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Properties

Label 18.0.82880965544...0832.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 7^{15}\cdot 11^{9}$
Root discriminant $145.26$
Ramified primes $2, 3, 7, 11$
Class number $12700800$ (GRH)
Class group $[2, 2, 2, 2, 630, 1260]$ (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![2131852243753, 133856748120, 761687723085, 18837567282, 126327354123, 1198422450, 13080744725, 49747296, 878462091, 1074518, 38661657, 2232, 1112048, -372, 20232, -6, 213, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 213*x^16 - 6*x^15 + 20232*x^14 - 372*x^13 + 1112048*x^12 + 2232*x^11 + 38661657*x^10 + 1074518*x^9 + 878462091*x^8 + 49747296*x^7 + 13080744725*x^6 + 1198422450*x^5 + 126327354123*x^4 + 18837567282*x^3 + 761687723085*x^2 + 133856748120*x + 2131852243753)
 
gp: K = bnfinit(x^18 + 213*x^16 - 6*x^15 + 20232*x^14 - 372*x^13 + 1112048*x^12 + 2232*x^11 + 38661657*x^10 + 1074518*x^9 + 878462091*x^8 + 49747296*x^7 + 13080744725*x^6 + 1198422450*x^5 + 126327354123*x^4 + 18837567282*x^3 + 761687723085*x^2 + 133856748120*x + 2131852243753, 1)
 

Normalized defining polynomial

\( x^{18} + 213 x^{16} - 6 x^{15} + 20232 x^{14} - 372 x^{13} + 1112048 x^{12} + 2232 x^{11} + 38661657 x^{10} + 1074518 x^{9} + 878462091 x^{8} + 49747296 x^{7} + 13080744725 x^{6} + 1198422450 x^{5} + 126327354123 x^{4} + 18837567282 x^{3} + 761687723085 x^{2} + 133856748120 x + 2131852243753 \)

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:  \(-828809655441826774321495340465219960832=-\,2^{18}\cdot 3^{24}\cdot 7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $145.26$
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:  $2, 3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2772=2^{2}\cdot 3^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2772}(1,·)$, $\chi_{2772}(2377,·)$, $\chi_{2772}(439,·)$, $\chi_{2772}(1231,·)$, $\chi_{2772}(2641,·)$, $\chi_{2772}(1363,·)$, $\chi_{2772}(793,·)$, $\chi_{2772}(1627,·)$, $\chi_{2772}(925,·)$, $\chi_{2772}(529,·)$, $\chi_{2772}(2155,·)$, $\chi_{2772}(1453,·)$, $\chi_{2772}(2287,·)$, $\chi_{2772}(307,·)$, $\chi_{2772}(1717,·)$, $\chi_{2772}(2551,·)$, $\chi_{2772}(1849,·)$, $\chi_{2772}(703,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11} a^{6} + \frac{5}{11} a^{4} - \frac{2}{11} a^{3} - \frac{2}{11} a^{2} - \frac{5}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{7} + \frac{5}{11} a^{5} - \frac{2}{11} a^{4} - \frac{2}{11} a^{3} - \frac{5}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{22} a^{8} - \frac{1}{11} a^{5} - \frac{5}{22} a^{4} - \frac{3}{11} a^{3} - \frac{1}{2} a^{2} - \frac{4}{11} a - \frac{5}{22}$, $\frac{1}{22} a^{9} - \frac{5}{22} a^{5} + \frac{2}{11} a^{4} + \frac{7}{22} a^{3} + \frac{5}{11} a^{2} + \frac{7}{22} a + \frac{1}{11}$, $\frac{1}{22} a^{10} - \frac{1}{22} a^{6} + \frac{2}{11} a^{5} + \frac{5}{22} a^{4} + \frac{1}{11} a^{3} - \frac{1}{22} a^{2} + \frac{2}{11} a + \frac{2}{11}$, $\frac{1}{22} a^{11} - \frac{1}{22} a^{7} + \frac{5}{22} a^{5} + \frac{2}{11} a^{4} + \frac{7}{22} a^{3} - \frac{5}{11} a^{2} + \frac{1}{11} a - \frac{2}{11}$, $\frac{1}{242} a^{12} - \frac{1}{242} a^{10} - \frac{2}{121} a^{9} - \frac{1}{242} a^{8} - \frac{4}{121} a^{7} + \frac{4}{121} a^{6} + \frac{34}{121} a^{5} + \frac{50}{121} a^{4} + \frac{30}{121} a^{3} + \frac{21}{242} a^{2} - \frac{16}{121} a + \frac{39}{121}$, $\frac{1}{242} a^{13} - \frac{1}{242} a^{11} - \frac{2}{121} a^{10} - \frac{1}{242} a^{9} + \frac{3}{242} a^{8} + \frac{4}{121} a^{7} + \frac{1}{121} a^{6} + \frac{39}{121} a^{5} - \frac{83}{242} a^{4} + \frac{87}{242} a^{3} - \frac{21}{242} a^{2} + \frac{39}{121} a - \frac{1}{2}$, $\frac{1}{242} a^{14} - \frac{2}{121} a^{11} - \frac{1}{121} a^{10} - \frac{1}{242} a^{9} - \frac{2}{121} a^{8} - \frac{3}{121} a^{7} - \frac{1}{121} a^{6} + \frac{7}{242} a^{5} + \frac{2}{11} a^{4} + \frac{39}{242} a^{3} - \frac{4}{11} a^{2} - \frac{109}{242} a + \frac{45}{242}$, $\frac{1}{242} a^{15} - \frac{1}{121} a^{11} - \frac{5}{242} a^{10} + \frac{1}{121} a^{9} + \frac{1}{242} a^{8} + \frac{5}{121} a^{7} - \frac{5}{242} a^{6} - \frac{40}{121} a^{5} - \frac{39}{121} a^{4} - \frac{1}{121} a^{3} - \frac{29}{121} a^{2} + \frac{5}{242} a + \frac{15}{242}$, $\frac{1}{12173565087255366542179899412} a^{16} - \frac{4835360133235793062256512}{3043391271813841635544974853} a^{15} - \frac{3949413047222307554798210}{3043391271813841635544974853} a^{14} - \frac{1429199451267660912068150}{3043391271813841635544974853} a^{13} + \frac{3951333338631761135079536}{3043391271813841635544974853} a^{12} + \frac{96544465788154275199294609}{6086782543627683271089949706} a^{11} - \frac{48826411204908202384701235}{3043391271813841635544974853} a^{10} + \frac{125144757077675281209025441}{6086782543627683271089949706} a^{9} - \frac{250299442879896977216818095}{12173565087255366542179899412} a^{8} + \frac{152956548192061869472711611}{6086782543627683271089949706} a^{7} + \frac{224104302851574084348229933}{6086782543627683271089949706} a^{6} - \frac{789888058815219457209296154}{3043391271813841635544974853} a^{5} - \frac{4349573430874179406198320513}{12173565087255366542179899412} a^{4} + \frac{1405655742668652647922066024}{3043391271813841635544974853} a^{3} - \frac{896079483740883095223154932}{3043391271813841635544974853} a^{2} + \frac{1347285255000692957546586481}{6086782543627683271089949706} a - \frac{2602174649539766545121439359}{12173565087255366542179899412}$, $\frac{1}{5450531300842732238694494196261070350708884708} a^{17} - \frac{48747056204234613}{5450531300842732238694494196261070350708884708} a^{16} + \frac{1862987855789072622856584713190021257769875}{1362632825210683059673623549065267587677221177} a^{15} + \frac{5477897124617135019650308565633227574022155}{2725265650421366119347247098130535175354442354} a^{14} + \frac{529408984146537059976267522753035433400936}{1362632825210683059673623549065267587677221177} a^{13} + \frac{3263960811336021172236249714203202025304467}{2725265650421366119347247098130535175354442354} a^{12} - \frac{16098575358987051744636562242446041720081117}{1362632825210683059673623549065267587677221177} a^{11} - \frac{2491025550432052614295858365704845788022591}{2725265650421366119347247098130535175354442354} a^{10} - \frac{92779489171089984063899543652406257716959591}{5450531300842732238694494196261070350708884708} a^{9} + \frac{113175915745326617525661113076727500262802755}{5450531300842732238694494196261070350708884708} a^{8} - \frac{67254764772791294884303092569529779233442117}{2725265650421366119347247098130535175354442354} a^{7} + \frac{49876669855489607738293630253236309759779835}{2725265650421366119347247098130535175354442354} a^{6} - \frac{6589084871492450766511479002904875052191637}{45045713230105225113177637985628680584370948} a^{5} - \frac{260654396269628174922558720879221132164626381}{5450531300842732238694494196261070350708884708} a^{4} - \frac{360399574596571283931426996229352043701603528}{1362632825210683059673623549065267587677221177} a^{3} + \frac{19317742245414067745925691917128908607648410}{1362632825210683059673623549065267587677221177} a^{2} + \frac{1703804561282036159322402483119544478861904081}{5450531300842732238694494196261070350708884708} a + \frac{1663158396438655801179367139848623735424368323}{5450531300842732238694494196261070350708884708}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{630}\times C_{1260}$, which has order $12700800$ (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:  \( 54408.48888868202 \) (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

\(\Q(\sqrt{-77}) \), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 6.0.9393301608768.5, 6.0.9393301608768.6, 6.0.191700032832.15, 6.0.1431687488.1, 9.9.62523502209.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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$$[2]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$