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Properties

Label 18.0.23267717097...0976.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{9}\cdot 19^{16}$
Root discriminant $376.64$
Ramified primes $2, 3, 7, 19$
Class number $2667636288$ (GRH)
Class group $[2, 2, 12, 55575756]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![30985776501229, 345736733040, 4852747373220, 246611628030, 398164960137, 22617048924, 21242025193, 1165841862, 787133106, 32240074, 21240672, 501372, 409951, 0, 6471, -76, 75, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 75*x^16 - 76*x^15 + 6471*x^14 + 409951*x^12 + 501372*x^11 + 21240672*x^10 + 32240074*x^9 + 787133106*x^8 + 1165841862*x^7 + 21242025193*x^6 + 22617048924*x^5 + 398164960137*x^4 + 246611628030*x^3 + 4852747373220*x^2 + 345736733040*x + 30985776501229)
 
gp: K = bnfinit(x^18 + 75*x^16 - 76*x^15 + 6471*x^14 + 409951*x^12 + 501372*x^11 + 21240672*x^10 + 32240074*x^9 + 787133106*x^8 + 1165841862*x^7 + 21242025193*x^6 + 22617048924*x^5 + 398164960137*x^4 + 246611628030*x^3 + 4852747373220*x^2 + 345736733040*x + 30985776501229, 1)
 

Normalized defining polynomial

\( x^{18} + 75 x^{16} - 76 x^{15} + 6471 x^{14} + 409951 x^{12} + 501372 x^{11} + 21240672 x^{10} + 32240074 x^{9} + 787133106 x^{8} + 1165841862 x^{7} + 21242025193 x^{6} + 22617048924 x^{5} + 398164960137 x^{4} + 246611628030 x^{3} + 4852747373220 x^{2} + 345736733040 x + 30985776501229 \)

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:  \(-23267717097701665308512958392073692044388990976=-\,2^{18}\cdot 3^{27}\cdot 7^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $376.64$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(3275,·)$, $\chi_{4788}(2183,·)$, $\chi_{4788}(841,·)$, $\chi_{4788}(3863,·)$, $\chi_{4788}(1681,·)$, $\chi_{4788}(1429,·)$, $\chi_{4788}(2519,·)$, $\chi_{4788}(1175,·)$, $\chi_{4788}(2015,·)$, $\chi_{4788}(3361,·)$, $\chi_{4788}(4451,·)$, $\chi_{4788}(1261,·)$, $\chi_{4788}(2353,·)$, $\chi_{4788}(2099,·)$, $\chi_{4788}(3445,·)$, $\chi_{4788}(1847,·)$, $\chi_{4788}(505,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{33} a^{10} - \frac{1}{33} a^{9} + \frac{4}{11} a^{8} + \frac{10}{33} a^{7} - \frac{10}{33} a^{6} - \frac{4}{11} a^{5} + \frac{4}{11} a^{4} - \frac{4}{11} a^{3} + \frac{4}{11} a^{2} - \frac{1}{33} a - \frac{1}{3}$, $\frac{1}{33} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{2} - \frac{4}{11} a$, $\frac{1}{33} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{4}{11} a^{2} - \frac{1}{3}$, $\frac{1}{33} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{4}{11} a^{3} - \frac{1}{3} a$, $\frac{1}{33} a^{14} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{4}{11} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{113553} a^{15} - \frac{1630}{113553} a^{14} - \frac{1196}{113553} a^{13} - \frac{523}{113553} a^{12} + \frac{419}{37851} a^{11} - \frac{920}{113553} a^{10} + \frac{14252}{113553} a^{9} - \frac{53555}{113553} a^{8} - \frac{3496}{12617} a^{7} - \frac{17650}{37851} a^{6} - \frac{27428}{113553} a^{5} + \frac{23183}{113553} a^{4} + \frac{6692}{113553} a^{3} - \frac{34288}{113553} a^{2} - \frac{11138}{37851} a - \frac{68}{333}$, $\frac{1}{1132852079601} a^{16} - \frac{332033}{1132852079601} a^{15} + \frac{9522607538}{1132852079601} a^{14} - \frac{14909769950}{1132852079601} a^{13} - \frac{7268085194}{1132852079601} a^{12} - \frac{10234087115}{1132852079601} a^{11} - \frac{13601243951}{1132852079601} a^{10} - \frac{37593960967}{1132852079601} a^{9} - \frac{277412722750}{1132852079601} a^{8} + \frac{140625445150}{377617359867} a^{7} - \frac{339572428469}{1132852079601} a^{6} - \frac{467066565068}{1132852079601} a^{5} + \frac{48789716691}{125872453289} a^{4} - \frac{18491201371}{377617359867} a^{3} + \frac{4940681443}{30617623773} a^{2} - \frac{282702230572}{1132852079601} a - \frac{159043792}{3322146861}$, $\frac{1}{5084405970395203476884011187683908799166059962423522656975811654417} a^{17} - \frac{266871233333532561695364694752080196177806207983744780}{1694801990131734492294670395894636266388686654141174218991937218139} a^{16} - \frac{17152499800322205803308014846327010985675110408354464751692133}{5084405970395203476884011187683908799166059962423522656975811654417} a^{15} - \frac{6424197534210994195233812371329568833776681530926083649736596238}{462218724581382134262182835243991709015096360220320241543255604947} a^{14} - \frac{64597358223675357170112434509690702710571779350930722325592175451}{5084405970395203476884011187683908799166059962423522656975811654417} a^{13} - \frac{71352155817884030898670707700814064265531024870755624314659433587}{5084405970395203476884011187683908799166059962423522656975811654417} a^{12} - \frac{2048281941696565870523910230728470713385500747858176140046693179}{564933996710578164098223465298212088796228884713724739663979072713} a^{11} - \frac{18000531438530520824265739312458592419582148298048347310823267052}{5084405970395203476884011187683908799166059962423522656975811654417} a^{10} - \frac{731542523649154904511617817101866824297108518957873082035335811266}{5084405970395203476884011187683908799166059962423522656975811654417} a^{9} + \frac{2223506736587121740250330142839044544230682583368942499069525692728}{5084405970395203476884011187683908799166059962423522656975811654417} a^{8} - \frac{1887951160866572167288033690231197552684465461113680569003431903385}{5084405970395203476884011187683908799166059962423522656975811654417} a^{7} + \frac{75682151031332781190789718475595824558076672906065337037943193441}{1694801990131734492294670395894636266388686654141174218991937218139} a^{6} + \frac{13636813712338919252952851293864628987051162318362967666556234455}{51357636064598014918020315027110189890566262246702249060361733883} a^{5} + \frac{137105731574219844172123561061564620230555396963162615874072220006}{5084405970395203476884011187683908799166059962423522656975811654417} a^{4} + \frac{46364371610719392813404087975226878642233900628866795222954394638}{154072908193794044754060945081330569671698786740106747181085201649} a^{3} + \frac{287210754441081289700318777944202880360061823082297367618507274131}{1694801990131734492294670395894636266388686654141174218991937218139} a^{2} - \frac{1731607789865658468263857811141087723203759181642678075813787639130}{5084405970395203476884011187683908799166059962423522656975811654417} a - \frac{5081484371634156052452911030082249865724647185779607430423311511}{14910281438109101105231704362709409968228914845816781985266309837}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{12}\times C_{55575756}$, which has order $2667636288$ (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:  \( 15010229.973756868 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-21}) \), 3.3.361.1, 6.0.77241777984.6, 9.9.9025761726072081.1

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

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ $18$ $18$

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
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$