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Properties

Label 18.0.24892779122...8272.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 7^{14}\cdot 29^{9}$
Root discriminant $257.57$
Ramified primes $2, 3, 7, 29$
Class number $137168640$ (GRH)
Class group $[6, 6, 504, 7560]$ (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![52082479009, -33061226361, 26131311780, -843969360, 2052116871, -346339797, 676780368, -44832291, 125898486, -3913117, 11001684, -243291, 513402, -15969, 12207, -264, 156, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 156*x^16 - 264*x^15 + 12207*x^14 - 15969*x^13 + 513402*x^12 - 243291*x^11 + 11001684*x^10 - 3913117*x^9 + 125898486*x^8 - 44832291*x^7 + 676780368*x^6 - 346339797*x^5 + 2052116871*x^4 - 843969360*x^3 + 26131311780*x^2 - 33061226361*x + 52082479009)
 
gp: K = bnfinit(x^18 - 3*x^17 + 156*x^16 - 264*x^15 + 12207*x^14 - 15969*x^13 + 513402*x^12 - 243291*x^11 + 11001684*x^10 - 3913117*x^9 + 125898486*x^8 - 44832291*x^7 + 676780368*x^6 - 346339797*x^5 + 2052116871*x^4 - 843969360*x^3 + 26131311780*x^2 - 33061226361*x + 52082479009, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 156 x^{16} - 264 x^{15} + 12207 x^{14} - 15969 x^{13} + 513402 x^{12} - 243291 x^{11} + 11001684 x^{10} - 3913117 x^{9} + 125898486 x^{8} - 44832291 x^{7} + 676780368 x^{6} - 346339797 x^{5} + 2052116871 x^{4} - 843969360 x^{3} + 26131311780 x^{2} - 33061226361 x + 52082479009 \)

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:  \(-24892779122496057347515050710073813084598272=-\,2^{12}\cdot 3^{31}\cdot 7^{14}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $257.57$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{2}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{63} a^{15} + \frac{1}{63} a^{14} + \frac{1}{21} a^{13} + \frac{1}{21} a^{12} + \frac{1}{63} a^{11} - \frac{1}{21} a^{10} - \frac{10}{63} a^{9} - \frac{1}{21} a^{8} - \frac{2}{21} a^{7} + \frac{4}{63} a^{6} - \frac{5}{63} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{63} a^{2} + \frac{1}{7} a + \frac{29}{63}$, $\frac{1}{19089} a^{16} - \frac{11}{2727} a^{15} + \frac{185}{6363} a^{14} + \frac{95}{2727} a^{13} + \frac{845}{19089} a^{12} - \frac{10}{707} a^{11} - \frac{407}{2727} a^{10} - \frac{98}{2727} a^{9} - \frac{124}{2121} a^{8} - \frac{3014}{19089} a^{7} + \frac{229}{19089} a^{6} + \frac{215}{909} a^{5} - \frac{3016}{19089} a^{4} + \frac{6848}{19089} a^{3} - \frac{60}{707} a^{2} + \frac{8581}{19089} a + \frac{44}{189}$, $\frac{1}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{17} + \frac{5866631328286552475741225390259839763755922806216808047919460475477695423765}{263081942853793809110507451642693729503426680424491142973445690211722387703744409} a^{16} - \frac{12068669613879118364711016547259140316586347140156412642875395596781466006819169}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{15} + \frac{685505302595663445666152412495564441495904687899778341067409767807343924124046}{28652488825660711883322593743263673510274192917518837353543590023058873908328599} a^{14} + \frac{94422386128857442445973198170314549863714456366985416774900729976450578341183875}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{13} + \frac{45395953979485150539019960471673809181896370643772879895910355465747588309774412}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{12} + \frac{11586820771857110737110288185105182157243105473521134981652900596736038616560831}{263081942853793809110507451642693729503426680424491142973445690211722387703744409} a^{11} - \frac{17465324821060574574474586925238119272827077214182491747953672781374818351537653}{413414481627390271459368852581375860648241926381343224672557513189849466391598357} a^{10} - \frac{17499325208641165883800290601463339313175288178284046099438935763042918869642105}{263081942853793809110507451642693729503426680424491142973445690211722387703744409} a^{9} + \frac{408110144302344222141104297464844330819316238331442162241294742826391377288069835}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{8} + \frac{117278474031005743872180511422568312852133925417754795647943767563965509727787549}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{7} - \frac{173829196350483317998344434865858455625157748905056540664381270087367218979009270}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{6} + \frac{116053124288079697644113756304274562162768754380964094278460397408468352264105824}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{5} - \frac{1291958475307833398930532851748659681500796297348254985770584384379202248331274083}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{4} - \frac{1006612376042047664561466025613888828804894647755320380132601636998994468869763014}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{3} - \frac{980883259962435055958378883125221688366275513895406906924486071637560657506256084}{2893901371391731900215581968069631024537693484669402572707902592328946264741188499} a^{2} - \frac{44363430778408505549382516182077743040960794672479080947459722556541687446166582}{413414481627390271459368852581375860648241926381343224672557513189849466391598357} a - \frac{1071194274853441503118458308592866779018792713833604265091929986404538231997868}{2604771711423701080302053976660333955479472083410803395776690002096261264393509}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{504}\times C_{7560}$, which has order $137168640$ (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:  \( 4695974.091249611 \) (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

\(\Q(\sqrt{-87}) \), 3.3.3969.2, 3.3.756.1, 6.0.1152596897487.9, 6.0.41817574512.4, 9.9.756284282720064.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.12.6.1$x^{12} + 146334 x^{6} - 20511149 x^{2} + 5353409889$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$