-\frac{83\cdots 93}{94\cdots 00}a^{16}-\frac{11\cdots 63}{37\cdots 40}a^{15}+\frac{91\cdots 17}{47\cdots 00}a^{14}+\frac{21\cdots 41}{75\cdots 80}a^{13}-\frac{54\cdots 89}{47\cdots 00}a^{12}-\frac{23\cdots 37}{18\cdots 00}a^{11}+\frac{92\cdots 73}{16\cdots 00}a^{10}+\frac{96\cdots 11}{37\cdots 00}a^{9}+\frac{19\cdots 51}{24\cdots 00}a^{8}-\frac{58\cdots 67}{24\cdots 00}a^{7}-\frac{19\cdots 67}{12\cdots 00}a^{6}+\frac{28\cdots 63}{48\cdots 80}a^{5}+\frac{47\cdots 53}{60\cdots 00}a^{4}+\frac{64\cdots 43}{30\cdots 00}a^{3}+\frac{16\cdots 19}{15\cdots 00}a^{2}-\frac{17\cdots 77}{77\cdots 50}a-\frac{68\cdots 99}{38\cdots 25}$, $\frac{45\cdots 23}{75\cdots 80}a^{17}-\frac{35\cdots 47}{37\cdots 40}a^{16}-\frac{12\cdots 59}{75\cdots 28}a^{15}+\frac{10\cdots 01}{18\cdots 20}a^{14}+\frac{11\cdots 53}{75\cdots 80}a^{13}+\frac{20\cdots 01}{75\cdots 28}a^{12}-\frac{25\cdots 09}{37\cdots 40}a^{11}-\frac{25\cdots 59}{80\cdots 20}a^{10}+\frac{10\cdots 07}{75\cdots 80}a^{9}+\frac{44\cdots 39}{38\cdots 24}a^{8}-\frac{29\cdots 83}{24\cdots 40}a^{7}-\frac{74\cdots 01}{48\cdots 80}a^{6}+\frac{19\cdots 47}{48\cdots 80}a^{5}+\frac{15\cdots 77}{24\cdots 40}a^{4}+\frac{16\cdots 19}{60\cdots 60}a^{3}+\frac{17\cdots 33}{77\cdots 45}a^{2}-\frac{45\cdots 57}{15\cdots 90}a-\frac{21\cdots 33}{77\cdots 45}$, $\frac{29\cdots 93}{94\cdots 00}a^{17}-\frac{97\cdots 43}{18\cdots 00}a^{16}-\frac{14\cdots 13}{18\cdots 20}a^{15}+\frac{11\cdots 17}{94\cdots 00}a^{14}+\frac{26\cdots 01}{37\cdots 64}a^{13}-\frac{18\cdots 33}{18\cdots 00}a^{12}-\frac{26\cdots 33}{94\cdots 00}a^{11}+\frac{48\cdots 89}{16\cdots 00}a^{10}+\frac{55\cdots 57}{94\cdots 00}a^{9}-\frac{37\cdots 83}{96\cdots 00}a^{8}-\frac{32\cdots 57}{48\cdots 00}a^{7}+\frac{45\cdots 73}{24\cdots 00}a^{6}+\frac{46\cdots 33}{12\cdots 12}a^{5}-\frac{40\cdots 07}{12\cdots 00}a^{4}-\frac{48\cdots 07}{60\cdots 00}a^{3}-\frac{18\cdots 33}{15\cdots 00}a^{2}+\frac{33\cdots 12}{38\cdots 25}a+\frac{42\cdots 03}{38\cdots 25}$, $\frac{13\cdots 63}{11\cdots 00}a^{17}-\frac{54\cdots 17}{56\cdots 00}a^{16}-\frac{34\cdots 83}{11\cdots 60}a^{15}+\frac{30\cdots 39}{14\cdots 00}a^{14}+\frac{64\cdots 57}{22\cdots 20}a^{13}-\frac{77\cdots 77}{56\cdots 00}a^{12}-\frac{67\cdots 89}{56\cdots 00}a^{11}+\frac{73\cdots 81}{47\cdots 00}a^{10}+\frac{28\cdots 47}{11\cdots 00}a^{9}+\frac{17\cdots 93}{28\cdots 00}a^{8}-\frac{17\cdots 59}{71\cdots 00}a^{7}-\frac{19\cdots 91}{14\cdots 00}a^{6}+\frac{96\cdots 09}{14\cdots 20}a^{5}+\frac{16\cdots 27}{22\cdots 75}a^{4}+\frac{15\cdots 41}{91\cdots 00}a^{3}+\frac{64\cdots 77}{11\cdots 75}a^{2}-\frac{90\cdots 74}{50\cdots 25}a-\frac{10\cdots 41}{88\cdots 75}$, $\frac{26\cdots 31}{31\cdots 00}a^{17}-\frac{10\cdots 93}{12\cdots 00}a^{16}-\frac{41\cdots 41}{19\cdots 30}a^{15}+\frac{31\cdots 07}{15\cdots 00}a^{14}+\frac{24\cdots 21}{12\cdots 32}a^{13}-\frac{20\cdots 29}{15\cdots 00}a^{12}-\frac{30\cdots 27}{39\cdots 00}a^{11}+\frac{58\cdots 29}{27\cdots 00}a^{10}+\frac{50\cdots 79}{31\cdots 00}a^{9}+\frac{17\cdots 21}{81\cdots 00}a^{8}-\frac{96\cdots 29}{62\cdots 00}a^{7}-\frac{62\cdots 99}{81\cdots 00}a^{6}+\frac{17\cdots 21}{41\cdots 60}a^{5}+\frac{86\cdots 83}{20\cdots 00}a^{4}+\frac{49\cdots 39}{51\cdots 00}a^{3}+\frac{42\cdots 71}{13\cdots 50}a^{2}-\frac{52\cdots 01}{50\cdots 25}a-\frac{27\cdots 73}{38\cdots 25}$, $\frac{49\cdots 79}{29\cdots 36}a^{17}-\frac{52\cdots 61}{22\cdots 72}a^{16}-\frac{20\cdots 69}{45\cdots 99}a^{15}+\frac{11\cdots 47}{14\cdots 68}a^{14}+\frac{64\cdots 21}{14\cdots 80}a^{13}+\frac{12\cdots 69}{14\cdots 80}a^{12}-\frac{69\cdots 57}{36\cdots 20}a^{11}-\frac{23\cdots 47}{24\cdots 52}a^{10}+\frac{58\cdots 39}{14\cdots 80}a^{9}+\frac{48\cdots 67}{14\cdots 88}a^{8}-\frac{97\cdots 23}{28\cdots 40}a^{7}-\frac{82\cdots 67}{18\cdots 60}a^{6}+\frac{78\cdots 91}{94\cdots 80}a^{5}+\frac{17\cdots 91}{94\cdots 80}a^{4}+\frac{72\cdots 77}{94\cdots 88}a^{3}+\frac{77\cdots 69}{12\cdots 60}a^{2}-\frac{82\cdots 56}{10\cdots 85}a-\frac{13\cdots 51}{17\cdots 35}$, $\frac{24\cdots 47}{28\cdots 00}a^{17}-\frac{39\cdots 97}{56\cdots 00}a^{16}-\frac{63\cdots 69}{28\cdots 40}a^{15}+\frac{43\cdots 23}{28\cdots 00}a^{14}+\frac{11\cdots 37}{56\cdots 80}a^{13}-\frac{54\cdots 47}{56\cdots 00}a^{12}-\frac{30\cdots 89}{35\cdots 00}a^{11}+\frac{48\cdots 31}{47\cdots 00}a^{10}+\frac{52\cdots 13}{28\cdots 00}a^{9}+\frac{10\cdots 71}{22\cdots 00}a^{8}-\frac{12\cdots 39}{71\cdots 00}a^{7}-\frac{74\cdots 33}{71\cdots 00}a^{6}+\frac{17\cdots 99}{36\cdots 80}a^{5}+\frac{19\cdots 17}{36\cdots 00}a^{4}+\frac{56\cdots 03}{45\cdots 50}a^{3}+\frac{51\cdots 22}{11\cdots 75}a^{2}-\frac{67\cdots 94}{50\cdots 25}a-\frac{83\cdots 71}{88\cdots 75}$, $\frac{59\cdots 61}{73\cdots 00}a^{17}-\frac{19\cdots 73}{28\cdots 00}a^{16}-\frac{19\cdots 41}{91\cdots 80}a^{15}-\frac{32\cdots 53}{36\cdots 00}a^{14}+\frac{30\cdots 19}{14\cdots 80}a^{13}+\frac{19\cdots 91}{36\cdots 00}a^{12}-\frac{20\cdots 53}{22\cdots 50}a^{11}-\frac{31\cdots 31}{62\cdots 00}a^{10}+\frac{13\cdots 29}{73\cdots 00}a^{9}+\frac{31\cdots 61}{18\cdots 00}a^{8}-\frac{16\cdots 23}{11\cdots 00}a^{7}-\frac{41\cdots 19}{18\cdots 00}a^{6}-\frac{36\cdots 49}{23\cdots 22}a^{5}+\frac{41\cdots 43}{47\cdots 00}a^{4}+\frac{10\cdots 73}{23\cdots 00}a^{3}+\frac{37\cdots 37}{60\cdots 00}a^{2}-\frac{15\cdots 22}{50\cdots 25}a-\frac{76\cdots 33}{88\cdots 75}$, $\frac{20\cdots 51}{29\cdots 36}a^{17}-\frac{62\cdots 03}{11\cdots 60}a^{16}-\frac{16\cdots 07}{91\cdots 80}a^{15}+\frac{18\cdots 19}{14\cdots 68}a^{14}+\frac{25\cdots 41}{14\cdots 80}a^{13}-\frac{11\cdots 17}{14\cdots 80}a^{12}-\frac{26\cdots 71}{36\cdots 20}a^{11}+\frac{86\cdots 09}{12\cdots 60}a^{10}+\frac{45\cdots 71}{29\cdots 36}a^{9}+\frac{59\cdots 53}{14\cdots 88}a^{8}-\frac{42\cdots 91}{28\cdots 40}a^{7}-\frac{81\cdots 19}{93\cdots 80}a^{6}+\frac{37\cdots 99}{94\cdots 80}a^{5}+\frac{83\cdots 23}{18\cdots 76}a^{4}+\frac{49\cdots 07}{47\cdots 40}a^{3}+\frac{45\cdots 49}{12\cdots 60}a^{2}-\frac{11\cdots 07}{10\cdots 85}a-\frac{28\cdots 51}{35\cdots 27}$, $\frac{14\cdots 89}{14\cdots 00}a^{17}-\frac{15\cdots 93}{14\cdots 00}a^{16}-\frac{36\cdots 51}{14\cdots 80}a^{15}+\frac{93\cdots 79}{36\cdots 00}a^{14}+\frac{64\cdots 67}{29\cdots 60}a^{13}-\frac{63\cdots 43}{36\cdots 00}a^{12}-\frac{64\cdots 57}{73\cdots 00}a^{11}+\frac{50\cdots 27}{15\cdots 00}a^{10}+\frac{26\cdots 61}{14\cdots 00}a^{9}+\frac{23\cdots 07}{18\cdots 00}a^{8}-\frac{25\cdots 53}{14\cdots 00}a^{7}-\frac{14\cdots 73}{18\cdots 00}a^{6}+\frac{95\cdots 73}{18\cdots 60}a^{5}+\frac{21\cdots 31}{47\cdots 00}a^{4}+\frac{11\cdots 03}{11\cdots 00}a^{3}+\frac{18\cdots 59}{60\cdots 00}a^{2}-\frac{10\cdots 33}{10\cdots 50}a-\frac{66\cdots 41}{88\cdots 75}$, $\frac{90\cdots 51}{14\cdots 00}a^{17}-\frac{82\cdots 57}{14\cdots 00}a^{16}-\frac{23\cdots 83}{14\cdots 80}a^{15}+\frac{24\cdots 63}{18\cdots 00}a^{14}+\frac{43\cdots 93}{29\cdots 60}a^{13}-\frac{32\cdots 77}{36\cdots 00}a^{12}-\frac{45\cdots 13}{73\cdots 00}a^{11}+\frac{93\cdots 77}{62\cdots 00}a^{10}+\frac{19\cdots 59}{14\cdots 00}a^{9}+\frac{31\cdots 93}{18\cdots 00}a^{8}-\frac{92\cdots 81}{71\cdots 00}a^{7}-\frac{54\cdots 01}{93\cdots 00}a^{6}+\frac{76\cdots 43}{18\cdots 60}a^{5}+\frac{19\cdots 53}{59\cdots 50}a^{4}+\frac{32\cdots 51}{59\cdots 50}a^{3}-\frac{11\cdots 27}{30\cdots 50}a^{2}-\frac{29\cdots 06}{50\cdots 25}a-\frac{25\cdots 89}{88\cdots 75}$, $\frac{16\cdots 71}{29\cdots 60}a^{17}-\frac{12\cdots 03}{28\cdots 40}a^{16}-\frac{21\cdots 91}{14\cdots 80}a^{15}+\frac{73\cdots 39}{73\cdots 40}a^{14}+\frac{40\cdots 61}{29\cdots 60}a^{13}-\frac{45\cdots 77}{73\cdots 40}a^{12}-\frac{83\cdots 93}{14\cdots 80}a^{11}+\frac{39\cdots 71}{62\cdots 80}a^{10}+\frac{35\cdots 87}{29\cdots 60}a^{9}+\frac{11\cdots 69}{37\cdots 20}a^{8}-\frac{64\cdots 43}{55\cdots 20}a^{7}-\frac{50\cdots 89}{74\cdots 44}a^{6}+\frac{59\cdots 53}{18\cdots 60}a^{5}+\frac{64\cdots 51}{18\cdots 76}a^{4}+\frac{97\cdots 69}{11\cdots 10}a^{3}+\frac{19\cdots 63}{60\cdots 30}a^{2}-\frac{17\cdots 13}{20\cdots 70}a-\frac{23\cdots 75}{35\cdots 27}$, $\frac{19\cdots 81}{29\cdots 60}a^{17}+\frac{74\cdots 29}{11\cdots 60}a^{16}-\frac{25\cdots 07}{14\cdots 80}a^{15}-\frac{72\cdots 33}{18\cdots 60}a^{14}+\frac{49\cdots 87}{29\cdots 60}a^{13}+\frac{10\cdots 33}{14\cdots 80}a^{12}-\frac{10\cdots 29}{14\cdots 80}a^{11}-\frac{67\cdots 87}{12\cdots 60}a^{10}+\frac{41\cdots 01}{29\cdots 60}a^{9}+\frac{12\cdots 19}{74\cdots 40}a^{8}-\frac{26\cdots 67}{28\cdots 40}a^{7}-\frac{74\cdots 03}{37\cdots 20}a^{6}-\frac{90\cdots 81}{18\cdots 60}a^{5}+\frac{31\cdots 29}{47\cdots 40}a^{4}+\frac{46\cdots 81}{94\cdots 88}a^{3}+\frac{78\cdots 41}{60\cdots 30}a^{2}+\frac{29\cdots 93}{20\cdots 70}a+\frac{10\cdots 17}{17\cdots 35}$, $\frac{65\cdots 57}{14\cdots 00}a^{17}-\frac{11\cdots 63}{28\cdots 00}a^{16}-\frac{17\cdots 41}{14\cdots 80}a^{15}+\frac{34\cdots 07}{36\cdots 00}a^{14}+\frac{33\cdots 31}{29\cdots 60}a^{13}-\frac{11\cdots 57}{18\cdots 00}a^{12}-\frac{35\cdots 11}{73\cdots 00}a^{11}+\frac{14\cdots 41}{15\cdots 00}a^{10}+\frac{15\cdots 33}{14\cdots 00}a^{9}+\frac{20\cdots 33}{93\cdots 00}a^{8}-\frac{41\cdots 01}{35\cdots 00}a^{7}-\frac{12\cdots 19}{18\cdots 00}a^{6}+\frac{64\cdots 43}{18\cdots 60}a^{5}+\frac{16\cdots 03}{47\cdots 00}a^{4}+\frac{25\cdots 36}{29\cdots 75}a^{3}+\frac{93\cdots 51}{30\cdots 50}a^{2}-\frac{92\cdots 29}{10\cdots 50}a-\frac{57\cdots 43}{88\cdots 75}$, $\frac{73\cdots 91}{14\cdots 00}a^{17}-\frac{99\cdots 53}{56\cdots 00}a^{16}-\frac{19\cdots 69}{14\cdots 80}a^{15}+\frac{54\cdots 03}{18\cdots 00}a^{14}+\frac{36\cdots 13}{29\cdots 60}a^{13}+\frac{18\cdots 51}{73\cdots 00}a^{12}-\frac{39\cdots 03}{73\cdots 00}a^{11}-\frac{11\cdots 43}{62\cdots 00}a^{10}+\frac{15\cdots 79}{14\cdots 00}a^{9}+\frac{28\cdots 21}{37\cdots 00}a^{8}-\frac{12\cdots 07}{14\cdots 00}a^{7}-\frac{19\cdots 47}{18\cdots 00}a^{6}-\frac{82\cdots 01}{18\cdots 60}a^{5}+\frac{47\cdots 61}{11\cdots 00}a^{4}+\frac{26\cdots 47}{11\cdots 00}a^{3}+\frac{30\cdots 61}{60\cdots 00}a^{2}+\frac{53\cdots 93}{10\cdots 50}a+\frac{17\cdots 71}{88\cdots 75}$ 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:  \( 54573832318800000000000 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 54573832318800000000000 \cdot 6}{2\cdot\sqrt{290870170556598330269359394760370258478274982077485285376}}\cr\approx \mathstrut & 2.51649479259392 \end{aligned}\] (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 - 2694*x^16 - 3336*x^15 + 2597787*x^14 + 8641692*x^13 - 1119005570*x^12 - 7198543116*x^11 + 225827964465*x^10 + 2300154078516*x^9 - 16615938449916*x^8 - 292200954101640*x^7 - 420745662756108*x^6 + 10758471758781408*x^5 + 64627251132351120*x^4 + 123813367121310144*x^3 + 26830278605886912*x^2 - 138989639991151872*x - 89792650683453952) 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 - 2694*x^16 - 3336*x^15 + 2597787*x^14 + 8641692*x^13 - 1119005570*x^12 - 7198543116*x^11 + 225827964465*x^10 + 2300154078516*x^9 - 16615938449916*x^8 - 292200954101640*x^7 - 420745662756108*x^6 + 10758471758781408*x^5 + 64627251132351120*x^4 + 123813367121310144*x^3 + 26830278605886912*x^2 - 138989639991151872*x - 89792650683453952, 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 - 2694*x^16 - 3336*x^15 + 2597787*x^14 + 8641692*x^13 - 1119005570*x^12 - 7198543116*x^11 + 225827964465*x^10 + 2300154078516*x^9 - 16615938449916*x^8 - 292200954101640*x^7 - 420745662756108*x^6 + 10758471758781408*x^5 + 64627251132351120*x^4 + 123813367121310144*x^3 + 26830278605886912*x^2 - 138989639991151872*x - 89792650683453952); 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 - 2694*x^16 - 3336*x^15 + 2597787*x^14 + 8641692*x^13 - 1119005570*x^12 - 7198543116*x^11 + 225827964465*x^10 + 2300154078516*x^9 - 16615938449916*x^8 - 292200954101640*x^7 - 420745662756108*x^6 + 10758471758781408*x^5 + 64627251132351120*x^4 + 123813367121310144*x^3 + 26830278605886912*x^2 - 138989639991151872*x - 89792650683453952); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_3^4:S_4$ (as 18T360):

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 1944
The 30 conjugacy class representatives for $C_3^4:S_4$
Character table for $C_3^4:S_4$

Intermediate fields

3.3.229.1, 6.6.768575296.1

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 24 siblings: data not computed
Degree 27 sibling: 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 R ${\href{/padicField/5.3.0.1}{3} }^{6}$ ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.9.0.1}{9} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.9.0.1}{9} }^{2}$ ${\href{/padicField/19.9.0.1}{9} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.1.2.2a1.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$$[2]$$
2.2.2.4a2.2$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$$2$$2$$4$$D_{4}$$$[2, 2]^{2}$$
2.2.2.4a1.1$x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$$2$$2$$4$$C_2^2$$$[2]^{2}$$
2.2.2.4a2.2$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$$2$$2$$4$$D_{4}$$$[2, 2]^{2}$$
2.2.2.4a2.2$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$$2$$2$$4$$D_{4}$$$[2, 2]^{2}$$
\(3\) Copy content Toggle raw display 3.3.3.12a9.3$x^{9} + 6 x^{8} + 9 x^{7} + 27 x^{6} + 36 x^{5} + 42 x^{4} + 47 x^{3} + 30 x^{2} + 9 x + 22$$3$$3$$12$$C_3 \wr C_3 $$$[2, 2, 2]^{3}$$
3.3.3.12a9.3$x^{9} + 6 x^{8} + 9 x^{7} + 27 x^{6} + 36 x^{5} + 42 x^{4} + 47 x^{3} + 30 x^{2} + 9 x + 22$$3$$3$$12$$C_3 \wr C_3 $$$[2, 2, 2]^{3}$$
\(197\) Copy content Toggle raw display 197.6.1.0a1.1$x^{6} + x^{4} + 124 x^{3} + 79 x^{2} + 173 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
197.2.3.4a1.1$x^{6} + 576 x^{5} + 110598 x^{4} + 7080192 x^{3} + 221196 x^{2} + 2501 x + 8$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
197.2.3.4a1.1$x^{6} + 576 x^{5} + 110598 x^{4} + 7080192 x^{3} + 221196 x^{2} + 2501 x + 8$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
\(229\) Copy content Toggle raw display 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 $2$$2$$1$$1$$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 $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Spectrum of ring of integers

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