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Properties

Label 18.0.11682260499...9239.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{15}\cdot 19^{9}$
Root discriminant $114.63$
Ramified primes $3, 7, 19$
Class number $3302656$ (GRH)
Class group $[2, 4, 4, 103208]$ (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![55442113336, -9990904116, 22625119512, -5625118235, 5915118711, -1005798375, 1138108250, -75110319, 146318868, -2483709, 11578902, -36945, 542514, -195, 14310, -1, 192, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 192*x^16 - x^15 + 14310*x^14 - 195*x^13 + 542514*x^12 - 36945*x^11 + 11578902*x^10 - 2483709*x^9 + 146318868*x^8 - 75110319*x^7 + 1138108250*x^6 - 1005798375*x^5 + 5915118711*x^4 - 5625118235*x^3 + 22625119512*x^2 - 9990904116*x + 55442113336)
 
gp: K = bnfinit(x^18 + 192*x^16 - x^15 + 14310*x^14 - 195*x^13 + 542514*x^12 - 36945*x^11 + 11578902*x^10 - 2483709*x^9 + 146318868*x^8 - 75110319*x^7 + 1138108250*x^6 - 1005798375*x^5 + 5915118711*x^4 - 5625118235*x^3 + 22625119512*x^2 - 9990904116*x + 55442113336, 1)
 

Normalized defining polynomial

\( x^{18} + 192 x^{16} - x^{15} + 14310 x^{14} - 195 x^{13} + 542514 x^{12} - 36945 x^{11} + 11578902 x^{10} - 2483709 x^{9} + 146318868 x^{8} - 75110319 x^{7} + 1138108250 x^{6} - 1005798375 x^{5} + 5915118711 x^{4} - 5625118235 x^{3} + 22625119512 x^{2} - 9990904116 x + 55442113336 \)

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:  \(-11682260499388589326243647883020829239=-\,3^{27}\cdot 7^{15}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.63$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1197=3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1197}(1,·)$, $\chi_{1197}(1025,·)$, $\chi_{1197}(172,·)$, $\chi_{1197}(970,·)$, $\chi_{1197}(398,·)$, $\chi_{1197}(400,·)$, $\chi_{1197}(341,·)$, $\chi_{1197}(856,·)$, $\chi_{1197}(797,·)$, $\chi_{1197}(799,·)$, $\chi_{1197}(227,·)$, $\chi_{1197}(740,·)$, $\chi_{1197}(1196,·)$, $\chi_{1197}(626,·)$, $\chi_{1197}(1139,·)$, $\chi_{1197}(457,·)$, $\chi_{1197}(58,·)$, $\chi_{1197}(571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{794632} a^{15} - \frac{35}{284} a^{14} + \frac{52319}{794632} a^{13} + \frac{48691}{397316} a^{12} - \frac{44919}{794632} a^{11} + \frac{77565}{397316} a^{10} + \frac{30447}{794632} a^{9} - \frac{85855}{397316} a^{8} - \frac{102683}{794632} a^{7} - \frac{20439}{198658} a^{6} + \frac{193097}{794632} a^{5} - \frac{167837}{397316} a^{4} - \frac{319359}{794632} a^{3} - \frac{22471}{198658} a^{2} + \frac{36995}{99329} a + \frac{13494}{99329}$, $\frac{1}{2038117447624} a^{16} + \frac{942887}{2038117447624} a^{15} - \frac{95654530141}{2038117447624} a^{14} + \frac{100092304345}{2038117447624} a^{13} + \frac{48842944481}{2038117447624} a^{12} - \frac{355253874317}{2038117447624} a^{11} - \frac{12016325501}{2038117447624} a^{10} + \frac{181062917585}{2038117447624} a^{9} - \frac{419103909363}{2038117447624} a^{8} - \frac{9338380875}{2038117447624} a^{7} + \frac{951859604291}{2038117447624} a^{6} - \frac{929546231089}{2038117447624} a^{5} + \frac{319038121393}{2038117447624} a^{4} - \frac{890017041971}{2038117447624} a^{3} - \frac{352340761057}{1019058723812} a^{2} + \frac{1583706707}{509529361906} a - \frac{33557844020}{254764680953}$, $\frac{1}{19708365919431061456278559484870396399580597307340376728371813496} a^{17} + \frac{1748902383705430277867910870436632810169552054581}{69395654645884019212248448890388719716833089110353439184407794} a^{16} + \frac{6202471685840506594849670783054256422816080121198239732693}{19708365919431061456278559484870396399580597307340376728371813496} a^{15} + \frac{657128835335620829843020170630762262425914593183389872247762}{13610749944358467856545966495076240607445163886284790558267827} a^{14} - \frac{1940233285111918887228130991899815857103307514934163116854629569}{19708365919431061456278559484870396399580597307340376728371813496} a^{13} - \frac{130221577880945152508835872304289804726255098143069291080984007}{4927091479857765364069639871217599099895149326835094182092953374} a^{12} - \frac{3663306585194064926642462518008747633969110976889465179423673311}{19708365919431061456278559484870396399580597307340376728371813496} a^{11} + \frac{413110550181584559747384292536514499976567319376508707127655233}{2463545739928882682034819935608799549947574663417547091046476687} a^{10} + \frac{3920161178830033437743494788405372141073863432377679652358695271}{19708365919431061456278559484870396399580597307340376728371813496} a^{9} - \frac{671232396971557627763340019572537093076329424831616067481884403}{9854182959715530728139279742435198199790298653670188364185906748} a^{8} + \frac{52079658279022610063675290303941500951745596362281617821099177}{277582618583536076848993795561554878867332356441413756737631176} a^{7} - \frac{1221263664257086168991586629975021929601336878709675155022152637}{2463545739928882682034819935608799549947574663417547091046476687} a^{6} + \frac{4090347397479562780385856029127462712585639040835638827065528071}{19708365919431061456278559484870396399580597307340376728371813496} a^{5} + \frac{2428712804431979563642549318865246385771480039048145371887227365}{9854182959715530728139279742435198199790298653670188364185906748} a^{4} - \frac{4104868239497203365624123546509192137599397422483435114459578479}{9854182959715530728139279742435198199790298653670188364185906748} a^{3} - \frac{186511062511419758851053057982745786551763952428708879296629322}{2463545739928882682034819935608799549947574663417547091046476687} a^{2} - \frac{651586585040252422811390757908079250686164061357056699161993581}{4927091479857765364069639871217599099895149326835094182092953374} a + \frac{1172964469595346792128620726477251485589771026224586573053356240}{2463545739928882682034819935608799549947574663417547091046476687}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{103208}$, which has order $3302656$ (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

\(\Q(\sqrt{-399}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, 6.0.46306954071.4, 6.0.3112538751.2, 6.0.2269040749479.8, 6.0.2269040749479.7, 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$19$19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$