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Properties

Label 18.0.17425940631...9891.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{15}\cdot 37^{9}$
Root discriminant $133.20$
Ramified primes $3, 7, 37$
Class number $5377456$ (GRH)
Class group $[38, 266, 532]$ (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![688036194091, -233842540932, 281667595275, -79873197249, 55287846336, -12797520429, 6227964435, -1125215883, 443405220, -59533211, 21090282, -1990740, 678152, -42069, 14229, -523, 177, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 177*x^16 - 523*x^15 + 14229*x^14 - 42069*x^13 + 678152*x^12 - 1990740*x^11 + 21090282*x^10 - 59533211*x^9 + 443405220*x^8 - 1125215883*x^7 + 6227964435*x^6 - 12797520429*x^5 + 55287846336*x^4 - 79873197249*x^3 + 281667595275*x^2 - 233842540932*x + 688036194091)
 
gp: K = bnfinit(x^18 - 3*x^17 + 177*x^16 - 523*x^15 + 14229*x^14 - 42069*x^13 + 678152*x^12 - 1990740*x^11 + 21090282*x^10 - 59533211*x^9 + 443405220*x^8 - 1125215883*x^7 + 6227964435*x^6 - 12797520429*x^5 + 55287846336*x^4 - 79873197249*x^3 + 281667595275*x^2 - 233842540932*x + 688036194091, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 177 x^{16} - 523 x^{15} + 14229 x^{14} - 42069 x^{13} + 678152 x^{12} - 1990740 x^{11} + 21090282 x^{10} - 59533211 x^{9} + 443405220 x^{8} - 1125215883 x^{7} + 6227964435 x^{6} - 12797520429 x^{5} + 55287846336 x^{4} - 79873197249 x^{3} + 281667595275 x^{2} - 233842540932 x + 688036194091 \)

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:  \(-174259406310011076626476389857780739891=-\,3^{24}\cdot 7^{15}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.20$
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:  $3, 7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2331=3^{2}\cdot 7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2331}(1,·)$, $\chi_{2331}(517,·)$, $\chi_{2331}(1222,·)$, $\chi_{2331}(73,·)$, $\chi_{2331}(778,·)$, $\chi_{2331}(1294,·)$, $\chi_{2331}(1999,·)$, $\chi_{2331}(850,·)$, $\chi_{2331}(1555,·)$, $\chi_{2331}(2071,·)$, $\chi_{2331}(1627,·)$, $\chi_{2331}(667,·)$, $\chi_{2331}(1444,·)$, $\chi_{2331}(2182,·)$, $\chi_{2331}(2221,·)$, $\chi_{2331}(445,·)$, $\chi_{2331}(628,·)$, $\chi_{2331}(1405,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1261} a^{12} - \frac{384}{1261} a^{11} + \frac{69}{1261} a^{10} - \frac{18}{97} a^{9} + \frac{441}{1261} a^{8} - \frac{46}{1261} a^{7} + \frac{375}{1261} a^{6} + \frac{614}{1261} a^{5} + \frac{580}{1261} a^{4} + \frac{543}{1261} a^{3} - \frac{201}{1261} a^{2} - \frac{575}{1261} a + \frac{233}{1261}$, $\frac{1}{1261} a^{13} + \frac{150}{1261} a^{11} - \frac{219}{1261} a^{10} + \frac{116}{1261} a^{9} + \frac{324}{1261} a^{8} + \frac{365}{1261} a^{7} - \frac{401}{1261} a^{6} + \frac{549}{1261} a^{5} + \frac{66}{1261} a^{4} + \frac{246}{1261} a^{3} + \frac{423}{1261} a^{2} + \frac{108}{1261} a - \frac{59}{1261}$, $\frac{1}{1261} a^{14} - \frac{625}{1261} a^{11} - \frac{146}{1261} a^{10} + \frac{116}{1261} a^{9} - \frac{213}{1261} a^{8} + \frac{2}{13} a^{7} - \frac{217}{1261} a^{6} + \frac{19}{1261} a^{5} + \frac{255}{1261} a^{4} - \frac{323}{1261} a^{3} - \frac{6}{1261} a^{2} + \frac{443}{1261} a + \frac{358}{1261}$, $\frac{1}{1261} a^{15} - \frac{556}{1261} a^{11} + \frac{367}{1261} a^{10} - \frac{187}{1261} a^{9} - \frac{340}{1261} a^{8} + \frac{36}{1261} a^{7} - \frac{152}{1261} a^{6} - \frac{600}{1261} a^{5} + \frac{270}{1261} a^{4} + \frac{160}{1261} a^{3} - \frac{343}{1261} a^{2} + \frac{368}{1261} a + \frac{610}{1261}$, $\frac{1}{1261} a^{16} - \frac{28}{1261} a^{11} + \frac{347}{1261} a^{10} - \frac{561}{1261} a^{9} + \frac{46}{97} a^{8} - \frac{508}{1261} a^{7} - \frac{165}{1261} a^{6} - \frac{77}{1261} a^{5} - \frac{176}{1261} a^{4} + \frac{186}{1261} a^{3} - \frac{420}{1261} a^{2} - \frac{57}{1261} a - \frac{335}{1261}$, $\frac{1}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{17} + \frac{1654376606691496891420126795636538839228730377910225653673428901687273}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{16} - \frac{1324018858359681296266127979581978438006536410851120526224028651347871}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{15} - \frac{29093416832221637210729852087628181517693156386120417379875216785039}{96643109614328590063453916335312996814600718960037905136827816008919413} a^{14} - \frac{1134261864816643858863499625342053416162881942298340997799258275343189}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{13} - \frac{2542845496555802641155218182571247055906959928336976627908903956342487}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{12} + \frac{1115676400746713118857368925868116271962234567465754068366530140268516761}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{11} + \frac{1687491729144764361330520390591929850466200217074777312983708423067724238}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{10} - \frac{423385238048168633126979198198488420667990038157721096141409049782219321}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{9} - \frac{1688250083698077818117920083024399640068495579589296184034697406375056269}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{8} - \frac{2357417776458191187282228854810828320020254452023604593271853419263590679}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{7} - \frac{733441603658604552137620890798093506139475122798809330869650197650704000}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{6} + \frac{133630038597050457496367122008170662973263745494512377791840135165175012}{527820060201333068808094466139017136448973157397130097285751918202559871} a^{5} - \frac{1980897029785771005369489502280371587931623403788662587756750465024725800}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{4} - \frac{1014781438281355603178122205608588389669872795970565105986797895688894642}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{3} + \frac{1766381789681908680724609749880429682117102709645139822599227314523976253}{6861660782617329894505228059807222773836651046162691264714774936633278323} a^{2} - \frac{352735678055236630881436839123658636540469064994645315625637154691061180}{6861660782617329894505228059807222773836651046162691264714774936633278323} a - \frac{91236192982914398611633869522595198089146660911094856627620710360181}{192133418716359025971081344603007945953480554592520686156715339977971}$

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

Class group and class number

$C_{38}\times C_{266}\times C_{532}$, which has order $5377456$ (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

\(\Q(\sqrt{-259}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.2, 3.3.3969.1, 6.0.113990676219.7, 6.0.851324971.1, 6.0.5585543134731.4, 6.0.5585543134731.3, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
7Data not computed
$37$37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$