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Properties

Label 18.2.458...125.2
Degree $18$
Signature $[2, 8]$
Discriminant $4.581\times 10^{27}$
Root discriminant \(34.41\)
Ramified primes $3,5$
Class number $2$ (GRH)
Class group [2] (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 - 12*x^15 + 45*x^12 + 40*x^9 - 45*x^6 + 228*x^3 - 1)
 
Copy content gp:K = bnfinit(y^18 - 12*y^15 + 45*y^12 + 40*y^9 - 45*y^6 + 228*y^3 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 12*x^15 + 45*x^12 + 40*x^9 - 45*x^6 + 228*x^3 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 12*x^15 + 45*x^12 + 40*x^9 - 45*x^6 + 228*x^3 - 1)
 

\( x^{18} - 12x^{15} + 45x^{12} + 40x^{9} - 45x^{6} + 228x^{3} - 1 \) 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:   \(4580524758819553253173828125\) \(\medspace = 3^{36}\cdot 5^{15}\) 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:  \(34.41\)
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:  $3^{37/18}5^{5/6}\approx 36.57836164674376$
Ramified primes:   \(3\), \(5\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{6}-\frac{1}{8}a^{3}+\frac{1}{8}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{7}-\frac{1}{8}a^{4}+\frac{1}{8}a$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{8}a^{5}+\frac{1}{8}a^{2}$, $\frac{1}{32}a^{12}+\frac{1}{16}a^{6}-\frac{3}{32}$, $\frac{1}{32}a^{13}+\frac{1}{16}a^{7}-\frac{3}{32}a$, $\frac{1}{32}a^{14}+\frac{1}{16}a^{8}-\frac{3}{32}a^{2}$, $\frac{1}{2176}a^{15}+\frac{1}{128}a^{12}-\frac{3}{1088}a^{9}-\frac{67}{1088}a^{6}+\frac{421}{2176}a^{3}-\frac{619}{2176}$, $\frac{1}{2176}a^{16}+\frac{1}{128}a^{13}-\frac{3}{1088}a^{10}-\frac{67}{1088}a^{7}+\frac{421}{2176}a^{4}-\frac{619}{2176}a$, $\frac{1}{2176}a^{17}+\frac{1}{128}a^{14}-\frac{3}{1088}a^{11}-\frac{67}{1088}a^{8}+\frac{421}{2176}a^{5}-\frac{619}{2176}a^{2}$ 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:  $C_{2}$, which has order $2$ (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:  $C_{2}$, which has order $2$ (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:   $\frac{21}{2176}a^{15}-\frac{15}{128}a^{12}+\frac{481}{1088}a^{9}+\frac{429}{1088}a^{6}-\frac{951}{2176}a^{3}+\frac{3525}{2176}$, $\frac{1349}{2176}a^{17}-\frac{951}{128}a^{14}+\frac{30225}{1088}a^{11}+\frac{27461}{1088}a^{8}-\frac{59847}{2176}a^{5}+\frac{306621}{2176}a^{2}$, $\frac{467}{2176}a^{17}-\frac{329}{128}a^{14}+\frac{10431}{1088}a^{11}+\frac{9715}{1088}a^{8}-\frac{21809}{2176}a^{5}+\frac{107027}{2176}a^{2}$, $\frac{43}{544}a^{17}+\frac{43}{1088}a^{16}-\frac{19}{2176}a^{15}-\frac{31}{32}a^{14}-\frac{31}{64}a^{13}+\frac{13}{128}a^{12}+\frac{1027}{272}a^{11}+\frac{1027}{544}a^{10}-\frac{351}{1088}a^{9}+\frac{689}{272}a^{8}+\frac{689}{544}a^{7}-\frac{767}{1088}a^{6}-\frac{2977}{544}a^{5}-\frac{3249}{1088}a^{4}+\frac{1521}{2176}a^{3}+\frac{10205}{544}a^{2}+\frac{9389}{1088}a+\frac{65}{2176}$, $\frac{959}{2176}a^{17}+\frac{9}{128}a^{16}+\frac{21}{2176}a^{15}-\frac{677}{128}a^{14}-\frac{107}{128}a^{13}-\frac{15}{128}a^{12}+\frac{21603}{1088}a^{11}+\frac{197}{64}a^{10}+\frac{481}{1088}a^{9}+\frac{18911}{1088}a^{8}+\frac{193}{64}a^{7}+\frac{429}{1088}a^{6}-\frac{41253}{2176}a^{5}-\frac{243}{128}a^{4}-\frac{951}{2176}a^{3}+\frac{220951}{2176}a^{2}+\frac{1929}{128}a+\frac{5701}{2176}$, $\frac{729}{2176}a^{17}-\frac{515}{128}a^{14}+\frac{16445}{1088}a^{11}+\frac{14465}{1088}a^{8}-\frac{33907}{2176}a^{5}+\frac{168977}{2176}a^{2}-2$, $\frac{1937}{2176}a^{17}+\frac{173}{1088}a^{16}+\frac{21}{2176}a^{15}-\frac{1367}{128}a^{14}-\frac{121}{64}a^{13}-\frac{15}{128}a^{12}+\frac{43557}{1088}a^{11}+\frac{3765}{544}a^{10}+\frac{481}{1088}a^{9}+\frac{38861}{1088}a^{8}+\frac{4015}{544}a^{7}+\frac{429}{1088}a^{6}-\frac{87291}{2176}a^{5}-\frac{8087}{1088}a^{4}-\frac{951}{2176}a^{3}+\frac{442381}{2176}a^{2}+\frac{36971}{1088}a+\frac{12229}{2176}$, $\frac{991}{1088}a^{17}-\frac{63}{2176}a^{16}-\frac{1}{68}a^{15}-\frac{699}{64}a^{14}+\frac{45}{128}a^{13}+\frac{3}{16}a^{12}+\frac{22255}{544}a^{11}-\frac{1443}{1088}a^{10}-\frac{107}{136}a^{9}+\frac{19861}{544}a^{8}-\frac{1559}{1088}a^{7}-\frac{9}{17}a^{6}-\frac{43421}{1088}a^{5}+\frac{6117}{2176}a^{4}+\frac{297}{136}a^{3}+\frac{226337}{1088}a^{2}-\frac{15471}{2176}a-\frac{1179}{272}$, $\frac{77}{1088}a^{16}-\frac{15}{1088}a^{15}-\frac{55}{64}a^{13}+\frac{11}{64}a^{12}+\frac{1809}{544}a^{10}-\frac{363}{544}a^{9}+\frac{1165}{544}a^{7}-\frac{321}{544}a^{6}-\frac{3215}{1088}a^{4}+\frac{2661}{1088}a^{3}+\frac{19181}{1088}a-\frac{3193}{1088}$ 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:  \( 22847802.33598889 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 22847802.33598889 \cdot 2}{2\cdot\sqrt{4580524758819553253173828125}}\cr\approx \mathstrut & 3.28008995044428 \end{aligned}\] (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 + 45*x^12 + 40*x^9 - 45*x^6 + 228*x^3 - 1) 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 + 45*x^12 + 40*x^9 - 45*x^6 + 228*x^3 - 1, 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 + 45*x^12 + 40*x^9 - 45*x^6 + 228*x^3 - 1); 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 + 45*x^12 + 40*x^9 - 45*x^6 + 228*x^3 - 1); 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

\(\Q(\sqrt{5}) \), 3.1.243.1, 3.1.6075.2, 3.1.675.1, 3.1.6075.1, 6.2.7381125.1, 6.2.2278125.1, 6.2.184528125.1, 6.2.184528125.2, 9.1.6053445140625.1

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(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: 18.0.13741574276458659759521484375.2
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 ${\href{/padicField/2.2.0.1}{2} }^{9}$ R R ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 3.2.9.36b6.16$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
\(5\) Copy content Toggle raw display 5.1.6.5a1.1$x^{6} + 5$$6$$1$$5$$D_{6}$$$[\ ]_{6}^{2}$$
5.2.6.10a1.2$x^{12} + 24 x^{11} + 252 x^{10} + 1520 x^{9} + 5820 x^{8} + 14784 x^{7} + 25376 x^{6} + 29568 x^{5} + 23280 x^{4} + 12160 x^{3} + 4032 x^{2} + 768 x + 69$$6$$2$$10$$D_6$$$[\ ]_{6}^{2}$$

Spectrum of ring of integers

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