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Properties

Label 18.2.213...309.1
Degree $18$
Signature $[2, 8]$
Discriminant $2.138\times 10^{30}$
Root discriminant \(48.42\)
Ramified primes $3,7$
Class number $12$ (GRH)
Class group [12] (GRH)
Galois group $C_6:S_3$ (as 18T12)

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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 - 30*x^15 + 228*x^12 - 1114*x^9 + 1059*x^6 + 192*x^3 + 64)
 
Copy content gp:K = bnfinit(y^18 - 30*y^15 + 228*y^12 - 1114*y^9 + 1059*y^6 + 192*y^3 + 64, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 30*x^15 + 228*x^12 - 1114*x^9 + 1059*x^6 + 192*x^3 + 64);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 30*x^15 + 228*x^12 - 1114*x^9 + 1059*x^6 + 192*x^3 + 64)
 

\( x^{18} - 30x^{15} + 228x^{12} - 1114x^{9} + 1059x^{6} + 192x^{3} + 64 \) 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:  $[2, 8]$
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:   \(2137750540154895153454511280309\) \(\medspace = 3^{37}\cdot 7^{15}\) 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:  \(48.42\)
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:  \(\Q(\sqrt{21}) \)
$\Aut(K/\Q)$:   $C_2$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{12}a^{9}+\frac{1}{4}a^{3}+\frac{1}{3}$, $\frac{1}{12}a^{10}-\frac{1}{4}a^{4}-\frac{1}{6}a$, $\frac{1}{12}a^{11}-\frac{1}{4}a^{5}-\frac{1}{6}a^{2}$, $\frac{1}{240}a^{12}+\frac{1}{240}a^{9}-\frac{3}{16}a^{6}-\frac{53}{240}a^{3}-\frac{13}{30}$, $\frac{1}{480}a^{13}+\frac{1}{480}a^{10}+\frac{5}{32}a^{7}+\frac{67}{480}a^{4}-\frac{13}{60}a$, $\frac{1}{2880}a^{14}+\frac{1}{1440}a^{13}+\frac{1}{720}a^{12}+\frac{1}{2880}a^{11}+\frac{1}{1440}a^{10}+\frac{1}{720}a^{9}+\frac{7}{64}a^{8}+\frac{7}{32}a^{7}-\frac{1}{16}a^{6}+\frac{547}{2880}a^{5}-\frac{173}{1440}a^{4}+\frac{187}{720}a^{3}+\frac{137}{360}a^{2}+\frac{47}{180}a-\frac{43}{90}$, $\frac{1}{57600}a^{15}-\frac{7}{57600}a^{12}+\frac{1267}{57600}a^{9}-\frac{9173}{57600}a^{6}-\frac{59}{720}a^{3}-\frac{437}{900}$, $\frac{1}{57600}a^{16}-\frac{7}{57600}a^{13}+\frac{1267}{57600}a^{10}-\frac{9173}{57600}a^{7}-\frac{59}{720}a^{4}-\frac{437}{900}a$, $\frac{1}{57600}a^{17}-\frac{7}{57600}a^{14}+\frac{1267}{57600}a^{11}+\frac{5227}{57600}a^{8}-\frac{59}{720}a^{5}+\frac{119}{450}a^{2}$ 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:  $C_{2}\times C_{12}$, which has order $24$ (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:  $9$
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:   $\frac{149}{57600}a^{15}-\frac{4403}{57600}a^{12}+\frac{31823}{57600}a^{9}-\frac{149977}{57600}a^{6}+\frac{343}{144}a^{3}+\frac{47}{900}$, $\frac{287}{57600}a^{15}-\frac{8249}{57600}a^{12}+\frac{54989}{57600}a^{9}-\frac{249451}{57600}a^{6}-\frac{19}{720}a^{3}-\frac{79}{900}$, $\frac{1}{19200}a^{15}-\frac{7}{19200}a^{12}-\frac{111}{6400}a^{9}+\frac{427}{19200}a^{6}+\frac{1}{240}a^{3}-\frac{479}{100}$, $\frac{599}{19200}a^{17}+\frac{259}{14400}a^{16}-\frac{17993}{19200}a^{14}-\frac{7663}{14400}a^{13}+\frac{45711}{6400}a^{11}+\frac{55903}{14400}a^{10}-\frac{668827}{19200}a^{8}-\frac{265757}{14400}a^{7}+\frac{15883}{480}a^{5}+\frac{16397}{1440}a^{4}+\frac{1079}{100}a^{2}+\frac{7393}{900}a+1$, $\frac{133}{19200}a^{16}-\frac{1337}{6400}a^{13}+\frac{31031}{19200}a^{10}-\frac{155009}{19200}a^{7}+\frac{1449}{160}a^{4}-\frac{154}{75}a+2$, $\frac{6617}{57600}a^{17}+\frac{1679}{19200}a^{16}+\frac{2507}{57600}a^{15}-\frac{197839}{57600}a^{14}-\frac{151939}{57600}a^{13}-\frac{75949}{57600}a^{12}+\frac{496073}{19200}a^{11}+\frac{1172399}{57600}a^{10}+\frac{197723}{19200}a^{9}-\frac{7208941}{57600}a^{8}-\frac{1924867}{19200}a^{7}-\frac{2944711}{57600}a^{6}+\frac{77479}{720}a^{5}+\frac{30511}{288}a^{4}+\frac{42457}{720}a^{3}+\frac{5911}{150}a^{2}+\frac{5593}{450}a-\frac{853}{300}$, $\frac{331}{14400}a^{17}-\frac{53}{1600}a^{16}+\frac{1397}{57600}a^{15}-\frac{2543}{3600}a^{14}+\frac{14419}{14400}a^{13}-\frac{14113}{19200}a^{12}+\frac{13787}{2400}a^{11}-\frac{112079}{14400}a^{10}+\frac{331039}{57600}a^{9}-\frac{105947}{3600}a^{8}+\frac{61969}{1600}a^{7}-\frac{1643881}{57600}a^{6}+\frac{124199}{2880}a^{5}-\frac{3169}{72}a^{4}+\frac{496}{15}a^{3}-\frac{8287}{600}a^{2}+\frac{769}{225}a+\frac{421}{900}$, $\frac{1787}{57600}a^{17}-\frac{17}{800}a^{16}+\frac{3}{200}a^{15}-\frac{53969}{57600}a^{14}+\frac{1517}{2400}a^{13}-\frac{263}{600}a^{12}+\frac{417869}{57600}a^{11}-\frac{11257}{2400}a^{10}+\frac{1853}{600}a^{9}-\frac{2063251}{57600}a^{8}+\frac{18141}{800}a^{7}-\frac{2919}{200}a^{6}+\frac{111857}{2880}a^{5}-\frac{271}{15}a^{4}+\frac{367}{60}a^{3}+\frac{8257}{1800}a^{2}-\frac{667}{150}a+\frac{211}{75}$, $\frac{1}{300}a^{17}-\frac{367}{28800}a^{16}-\frac{533}{57600}a^{15}-\frac{733}{7200}a^{14}+\frac{10949}{28800}a^{13}+\frac{1739}{6400}a^{12}+\frac{5843}{7200}a^{11}-\frac{27403}{9600}a^{10}-\frac{111391}{57600}a^{9}-\frac{9709}{2400}a^{8}+\frac{404591}{28800}a^{7}+\frac{522409}{57600}a^{6}+\frac{6973}{1440}a^{5}-\frac{3655}{288}a^{4}-\frac{43}{8}a^{3}+\frac{829}{225}a^{2}+\frac{241}{300}a-\frac{949}{900}$ 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:  \( 160334625.348843 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 160334625.348843 \cdot 12}{2\cdot\sqrt{2137750540154895153454511280309}}\cr\approx \mathstrut & 6.39291599168281 \end{aligned}\] (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 - 30*x^15 + 228*x^12 - 1114*x^9 + 1059*x^6 + 192*x^3 + 64) 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 - 30*x^15 + 228*x^12 - 1114*x^9 + 1059*x^6 + 192*x^3 + 64, 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 - 30*x^15 + 228*x^12 - 1114*x^9 + 1059*x^6 + 192*x^3 + 64); 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 - 30*x^15 + 228*x^12 - 1114*x^9 + 1059*x^6 + 192*x^3 + 64); 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

\(\Q(\sqrt{21}) \), 3.1.11907.2, 3.1.1323.1, 3.1.243.1, 3.1.11907.1, 6.2.2977309629.3, 6.2.60761421.1, 6.2.36756909.1, 6.2.2977309629.4, 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: 18.0.712583513384965051151503760103.1
Minimal sibling: 18.0.712583513384965051151503760103.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{9}$ R ${\href{/padicField/5.2.0.1}{2} }^{8}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ R ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(3\) Copy content Toggle raw display 3.1.18.37c4.77$x^{18} + 9 x^{8} + 3 x^{6} + 18 x^{4} + 9 x^{2} + 6$$18$$1$$37$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)