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Properties

Label 18.0.113...672.2
Degree $18$
Signature $[0, 9]$
Discriminant $-1.138\times 10^{71}$
Root discriminant \(8862.48\)
Ramified primes $2,3,2287$
Class number not computed
Class group not computed
Galois group $S_3^2$ (as 18T11)

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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 - 9*x^17 + 153*x^16 - 906*x^15 + 18369*x^14 + 937467*x^13 - 814452*x^12 + 6424119*x^11 - 850921065*x^10 + 4107177586*x^9 + 127680039693*x^8 + 430642206315*x^7 + 3937560681471*x^6 + 13266832603428*x^5 - 1587871390100094*x^4 - 3591144618213000*x^3 + 200395787039781588*x^2 - 1825080325001722584*x + 6277498164292346128)
 
Copy content gp:K = bnfinit(y^18 - 9*y^17 + 153*y^16 - 906*y^15 + 18369*y^14 + 937467*y^13 - 814452*y^12 + 6424119*y^11 - 850921065*y^10 + 4107177586*y^9 + 127680039693*y^8 + 430642206315*y^7 + 3937560681471*y^6 + 13266832603428*y^5 - 1587871390100094*y^4 - 3591144618213000*y^3 + 200395787039781588*y^2 - 1825080325001722584*y + 6277498164292346128, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 153*x^16 - 906*x^15 + 18369*x^14 + 937467*x^13 - 814452*x^12 + 6424119*x^11 - 850921065*x^10 + 4107177586*x^9 + 127680039693*x^8 + 430642206315*x^7 + 3937560681471*x^6 + 13266832603428*x^5 - 1587871390100094*x^4 - 3591144618213000*x^3 + 200395787039781588*x^2 - 1825080325001722584*x + 6277498164292346128);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 9*x^17 + 153*x^16 - 906*x^15 + 18369*x^14 + 937467*x^13 - 814452*x^12 + 6424119*x^11 - 850921065*x^10 + 4107177586*x^9 + 127680039693*x^8 + 430642206315*x^7 + 3937560681471*x^6 + 13266832603428*x^5 - 1587871390100094*x^4 - 3591144618213000*x^3 + 200395787039781588*x^2 - 1825080325001722584*x + 6277498164292346128)
 

\( x^{18} - 9 x^{17} + 153 x^{16} - 906 x^{15} + 18369 x^{14} + 937467 x^{13} - 814452 x^{12} + \cdots + 62\!\cdots\!28 \) 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:   \(-113761460893702510484735982000599907732061184340694743130306972736028672\) \(\medspace = -\,2^{18}\cdot 3^{39}\cdot 2287^{14}\) 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:  \(8862.48\)
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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $S_3$
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:  \(\Q(\sqrt{-3}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{4574}a^{12}+\frac{1077}{4574}a^{11}+\frac{513}{2287}a^{10}+\frac{547}{2287}a^{9}-\frac{621}{4574}a^{8}+\frac{842}{2287}a^{7}-\frac{1017}{4574}a^{6}+\frac{559}{4574}a^{5}-\frac{615}{2287}a^{4}-\frac{751}{2287}a^{3}+\frac{1451}{4574}a^{2}+\frac{114}{2287}a+\frac{913}{2287}$, $\frac{1}{4574}a^{13}+\frac{303}{2287}a^{11}+\frac{713}{4574}a^{10}-\frac{527}{2287}a^{9}+\frac{205}{2287}a^{8}+\frac{1193}{4574}a^{7}-\frac{946}{2287}a^{6}-\frac{896}{2287}a^{5}-\frac{965}{4574}a^{4}+\frac{1098}{2287}a^{3}-\frac{239}{2287}a^{2}-\frac{654}{2287}a+\frac{109}{2287}$, $\frac{1}{100628}a^{14}+\frac{3}{100628}a^{13}-\frac{2}{25157}a^{12}+\frac{9057}{100628}a^{11}+\frac{11481}{100628}a^{10}+\frac{621}{25157}a^{9}+\frac{13223}{100628}a^{8}+\frac{19731}{100628}a^{7}-\frac{2417}{25157}a^{6}+\frac{25501}{100628}a^{5}+\frac{40977}{100628}a^{4}+\frac{2241}{25157}a^{3}-\frac{14585}{50314}a^{2}+\frac{10959}{25157}a-\frac{9118}{25157}$, $\frac{1}{100628}a^{15}+\frac{5}{100628}a^{13}-\frac{5}{100628}a^{12}+\frac{11625}{50314}a^{11}+\frac{19895}{100628}a^{10}+\frac{4671}{100628}a^{9}-\frac{920}{25157}a^{8}-\frac{47983}{100628}a^{7}-\frac{403}{9148}a^{6}+\frac{14159}{50314}a^{5}-\frac{28497}{100628}a^{4}-\frac{22993}{50314}a^{3}+\frac{4671}{25157}a^{2}+\frac{11520}{25157}a+\frac{6542}{25157}$, $\frac{1}{40\cdots 64}a^{16}-\frac{56\cdots 21}{50\cdots 33}a^{15}+\frac{18\cdots 51}{40\cdots 64}a^{14}-\frac{40\cdots 87}{40\cdots 64}a^{13}+\frac{21\cdots 46}{50\cdots 33}a^{12}-\frac{13\cdots 79}{40\cdots 64}a^{11}+\frac{91\cdots 93}{40\cdots 64}a^{10}-\frac{15\cdots 71}{20\cdots 32}a^{9}+\frac{90\cdots 41}{36\cdots 24}a^{8}-\frac{14\cdots 55}{40\cdots 64}a^{7}-\frac{27\cdots 65}{10\cdots 66}a^{6}+\frac{62\cdots 69}{40\cdots 64}a^{5}-\frac{15\cdots 73}{10\cdots 66}a^{4}+\frac{17\cdots 47}{20\cdots 32}a^{3}+\frac{23\cdots 82}{50\cdots 33}a^{2}+\frac{42\cdots 05}{10\cdots 66}a-\frac{68\cdots 99}{50\cdots 33}$, $\frac{1}{12\cdots 52}a^{17}+\frac{81\cdots 23}{12\cdots 52}a^{16}+\frac{45\cdots 67}{12\cdots 52}a^{15}+\frac{55\cdots 79}{18\cdots 14}a^{14}+\frac{11\cdots 89}{12\cdots 52}a^{13}-\frac{20\cdots 39}{12\cdots 52}a^{12}+\frac{58\cdots 61}{25\cdots 84}a^{11}-\frac{93\cdots 81}{12\cdots 52}a^{10}-\frac{17\cdots 67}{12\cdots 52}a^{9}-\frac{64\cdots 13}{15\cdots 19}a^{8}-\frac{59\cdots 51}{12\cdots 52}a^{7}+\frac{57\cdots 17}{12\cdots 52}a^{6}+\frac{47\cdots 39}{11\cdots 32}a^{5}+\frac{12\cdots 97}{13\cdots 29}a^{4}+\frac{18\cdots 71}{55\cdots 16}a^{3}-\frac{31\cdots 04}{15\cdots 19}a^{2}-\frac{27\cdots 55}{27\cdots 58}a+\frac{92\cdots 15}{15\cdots 19}$ 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:   \( -\frac{16964713181417673998820510650069695139406267}{3890106218790910040164489840002112331633252930438231498131196} a^{17} + \frac{47486786130169924345766533496866998897134895}{7780212437581820080328979680004224663266505860876462996262392} a^{16} - \frac{1223895079236207773232208583923504287129468675}{1945053109395455020082244920001056165816626465219115749065598} a^{15} - \frac{5592928327592118609232550362499373793700757}{46588098428633653175622632814396554869859316532194389199176} a^{14} - \frac{697105900080186791258298859868645405394505611969}{7780212437581820080328979680004224663266505860876462996262392} a^{13} - \frac{407192495527117061518394812126714069968494854443}{88411504972520682731011132727320734809846657509959806775709} a^{12} - \frac{23350267054317397014234542091621675328133792241567}{707292039780165461848089061818565878478773260079678454205672} a^{11} - \frac{2031367481739002141562691744740583605448831437345405}{7780212437581820080328979680004224663266505860876462996262392} a^{10} + \frac{8231224539439016356049204153203408445352574066565365}{3890106218790910040164489840002112331633252930438231498131196} a^{9} - \frac{6598828580735240299659706678070227135232667310935207}{7780212437581820080328979680004224663266505860876462996262392} a^{8} - \frac{4212982142506620026142836806176226744476312367412705409}{7780212437581820080328979680004224663266505860876462996262392} a^{7} - \frac{1132560936617963282707347968327544278017118770972712253}{176823009945041365462022265454641469619693315019919613551418} a^{6} - \frac{493240715993382851502637786912116496920011303082035211939}{7780212437581820080328979680004224663266505860876462996262392} a^{5} - \frac{1025158687112080948259318030981659082084647275227681118699}{1945053109395455020082244920001056165816626465219115749065598} a^{4} + \frac{12331176929613578372067751589474395504424180006951198117097}{3890106218790910040164489840002112331633252930438231498131196} a^{3} + \frac{88015590263834729667637220013043548786299592949516044781239}{1945053109395455020082244920001056165816626465219115749065598} a^{2} - \frac{911857170294343024435477180256801714504849827968211077514443}{1945053109395455020082244920001056165816626465219115749065598} a + \frac{341073261888530943961829926525727189500893564862765099030776}{88411504972520682731011132727320734809846657509959806775709} \)  (order $6$) 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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot R \cdot h}{6\cdot\sqrt{113761460893702510484735982000599907732061184340694743130306972736028672}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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 - 9*x^17 + 153*x^16 - 906*x^15 + 18369*x^14 + 937467*x^13 - 814452*x^12 + 6424119*x^11 - 850921065*x^10 + 4107177586*x^9 + 127680039693*x^8 + 430642206315*x^7 + 3937560681471*x^6 + 13266832603428*x^5 - 1587871390100094*x^4 - 3591144618213000*x^3 + 200395787039781588*x^2 - 1825080325001722584*x + 6277498164292346128) 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 - 9*x^17 + 153*x^16 - 906*x^15 + 18369*x^14 + 937467*x^13 - 814452*x^12 + 6424119*x^11 - 850921065*x^10 + 4107177586*x^9 + 127680039693*x^8 + 430642206315*x^7 + 3937560681471*x^6 + 13266832603428*x^5 - 1587871390100094*x^4 - 3591144618213000*x^3 + 200395787039781588*x^2 - 1825080325001722584*x + 6277498164292346128, 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 - 9*x^17 + 153*x^16 - 906*x^15 + 18369*x^14 + 937467*x^13 - 814452*x^12 + 6424119*x^11 - 850921065*x^10 + 4107177586*x^9 + 127680039693*x^8 + 430642206315*x^7 + 3937560681471*x^6 + 13266832603428*x^5 - 1587871390100094*x^4 - 3591144618213000*x^3 + 200395787039781588*x^2 - 1825080325001722584*x + 6277498164292346128); 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 - 9*x^17 + 153*x^16 - 906*x^15 + 18369*x^14 + 937467*x^13 - 814452*x^12 + 6424119*x^11 - 850921065*x^10 + 4107177586*x^9 + 127680039693*x^8 + 430642206315*x^7 + 3937560681471*x^6 + 13266832603428*x^5 - 1587871390100094*x^4 - 3591144618213000*x^3 + 200395787039781588*x^2 - 1825080325001722584*x + 6277498164292346128); 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

\(\Q(\sqrt{-3}) \), 3.1.1270979667.1 x3, 3.1.1481976.3, 6.0.6588758593728.2, 6.0.4846167941782292667.1, 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: 18.0.44402766688238530812346229902570152786470217145543488433324722627805452435456.2, deg 18
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 ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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 2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
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$$$[3]^{2}$$
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 3.1.18.39c2.19$x^{18} + 6 x^{12} + 18 x^{4} + 12$$18$$1$$39$not computednot computed
\(2287\) Copy content Toggle raw display Deg $3$$3$$1$$2$
Deg $3$$3$$1$$2$
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)