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Properties

Label 18.0.11529697427...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{27}\cdot 5^{9}\cdot 37^{14}$
Root discriminant $686.70$
Ramified primes $2, 3, 5, 37$
Class number $9299419272$ (GRH)
Class group $[6, 6, 258317202]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2359097078790079, 954995916305826, 349997531677587, 60709569882666, 11209338881445, 1003108842378, 222938790160, 23739397950, 5992436430, 334422428, 62693295, -49284, 1053253, 43290, 12087, -518, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 48*x^16 - 518*x^15 + 12087*x^14 + 43290*x^13 + 1053253*x^12 - 49284*x^11 + 62693295*x^10 + 334422428*x^9 + 5992436430*x^8 + 23739397950*x^7 + 222938790160*x^6 + 1003108842378*x^5 + 11209338881445*x^4 + 60709569882666*x^3 + 349997531677587*x^2 + 954995916305826*x + 2359097078790079)
 
gp: K = bnfinit(x^18 + 48*x^16 - 518*x^15 + 12087*x^14 + 43290*x^13 + 1053253*x^12 - 49284*x^11 + 62693295*x^10 + 334422428*x^9 + 5992436430*x^8 + 23739397950*x^7 + 222938790160*x^6 + 1003108842378*x^5 + 11209338881445*x^4 + 60709569882666*x^3 + 349997531677587*x^2 + 954995916305826*x + 2359097078790079, 1)
 

Normalized defining polynomial

\( x^{18} + 48 x^{16} - 518 x^{15} + 12087 x^{14} + 43290 x^{13} + 1053253 x^{12} - 49284 x^{11} + 62693295 x^{10} + 334422428 x^{9} + 5992436430 x^{8} + 23739397950 x^{7} + 222938790160 x^{6} + 1003108842378 x^{5} + 11209338881445 x^{4} + 60709569882666 x^{3} + 349997531677587 x^{2} + 954995916305826 x + 2359097078790079 \)

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:  \(-1152969742748338361522490256327499776524288000000000=-\,2^{33}\cdot 3^{27}\cdot 5^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $686.70$
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:  $2, 3, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{222} a^{12} - \frac{5}{222} a^{10} + \frac{23}{74} a^{8} - \frac{1}{6} a^{7} + \frac{59}{222} a^{6} + \frac{17}{74} a^{4} + \frac{1}{6} a^{3} + \frac{19}{74} a^{2} + \frac{1}{6} a - \frac{11}{222}$, $\frac{1}{222} a^{13} - \frac{5}{222} a^{11} - \frac{5}{222} a^{9} - \frac{1}{6} a^{8} + \frac{59}{222} a^{7} + \frac{1}{3} a^{6} + \frac{17}{74} a^{5} + \frac{1}{6} a^{4} + \frac{19}{74} a^{3} + \frac{1}{6} a^{2} - \frac{11}{222} a - \frac{1}{3}$, $\frac{1}{222} a^{14} + \frac{7}{222} a^{10} + \frac{71}{222} a^{8} - \frac{1}{6} a^{7} - \frac{4}{37} a^{6} - \frac{1}{3} a^{5} + \frac{15}{37} a^{4} - \frac{1}{2} a^{3} + \frac{26}{111} a^{2} + \frac{1}{6} a - \frac{3}{37}$, $\frac{1}{666} a^{15} + \frac{1}{666} a^{14} + \frac{1}{666} a^{13} + \frac{1}{666} a^{12} + \frac{13}{222} a^{11} + \frac{1}{333} a^{10} + \frac{29}{666} a^{9} + \frac{251}{666} a^{8} + \frac{4}{37} a^{7} - \frac{13}{222} a^{6} + \frac{67}{666} a^{5} - \frac{22}{333} a^{4} - \frac{38}{333} a^{3} + \frac{110}{333} a^{2} - \frac{59}{222} a - \frac{251}{666}$, $\frac{1}{666} a^{16} - \frac{1}{666} a^{12} - \frac{1}{18} a^{11} - \frac{103}{333} a^{8} + \frac{1}{3} a^{7} + \frac{25}{666} a^{6} - \frac{1}{6} a^{5} - \frac{23}{666} a^{4} + \frac{5}{18} a^{3} + \frac{22}{333} a^{2} + \frac{7}{18} a - \frac{104}{333}$, $\frac{1}{47891350184100167802884414800644795540430966445934548023052394822889219579734} a^{17} + \frac{7924883524436320319730286295418196715084442894909523865896613948843668343}{15963783394700055934294804933548265180143655481978182674350798274296406526578} a^{16} + \frac{5889091743142934693796923691746901061070615171130535450050924647017737023}{23945675092050083901442207400322397770215483222967274011526197411444609789867} a^{15} - \frac{48423202500018863179140385248138785802900537394975280683216715651405937485}{47891350184100167802884414800644795540430966445934548023052394822889219579734} a^{14} - \frac{7924674976741282264073223325074914676610727918197112664241269066170962783}{15963783394700055934294804933548265180143655481978182674350798274296406526578} a^{13} - \frac{18619441486642573040321953726655407601827093616309016579232209936721994065}{15963783394700055934294804933548265180143655481978182674350798274296406526578} a^{12} - \frac{70040640719590798410624312865361426238308252603477419778571057799983937129}{5321261131566685311431601644516088393381218493992727558116932758098802175526} a^{11} - \frac{1950200183059470978788497657920759374638776795335589722975641689587169969955}{47891350184100167802884414800644795540430966445934548023052394822889219579734} a^{10} - \frac{243700469990439437505171117039070037007761727612305386903931601252552947455}{15963783394700055934294804933548265180143655481978182674350798274296406526578} a^{9} + \frac{19596032794494467042476958559480230580708879493555058757156379759086564015305}{47891350184100167802884414800644795540430966445934548023052394822889219579734} a^{8} - \frac{2323851081935772643532005023455330021708197894688616216451929788732873593463}{47891350184100167802884414800644795540430966445934548023052394822889219579734} a^{7} + \frac{2261506266886079719100019722874711604078566260931879022912867668120911007517}{15963783394700055934294804933548265180143655481978182674350798274296406526578} a^{6} - \frac{12619639069006813782018362884539426223096590544116245983607488078736075734417}{47891350184100167802884414800644795540430966445934548023052394822889219579734} a^{5} + \frac{2395176888865205427004351811901480894867557416584340739089100317991120542957}{5321261131566685311431601644516088393381218493992727558116932758098802175526} a^{4} - \frac{9476213505737362129617360582476571479004541829082947207027789592990784428463}{23945675092050083901442207400322397770215483222967274011526197411444609789867} a^{3} - \frac{7717530800264129030131115557209100484552558480264835199090049343147990203570}{23945675092050083901442207400322397770215483222967274011526197411444609789867} a^{2} + \frac{22264668593029357722763525195031026150375555503007097990046526400125051164641}{47891350184100167802884414800644795540430966445934548023052394822889219579734} a + \frac{11897114580322744438430337526868401088197242821326415857140181399582868477021}{23945675092050083901442207400322397770215483222967274011526197411444609789867}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{258317202}$, which has order $9299419272$ (assuming GRH)

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:  \( -1 \) (order $2$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 118546543.87559307 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-30}) \), 3.3.110889.1, 3.3.148.1, 6.0.2360903101632000.1, 6.0.9462528000.12, 9.9.3228844269788073792.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: 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 R R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
2Data not computed
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$