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Properties

Label 18.0.712...103.1
Degree $18$
Signature $[0, 9]$
Discriminant $-7.126\times 10^{29}$
Root discriminant \(45.55\)
Ramified primes $3,7$
Class number $12$ (GRH)
Class group [12] (GRH)
Galois group $C_6:S_3$ (as 18T12)

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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 - 12*x^15 + 81*x^12 + 680*x^9 + 891*x^6 + 228*x^3 + 1331)
 
Copy content gp:K = bnfinit(y^18 - 12*y^15 + 81*y^12 + 680*y^9 + 891*y^6 + 228*y^3 + 1331, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 12*x^15 + 81*x^12 + 680*x^9 + 891*x^6 + 228*x^3 + 1331);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 12*x^15 + 81*x^12 + 680*x^9 + 891*x^6 + 228*x^3 + 1331)
 

\( x^{18} - 12x^{15} + 81x^{12} + 680x^{9} + 891x^{6} + 228x^{3} + 1331 \) 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:   \(-712583513384965051151503760103\) \(\medspace = -\,3^{36}\cdot 7^{15}\) 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:  \(45.55\)
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:  $3^{37/18}7^{5/6}\approx 48.416970588180874$
Ramified primes:   \(3\), \(7\) 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:  \(\Q(\sqrt{-7}) \)
$\Aut(K/\Q)$:   $C_2$
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:  \(\Q(\sqrt{-7}) \)

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{6}-\frac{1}{8}a^{3}+\frac{1}{8}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{7}-\frac{1}{8}a^{4}+\frac{1}{8}a$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{8}a^{5}+\frac{1}{8}a^{2}$, $\frac{1}{32}a^{12}+\frac{1}{16}a^{6}-\frac{3}{32}$, $\frac{1}{32}a^{13}+\frac{1}{16}a^{7}-\frac{3}{32}a$, $\frac{1}{32}a^{14}+\frac{1}{16}a^{8}-\frac{3}{32}a^{2}$, $\frac{1}{380480}a^{15}-\frac{5791}{380480}a^{12}+\frac{781}{38048}a^{9}-\frac{4643}{38048}a^{6}-\frac{2111}{13120}a^{3}+\frac{128669}{380480}$, $\frac{1}{4185280}a^{16}-\frac{53351}{4185280}a^{13}-\frac{3975}{418528}a^{10}+\frac{38161}{418528}a^{7}+\frac{2939}{13120}a^{4}+\frac{318909}{4185280}a$, $\frac{1}{46038080}a^{17}+\frac{208229}{46038080}a^{14}+\frac{257605}{4603808}a^{11}+\frac{352057}{4603808}a^{8}+\frac{34099}{144320}a^{5}-\frac{7266911}{46038080}a^{2}$ 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:  $C_{12}$, which has order $12$ (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:  $C_{12}$, 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:  $8$
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:   $\frac{49}{38048}a^{15}-\frac{777}{38048}a^{12}+\frac{3483}{19024}a^{9}+\frac{3905}{19024}a^{6}+\frac{45}{1312}a^{3}+\frac{10215}{38048}$, $\frac{247}{380480}a^{15}-\frac{3577}{380480}a^{12}+\frac{2667}{38048}a^{9}+\frac{13643}{38048}a^{6}-\frac{9737}{13120}a^{3}-\frac{1035157}{380480}$, $\frac{3493}{2092640}a^{16}-\frac{2793}{130790}a^{13}+\frac{31381}{209264}a^{10}+\frac{54387}{52316}a^{7}+\frac{2807}{6560}a^{4}-\frac{203021}{261580}a+2$, $\frac{382989}{46038080}a^{17}-\frac{42521}{4185280}a^{16}+\frac{1063}{76096}a^{15}-\frac{5802359}{46038080}a^{14}+\frac{639271}{4185280}a^{13}-\frac{15837}{76096}a^{12}+\frac{4879713}{4603808}a^{11}-\frac{535181}{418528}a^{10}+\frac{65611}{38048}a^{9}+\frac{11441093}{4603808}a^{8}-\frac{1318725}{418528}a^{7}+\frac{177311}{38048}a^{6}-\frac{263529}{144320}a^{5}+\frac{20101}{13120}a^{4}-\frac{6049}{2624}a^{3}+\frac{194489861}{46038080}a^{2}-\frac{18856149}{4185280}a+\frac{453919}{76096}$, $\frac{333969}{46038080}a^{17}+\frac{1259}{380480}a^{16}-\frac{147}{38048}a^{15}-\frac{4243699}{46038080}a^{14}-\frac{14189}{380480}a^{13}+\frac{2331}{38048}a^{12}+\frac{3026053}{4603808}a^{11}+\frac{8299}{38048}a^{10}-\frac{10449}{19024}a^{9}+\frac{20188857}{4603808}a^{8}+\frac{104219}{38048}a^{7}-\frac{11715}{19024}a^{6}+\frac{583651}{144320}a^{5}+\frac{26931}{13120}a^{4}-\frac{135}{1312}a^{3}-\frac{42876839}{46038080}a^{2}-\frac{1802369}{380480}a-\frac{373077}{38048}$, $\frac{5857}{9207616}a^{17}+\frac{819}{261580}a^{16}-\frac{1823}{380480}a^{15}-\frac{124129}{9207616}a^{14}-\frac{10609}{261580}a^{13}+\frac{22453}{380480}a^{12}+\frac{623017}{4603808}a^{11}+\frac{15371}{52316}a^{10}-\frac{15987}{38048}a^{9}-\frac{1101985}{4603808}a^{8}+\frac{47365}{26158}a^{7}-\frac{110879}{38048}a^{6}-\frac{49901}{28864}a^{5}+\frac{683}{410}a^{4}-\frac{61407}{13120}a^{3}+\frac{1345923}{9207616}a^{2}-\frac{655109}{261580}a-\frac{2066527}{380480}$, $\frac{115569}{46038080}a^{17}-\frac{17}{3520}a^{16}-\frac{4153}{380480}a^{15}-\frac{1589019}{46038080}a^{14}+\frac{237}{3520}a^{13}+\frac{56003}{380480}a^{12}+\frac{1203289}{4603808}a^{11}-\frac{185}{352}a^{10}-\frac{42705}{38048}a^{9}+\frac{5901517}{4603808}a^{8}-\frac{771}{352}a^{7}-\frac{207709}{38048}a^{6}-\frac{32629}{144320}a^{5}-\frac{287}{320}a^{4}-\frac{51297}{13120}a^{3}-\frac{20246279}{46038080}a^{2}+\frac{7377}{3520}a-\frac{810497}{380480}$, $\frac{22993}{23019040}a^{17}-\frac{2253}{837056}a^{16}+\frac{37}{190240}a^{15}-\frac{150923}{23019040}a^{14}+\frac{29951}{837056}a^{13}-\frac{247}{190240}a^{12}+\frac{25035}{2301904}a^{11}-\frac{108753}{418528}a^{10}+\frac{361}{19024}a^{9}+\frac{2790827}{2301904}a^{8}-\frac{630885}{418528}a^{7}-\frac{575}{19024}a^{6}+\frac{245407}{72160}a^{5}-\frac{4495}{2624}a^{4}-\frac{2667}{6560}a^{3}+\frac{15758877}{23019040}a^{2}+\frac{58283}{837056}a+\frac{171213}{190240}$ 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:  \( 102332590.90594403 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 102332590.90594403 \cdot 12}{2\cdot\sqrt{712583513384965051151503760103}}\cr\approx \mathstrut & 11.1011034672095 \end{aligned}\] (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 - 12*x^15 + 81*x^12 + 680*x^9 + 891*x^6 + 228*x^3 + 1331) 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 - 12*x^15 + 81*x^12 + 680*x^9 + 891*x^6 + 228*x^3 + 1331, 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 - 12*x^15 + 81*x^12 + 680*x^9 + 891*x^6 + 228*x^3 + 1331); 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 - 12*x^15 + 81*x^12 + 680*x^9 + 891*x^6 + 228*x^3 + 1331); 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

$C_6:S_3$ (as 18T12):

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 12 conjugacy class representatives for $C_6:S_3$
Character table for $C_6:S_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.11907.2, 3.1.243.1, 3.1.1323.1, 3.1.11907.1, 6.0.992436543.3, 6.0.12252303.1, 6.0.20253807.1, 6.0.992436543.1, 9.1.45579633110361.5

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 18 sibling: 18.2.2137750540154895153454511280309.1
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 ${\href{/padicField/2.2.0.1}{2} }^{8}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ R ${\href{/padicField/5.2.0.1}{2} }^{9}$ R ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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
\(3\) Copy content Toggle raw display 3.2.9.36b6.16$x^{18} + 18 x^{17} + 162 x^{16} + 960 x^{15} + 4176 x^{14} + 14112 x^{13} + 38304 x^{12} + 85248 x^{11} + 157536 x^{10} + 243392 x^{9} + 315072 x^{8} + 340992 x^{7} + 306435 x^{6} + 225810 x^{5} + 133695 x^{4} + 61572 x^{3} + 20925 x^{2} + 4770 x + 593$$9$$2$$36$not computednot computed
\(7\) Copy content Toggle raw display 7.3.6.15a1.3$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98736 x^{12} + 161280 x^{11} + 238464 x^{10} + 208640 x^{9} + 334080 x^{8} + 138240 x^{7} + 280320 x^{6} + 46080 x^{5} + 138240 x^{4} + 6144 x^{3} + 36864 x^{2} + 4103$$6$$3$$15$$C_6 \times C_3$$$[\ ]_{6}^{3}$$

Spectrum of ring of integers

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