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Properties

Label 18.0.444...456.2
Degree $18$
Signature $[0, 9]$
Discriminant $-4.440\times 10^{76}$
Root discriminant \($18\,121$.30\)
Ramified primes $2,3,2287$
Class number not computed
Class group not computed
Galois group $S_3^2$ (as 18T11)

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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 + 684*x^16 - 48*x^15 + 194940*x^14 - 27360*x^13 + 34077768*x^12 - 6238080*x^11 - 1520808336*x^10 + 1067219840*x^9 + 982288936224*x^8 - 162179136000*x^7 + 48350568179328*x^6 + 10169023578624*x^5 - 3755551884831360*x^4 - 527155694220288*x^3 + 1027535627818450944*x^2 + 72105946777755648*x + 1265090591033344)
 
Copy content gp:K = bnfinit(y^18 + 684*y^16 - 48*y^15 + 194940*y^14 - 27360*y^13 + 34077768*y^12 - 6238080*y^11 - 1520808336*y^10 + 1067219840*y^9 + 982288936224*y^8 - 162179136000*y^7 + 48350568179328*y^6 + 10169023578624*y^5 - 3755551884831360*y^4 - 527155694220288*y^3 + 1027535627818450944*y^2 + 72105946777755648*y + 1265090591033344, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 684*x^16 - 48*x^15 + 194940*x^14 - 27360*x^13 + 34077768*x^12 - 6238080*x^11 - 1520808336*x^10 + 1067219840*x^9 + 982288936224*x^8 - 162179136000*x^7 + 48350568179328*x^6 + 10169023578624*x^5 - 3755551884831360*x^4 - 527155694220288*x^3 + 1027535627818450944*x^2 + 72105946777755648*x + 1265090591033344);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + 684*x^16 - 48*x^15 + 194940*x^14 - 27360*x^13 + 34077768*x^12 - 6238080*x^11 - 1520808336*x^10 + 1067219840*x^9 + 982288936224*x^8 - 162179136000*x^7 + 48350568179328*x^6 + 10169023578624*x^5 - 3755551884831360*x^4 - 527155694220288*x^3 + 1027535627818450944*x^2 + 72105946777755648*x + 1265090591033344)
 

\( x^{18} + 684 x^{16} - 48 x^{15} + 194940 x^{14} - 27360 x^{13} + 34077768 x^{12} - 6238080 x^{11} + \cdots + 12\!\cdots\!44 \) 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:  $[0, 9]$
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:   \(-44402766688238530812346229902570152786470217145543488433324722627805452435456\) \(\medspace = -\,2^{27}\cdot 3^{38}\cdot 2287^{15}\) 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:  \($18\,121$.30\)
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:  $2^{3/2}3^{13/6}2287^{5/6}\approx 19261.76954796617$
Ramified primes:   \(2\), \(3\), \(2287\) 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:  \(\Q(\sqrt{-4574}) \)
$\Aut(K/\Q)$:   $S_3$
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.
Maximal CM subfield:  \(\Q(\sqrt{-4574}) \)

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{18}a^{3}+\frac{1}{3}a-\frac{4}{9}$, $\frac{1}{36}a^{4}+\frac{1}{6}a^{2}-\frac{2}{9}a$, $\frac{1}{36}a^{5}-\frac{2}{9}a^{2}+\frac{1}{3}$, $\frac{1}{648}a^{6}-\frac{1}{108}a^{4}-\frac{2}{81}a^{3}-\frac{1}{9}a^{2}+\frac{2}{27}a+\frac{8}{81}$, $\frac{1}{1296}a^{7}+\frac{1}{108}a^{5}-\frac{1}{81}a^{4}-\frac{2}{27}a^{2}-\frac{19}{162}a+\frac{2}{9}$, $\frac{1}{7776}a^{8}+\frac{1}{3888}a^{7}+\frac{1}{1944}a^{6}+\frac{1}{972}a^{5}+\frac{1}{486}a^{4}-\frac{7}{486}a^{3}+\frac{29}{972}a^{2}-\frac{79}{486}a+\frac{38}{243}$, $\frac{1}{23328}a^{9}+\frac{1}{1944}a^{6}+\frac{1}{108}a^{5}-\frac{1}{108}a^{4}-\frac{7}{972}a^{3}-\frac{5}{27}a^{2}-\frac{7}{27}a-\frac{349}{729}$, $\frac{1}{139968}a^{10}-\frac{1}{69984}a^{9}+\frac{1}{11664}a^{7}-\frac{1}{5832}a^{6}-\frac{1}{108}a^{5}+\frac{65}{5832}a^{4}+\frac{43}{2916}a^{3}+\frac{1}{6}a^{2}+\frac{1649}{4374}a+\frac{781}{2187}$, $\frac{1}{139968}a^{11}+\frac{1}{69984}a^{9}-\frac{1}{23328}a^{8}-\frac{1}{3888}a^{7}-\frac{1}{2916}a^{6}+\frac{5}{5832}a^{5}-\frac{1}{486}a^{4}-\frac{1}{2916}a^{3}-\frac{41}{8748}a^{2}-\frac{47}{486}a+\frac{254}{2187}$, $\frac{1}{3841281792}a^{12}+\frac{107}{35567424}a^{11}-\frac{19}{640213632}a^{10}-\frac{6841}{480160224}a^{9}+\frac{923}{35567424}a^{8}+\frac{133}{26675568}a^{7}+\frac{34399}{160053408}a^{6}-\frac{24997}{4445928}a^{5}+\frac{309505}{26675568}a^{4}+\frac{143761}{60020028}a^{3}+\frac{530597}{4445928}a^{2}-\frac{1297945}{10003338}a+\frac{6826823}{30010014}$, $\frac{1}{23047690752}a^{13}-\frac{1}{11523845376}a^{12}-\frac{12791}{3841281792}a^{11}+\frac{115}{5761922688}a^{10}+\frac{111481}{5761922688}a^{9}-\frac{11081}{320106816}a^{8}+\frac{59743}{960320448}a^{7}+\frac{111449}{480160224}a^{6}+\frac{255461}{160053408}a^{5}-\frac{3158839}{720240336}a^{4}+\frac{10196759}{720240336}a^{3}-\frac{22089469}{120040056}a^{2}-\frac{57231595}{180060084}a+\frac{36302827}{90030042}$, $\frac{1}{46095381504}a^{14}+\frac{1}{23047690752}a^{12}-\frac{10903}{5761922688}a^{11}-\frac{5807}{3841281792}a^{10}+\frac{4051}{360120168}a^{9}+\frac{75517}{1920640896}a^{8}-\frac{20327}{80026704}a^{7}+\frac{644689}{960320448}a^{6}-\frac{8901203}{720240336}a^{5}+\frac{1905773}{160053408}a^{4}+\frac{1145333}{180060084}a^{3}-\frac{1993685}{45015021}a^{2}-\frac{2247521}{15005007}a+\frac{42463831}{90030042}$, $\frac{1}{414858433536}a^{15}-\frac{1}{69143072256}a^{13}+\frac{1}{12964326048}a^{12}+\frac{17131}{11523845376}a^{11}-\frac{4361}{8642884032}a^{10}+\frac{1031435}{51857304192}a^{9}+\frac{8467}{240080112}a^{8}+\frac{733115}{2880961344}a^{7}-\frac{1021619}{1620540756}a^{6}+\frac{320639}{480160224}a^{5}+\frac{5976617}{1080360504}a^{4}-\frac{44045893}{1620540756}a^{3}+\frac{6597749}{90030042}a^{2}-\frac{46223783}{135045063}a+\frac{25612618}{405135189}$, $\frac{1}{56\cdots 24}a^{16}+\frac{51\cdots 23}{70\cdots 28}a^{15}-\frac{22\cdots 99}{23\cdots 76}a^{14}+\frac{16\cdots 09}{88\cdots 16}a^{13}-\frac{18\cdots 97}{35\cdots 64}a^{12}-\frac{13\cdots 45}{29\cdots 72}a^{11}+\frac{41\cdots 61}{35\cdots 64}a^{10}+\frac{80\cdots 19}{17\cdots 32}a^{9}-\frac{74\cdots 09}{19\cdots 48}a^{8}+\frac{15\cdots 05}{44\cdots 08}a^{7}+\frac{83\cdots 87}{44\cdots 08}a^{6}-\frac{95\cdots 21}{18\cdots 42}a^{5}+\frac{70\cdots 21}{88\cdots 16}a^{4}-\frac{10\cdots 33}{22\cdots 04}a^{3}+\frac{13\cdots 25}{73\cdots 68}a^{2}+\frac{27\cdots 87}{55\cdots 26}a-\frac{72\cdots 44}{27\cdots 63}$, $\frac{1}{63\cdots 28}a^{17}-\frac{15589}{31\cdots 64}a^{16}+\frac{20\cdots 97}{19\cdots 04}a^{15}-\frac{22\cdots 41}{49\cdots 76}a^{14}+\frac{21\cdots 59}{15\cdots 32}a^{13}-\frac{25\cdots 11}{79\cdots 16}a^{12}+\frac{32\cdots 23}{79\cdots 16}a^{11}+\frac{63\cdots 77}{39\cdots 08}a^{10}-\frac{33\cdots 31}{39\cdots 08}a^{9}+\frac{74\cdots 55}{19\cdots 04}a^{8}+\frac{30\cdots 83}{19\cdots 04}a^{7}+\frac{14\cdots 37}{99\cdots 52}a^{6}+\frac{15\cdots 85}{61\cdots 72}a^{5}+\frac{10\cdots 11}{12\cdots 44}a^{4}-\frac{46\cdots 09}{49\cdots 76}a^{3}+\frac{50\cdots 73}{24\cdots 88}a^{2}+\frac{43\cdots 11}{12\cdots 44}a-\frac{10\cdots 95}{61\cdots 72}$ 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:  not computed
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:  not computed
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:  $8$
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:  not computed
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:  not computed
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot R \cdot h}{2\cdot\sqrt{44402766688238530812346229902570152786470217145543488433324722627805452435456}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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 + 684*x^16 - 48*x^15 + 194940*x^14 - 27360*x^13 + 34077768*x^12 - 6238080*x^11 - 1520808336*x^10 + 1067219840*x^9 + 982288936224*x^8 - 162179136000*x^7 + 48350568179328*x^6 + 10169023578624*x^5 - 3755551884831360*x^4 - 527155694220288*x^3 + 1027535627818450944*x^2 + 72105946777755648*x + 1265090591033344) 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 + 684*x^16 - 48*x^15 + 194940*x^14 - 27360*x^13 + 34077768*x^12 - 6238080*x^11 - 1520808336*x^10 + 1067219840*x^9 + 982288936224*x^8 - 162179136000*x^7 + 48350568179328*x^6 + 10169023578624*x^5 - 3755551884831360*x^4 - 527155694220288*x^3 + 1027535627818450944*x^2 + 72105946777755648*x + 1265090591033344, 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 + 684*x^16 - 48*x^15 + 194940*x^14 - 27360*x^13 + 34077768*x^12 - 6238080*x^11 - 1520808336*x^10 + 1067219840*x^9 + 982288936224*x^8 - 162179136000*x^7 + 48350568179328*x^6 + 10169023578624*x^5 - 3755551884831360*x^4 - 527155694220288*x^3 + 1027535627818450944*x^2 + 72105946777755648*x + 1265090591033344); 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 + 684*x^16 - 48*x^15 + 194940*x^14 - 27360*x^13 + 34077768*x^12 - 6238080*x^11 - 1520808336*x^10 + 1067219840*x^9 + 982288936224*x^8 - 162179136000*x^7 + 48350568179328*x^6 + 10169023578624*x^5 - 3755551884831360*x^4 - 527155694220288*x^3 + 1027535627818450944*x^2 + 72105946777755648*x + 1265090591033344); 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

$S_3^2$ (as 18T11):

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 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-4574}) \), 3.1.1270979667.1, 3.1.1481976.3 x3, 6.0.1891530424807441634889216.1, 6.0.40182642410282496.2, 9.1.194731833464812587366907938019256832.1 x3

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

Galois closure: data not computed
Degree 6 sibling: 6.2.5674591274422324904667648.2
Degree 9 sibling: 9.1.194731833464812587366907938019256832.1
Degree 12 sibling: deg 12
Degree 18 siblings: deg 18, 18.0.113761460893702510484735982000599907732061184340694743130306972736028672.2
Minimal sibling: 6.2.5674591274422324904667648.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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 2.1.2.3a1.3$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$$[3]$$
2.1.2.3a1.3$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$$[3]$$
2.1.2.3a1.3$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$$[3]$$
2.2.2.6a1.5$x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.2.2.6a1.5$x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.2.2.6a1.5$x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$$2$$2$$6$$C_2^2$$$[3]^{2}$$
\(3\) Copy content Toggle raw display 3.1.9.19c2.6$x^{9} + 6 x^{6} + 18 x^{2} + 12$$9$$1$$19$$S_3\times C_3$$$[2, \frac{5}{2}]_{2}$$
3.1.9.19c2.6$x^{9} + 6 x^{6} + 18 x^{2} + 12$$9$$1$$19$$S_3\times C_3$$$[2, \frac{5}{2}]_{2}$$
\(2287\) Copy content Toggle raw display Deg $6$$6$$1$$5$
Deg $6$$6$$1$$5$
Deg $6$$6$$1$$5$

Spectrum of ring of integers

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