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Properties

Label 18.0.58082758241...1303.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 13^{12}\cdot 23^{9}$
Root discriminant $97.03$
Ramified primes $7, 13, 23$
Class number $240084$ (GRH)
Class group $[3, 6, 13338]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![478852184, -20839308, 281458960, -10655757, 76232667, -3226096, 12750158, -834166, 1475240, -130764, 127488, -12672, 8852, -914, 490, -48, 16, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 16*x^16 - 48*x^15 + 490*x^14 - 914*x^13 + 8852*x^12 - 12672*x^11 + 127488*x^10 - 130764*x^9 + 1475240*x^8 - 834166*x^7 + 12750158*x^6 - 3226096*x^5 + 76232667*x^4 - 10655757*x^3 + 281458960*x^2 - 20839308*x + 478852184)
 
gp: K = bnfinit(x^18 - 3*x^17 + 16*x^16 - 48*x^15 + 490*x^14 - 914*x^13 + 8852*x^12 - 12672*x^11 + 127488*x^10 - 130764*x^9 + 1475240*x^8 - 834166*x^7 + 12750158*x^6 - 3226096*x^5 + 76232667*x^4 - 10655757*x^3 + 281458960*x^2 - 20839308*x + 478852184, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 16 x^{16} - 48 x^{15} + 490 x^{14} - 914 x^{13} + 8852 x^{12} - 12672 x^{11} + 127488 x^{10} - 130764 x^{9} + 1475240 x^{8} - 834166 x^{7} + 12750158 x^{6} - 3226096 x^{5} + 76232667 x^{4} - 10655757 x^{3} + 281458960 x^{2} - 20839308 x + 478852184 \)

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:  \(-580827582411592412573117020901351303=-\,7^{12}\cdot 13^{12}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.03$
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:  $7, 13, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2093=7\cdot 13\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{2093}(1,·)$, $\chi_{2093}(898,·)$, $\chi_{2093}(919,·)$, $\chi_{2093}(781,·)$, $\chi_{2093}(1933,·)$, $\chi_{2093}(22,·)$, $\chi_{2093}(599,·)$, $\chi_{2093}(666,·)$, $\chi_{2093}(1563,·)$, $\chi_{2093}(737,·)$, $\chi_{2093}(484,·)$, $\chi_{2093}(1381,·)$, $\chi_{2093}(620,·)$, $\chi_{2093}(1264,·)$, $\chi_{2093}(438,·)$, $\chi_{2093}(183,·)$, $\chi_{2093}(1080,·)$, $\chi_{2093}(1082,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a$, $\frac{1}{11195869914} a^{15} - \frac{238593709}{11195869914} a^{14} - \frac{280019323}{11195869914} a^{13} - \frac{321499994}{5597934957} a^{12} + \frac{766711681}{11195869914} a^{11} - \frac{29527651}{11195869914} a^{10} + \frac{88754407}{5597934957} a^{9} - \frac{555501947}{5597934957} a^{8} - \frac{1088226199}{3731956638} a^{7} + \frac{4883421035}{11195869914} a^{6} - \frac{1562129564}{5597934957} a^{5} + \frac{10539063}{1243985546} a^{4} + \frac{3316732063}{11195869914} a^{3} + \frac{93765858}{621992773} a^{2} - \frac{131210609}{1243985546} a - \frac{1773297826}{5597934957}$, $\frac{1}{4052904908868} a^{16} + \frac{17}{675484151478} a^{15} - \frac{555771451}{11195869914} a^{14} - \frac{14040808964}{1013226227217} a^{13} - \frac{54741998989}{675484151478} a^{12} + \frac{4274030243}{675484151478} a^{11} - \frac{12880046827}{2026452454434} a^{10} - \frac{2412578411}{1013226227217} a^{9} - \frac{1242486599}{2026452454434} a^{8} + \frac{549387783625}{2026452454434} a^{7} + \frac{183247930036}{1013226227217} a^{6} + \frac{807783833437}{2026452454434} a^{5} + \frac{340041336599}{2026452454434} a^{4} - \frac{692404603849}{2026452454434} a^{3} - \frac{6157709215}{14526540892} a^{2} - \frac{194868440450}{1013226227217} a + \frac{93985622269}{1013226227217}$, $\frac{1}{19415620758072797889009544931185105956} a^{17} - \frac{393476967097673299750093}{4853905189518199472252386232796276489} a^{16} - \frac{33548296442611663500733256}{1617968396506066490750795410932092163} a^{15} - \frac{446542968877577012267359532982639259}{9707810379036398944504772465592552978} a^{14} + \frac{214586938836326241371865666698578199}{3235936793012132981501590821864184326} a^{13} - \frac{336303578009383412372872449687725504}{4853905189518199472252386232796276489} a^{12} + \frac{27986152217352098238871272621750283}{1078645597670710993833863607288061442} a^{11} + \frac{223720352930753449297799364476943499}{4853905189518199472252386232796276489} a^{10} + \frac{185388270966223173053893041546214691}{4853905189518199472252386232796276489} a^{9} - \frac{1301557974107856342897985611300134437}{9707810379036398944504772465592552978} a^{8} + \frac{3553281009798153224361279866690796535}{9707810379036398944504772465592552978} a^{7} - \frac{123300481377761693371714701874469543}{9707810379036398944504772465592552978} a^{6} + \frac{578107011202184620965107402256464906}{1617968396506066490750795410932092163} a^{5} - \frac{243322068822537070668170509504717127}{1078645597670710993833863607288061442} a^{4} + \frac{4854628471906431390730066668676618339}{19415620758072797889009544931185105956} a^{3} - \frac{36039021268013001706377334577455961}{4853905189518199472252386232796276489} a^{2} + \frac{1466535657982275679278639396640681769}{3235936793012132981501590821864184326} a - \frac{1088170592258589917335832416351560380}{4853905189518199472252386232796276489}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{13338}$, which has order $240084$ (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:  \( 205236.825908 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.3.8281.1, 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 6.0.834351550487.2, 6.0.347501687.1, 6.0.29212967.1, 6.0.834351550487.1, 9.9.567869252041.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
7Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$