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Properties

Label 18.0.11627251210...3216.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 127^{17}$
Root discriminant $1008.41$
Ramified primes $2, 3, 127$
Class number $17737521360$ (GRH)
Class group $[2, 2, 2, 2217190170]$ (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![9792653414400, 0, 39992306079744, 0, 14140144366848, 0, 1757405549952, 0, 89192075616, 0, 2249058240, 0, 30376749, 0, 218313, 0, 762, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 762*x^16 + 218313*x^14 + 30376749*x^12 + 2249058240*x^10 + 89192075616*x^8 + 1757405549952*x^6 + 14140144366848*x^4 + 39992306079744*x^2 + 9792653414400)
 
gp: K = bnfinit(x^18 + 762*x^16 + 218313*x^14 + 30376749*x^12 + 2249058240*x^10 + 89192075616*x^8 + 1757405549952*x^6 + 14140144366848*x^4 + 39992306079744*x^2 + 9792653414400, 1)
 

Normalized defining polynomial

\( x^{18} + 762 x^{16} + 218313 x^{14} + 30376749 x^{12} + 2249058240 x^{10} + 89192075616 x^{8} + 1757405549952 x^{6} + 14140144366848 x^{4} + 39992306079744 x^{2} + 9792653414400 \)

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:  \(-1162725121036380649066150832445210229839758684897673216=-\,2^{18}\cdot 3^{27}\cdot 127^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1008.41$
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:  $2, 3, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4572=2^{2}\cdot 3^{2}\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{4572}(1,·)$, $\chi_{4572}(2183,·)$, $\chi_{4572}(2893,·)$, $\chi_{4572}(3085,·)$, $\chi_{4572}(1487,·)$, $\chi_{4572}(2389,·)$, $\chi_{4572}(1883,·)$, $\chi_{4572}(3107,·)$, $\chi_{4572}(1957,·)$, $\chi_{4572}(361,·)$, $\chi_{4572}(2305,·)$, $\chi_{4572}(4211,·)$, $\chi_{4572}(2267,·)$, $\chi_{4572}(2615,·)$, $\chi_{4572}(2689,·)$, $\chi_{4572}(1465,·)$, $\chi_{4572}(1679,·)$, $\chi_{4572}(4571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{15} a^{6} - \frac{2}{5} a^{2}$, $\frac{1}{30} a^{7} + \frac{3}{10} a^{3} - \frac{1}{2} a$, $\frac{1}{60} a^{8} - \frac{1}{30} a^{6} + \frac{3}{20} a^{4} - \frac{1}{20} a^{2}$, $\frac{1}{120} a^{9} - \frac{1}{60} a^{7} + \frac{3}{40} a^{5} + \frac{19}{40} a^{3} - \frac{1}{2} a$, $\frac{1}{1200} a^{10} - \frac{1}{120} a^{8} + \frac{11}{400} a^{6} + \frac{39}{80} a^{4} - \frac{7}{100} a^{2}$, $\frac{1}{2400} a^{11} - \frac{1}{240} a^{9} + \frac{11}{800} a^{7} + \frac{7}{160} a^{5} - \frac{7}{200} a^{3} + \frac{1}{5} a$, $\frac{1}{288000} a^{12} + \frac{11}{48000} a^{10} + \frac{451}{96000} a^{8} + \frac{1991}{96000} a^{6} - \frac{357}{4000} a^{4} - \frac{831}{2000} a^{2} + \frac{9}{20}$, $\frac{1}{576000} a^{13} + \frac{11}{96000} a^{11} + \frac{451}{192000} a^{9} + \frac{1991}{192000} a^{7} - \frac{357}{8000} a^{5} + \frac{1169}{4000} a^{3} + \frac{9}{40} a$, $\frac{1}{5760000} a^{14} - \frac{1}{576000} a^{12} + \frac{59}{1920000} a^{10} - \frac{2617}{384000} a^{8} - \frac{3811}{480000} a^{6} + \frac{67}{8000} a^{4} - \frac{2123}{5000} a^{2} + \frac{29}{100}$, $\frac{1}{1923840000} a^{15} - \frac{7}{21376000} a^{13} + \frac{95219}{641280000} a^{11} - \frac{15247}{42752000} a^{9} - \frac{21713}{5010000} a^{7} + \frac{61801}{2672000} a^{5} + \frac{1426959}{3340000} a^{3} + \frac{7297}{16700} a$, $\frac{1}{15477301993753248548578672320000000} a^{16} - \frac{304423584645178740436123217}{7738650996876624274289336160000000} a^{14} - \frac{560036284615226108520517541}{5159100664584416182859557440000000} a^{12} + \frac{49356926101961853216910936873}{1719700221528138727619852480000000} a^{10} + \frac{1255393027509118252337479841813}{429925055382034681904963120000000} a^{8} + \frac{120582860642500684132816802247}{6717578990344291904765048750000} a^{6} + \frac{969008884632599105303888730577}{13435157980688583809530097500000} a^{4} - \frac{890714976241433654775792659}{2115106734995054126185468750} a^{2} - \frac{180624958659359347005638477}{402250238942771970345212500}$, $\frac{1}{30954603987506497097157344640000000} a^{17} + \frac{1286596951327957026238283}{15477301993753248548578672320000000} a^{15} - \frac{10690514206163770461294312623}{30954603987506497097157344640000000} a^{13} - \frac{715015067946651155711875664127}{3439400443056277455239704960000000} a^{11} - \frac{3424667827518350791037964031937}{859850110764069363809926240000000} a^{9} + \frac{8732396767370061100946017130231}{644887583073052022857444680000000} a^{7} + \frac{741455924462673001679602019327}{26870315961377167619060195000000} a^{5} - \frac{4905268094223871026991170397}{33841707759920866018967500000} a^{3} + \frac{30526959294469694382621106483}{67175789903442919047650487500} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2217190170}$, which has order $17737521360$ (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:  \( 88513037217.36226 \) (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

\(\Q(\sqrt{-381}) \), 3.3.16129.1, 6.0.57090302335296.2, 9.9.35965394160281315137521.3

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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
127Data not computed