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Properties

Label 18.0.13571925211...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{33}\cdot 5^{12}$
Root discriminant $21.91$
Ramified primes $3, 5$
Class number $1$
Class group Trivial
Galois group $C_2\times C_3:S_3.C_2$ (as 18T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![144, -540, 1053, -1917, 2997, -3294, 3186, -2997, 2070, -1164, 765, -378, 144, -81, 36, -18, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 18*x^15 + 36*x^14 - 81*x^13 + 144*x^12 - 378*x^11 + 765*x^10 - 1164*x^9 + 2070*x^8 - 2997*x^7 + 3186*x^6 - 3294*x^5 + 2997*x^4 - 1917*x^3 + 1053*x^2 - 540*x + 144)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 - 18*x^15 + 36*x^14 - 81*x^13 + 144*x^12 - 378*x^11 + 765*x^10 - 1164*x^9 + 2070*x^8 - 2997*x^7 + 3186*x^6 - 3294*x^5 + 2997*x^4 - 1917*x^3 + 1053*x^2 - 540*x + 144, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} - 18 x^{15} + 36 x^{14} - 81 x^{13} + 144 x^{12} - 378 x^{11} + 765 x^{10} - 1164 x^{9} + 2070 x^{8} - 2997 x^{7} + 3186 x^{6} - 3294 x^{5} + 2997 x^{4} - 1917 x^{3} + 1053 x^{2} - 540 x + 144 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1357192521131719482421875=-\,3^{33}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.91$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{9} a^{10} + \frac{1}{3} a$, $\frac{1}{9} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{10} - \frac{1}{2} a^{7} + \frac{1}{6} a^{4} - \frac{1}{6} a$, $\frac{1}{90} a^{14} - \frac{1}{45} a^{13} - \frac{1}{18} a^{11} + \frac{1}{45} a^{10} - \frac{1}{45} a^{9} + \frac{1}{30} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{11}{30} a^{5} + \frac{1}{3} a^{4} + \frac{2}{5} a^{3} + \frac{13}{30} a^{2} + \frac{1}{15} a - \frac{1}{15}$, $\frac{1}{270} a^{15} + \frac{1}{45} a^{13} - \frac{1}{18} a^{12} + \frac{2}{45} a^{11} + \frac{2}{45} a^{10} + \frac{1}{30} a^{9} + \frac{2}{5} a^{8} - \frac{1}{3} a^{7} + \frac{13}{90} a^{6} - \frac{2}{15} a^{5} + \frac{7}{15} a^{4} - \frac{11}{30} a^{3} - \frac{7}{15} a^{2} + \frac{2}{15} a + \frac{2}{5}$, $\frac{1}{5130} a^{16} - \frac{4}{2565} a^{15} + \frac{4}{855} a^{14} + \frac{3}{190} a^{13} - \frac{2}{45} a^{12} - \frac{1}{95} a^{11} + \frac{43}{1710} a^{10} + \frac{1}{19} a^{9} + \frac{2}{57} a^{8} + \frac{47}{342} a^{7} + \frac{1}{171} a^{6} + \frac{1}{57} a^{5} - \frac{51}{190} a^{4} + \frac{103}{285} a^{3} - \frac{36}{95} a^{2} - \frac{4}{285} a - \frac{8}{95}$, $\frac{1}{1834359657660} a^{17} - \frac{57506683}{1834359657660} a^{16} - \frac{787161709}{611453219220} a^{15} + \frac{112776859}{61145321922} a^{14} + \frac{2483440021}{101908869870} a^{13} + \frac{678727577}{611453219220} a^{12} + \frac{3231832831}{152863304805} a^{11} - \frac{392772163}{16984811645} a^{10} + \frac{19795476091}{611453219220} a^{9} + \frac{25577046997}{152863304805} a^{8} + \frac{67054118564}{152863304805} a^{7} + \frac{1200492861}{13587849316} a^{6} + \frac{6134531519}{101908869870} a^{5} - \frac{1938755809}{16984811645} a^{4} - \frac{54175531099}{203817739740} a^{3} - \frac{16186106341}{40763547948} a^{2} + \frac{2739691493}{67939246580} a + \frac{5131205593}{50954434935}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{52746977}{2145449892} a^{17} - \frac{119291039}{2145449892} a^{16} + \frac{390991081}{2145449892} a^{15} - \frac{1001856473}{3218174838} a^{14} + \frac{239149687}{357574982} a^{13} - \frac{9757006351}{6436349676} a^{12} + \frac{441577319}{178787491} a^{11} - \frac{12181855252}{1609087419} a^{10} + \frac{9567266203}{715149964} a^{9} - \frac{3450465400}{178787491} a^{8} + \frac{6699319196}{178787491} a^{7} - \frac{33823767875}{715149964} a^{6} + \frac{49327961971}{1072724946} a^{5} - \frac{8913561072}{178787491} a^{4} + \frac{85484138417}{2145449892} a^{3} - \frac{15012043171}{715149964} a^{2} + \frac{28238962031}{2145449892} a - \frac{856241244}{178787491} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 327769.649962 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3.C_2$ (as 18T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $C_2\times C_3:S_3.C_2$
Character table for $C_2\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 9.1.672605015625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 18 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$