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Properties

Label 18.0.29677268431...1863.7
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{15}\cdot 19^{12}$
Root discriminant $155.92$
Ramified primes $3, 7, 19$
Class number $29488368$ (GRH)
Class group $[4, 7372092]$ (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![10835984384, 12286132224, 4177096704, 533722240, -4174416, -104708448, -77344856, -35960496, -823257, 4611968, 1071381, -228384, -91658, 6240, 4050, -48, -93, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 93*x^16 - 48*x^15 + 4050*x^14 + 6240*x^13 - 91658*x^12 - 228384*x^11 + 1071381*x^10 + 4611968*x^9 - 823257*x^8 - 35960496*x^7 - 77344856*x^6 - 104708448*x^5 - 4174416*x^4 + 533722240*x^3 + 4177096704*x^2 + 12286132224*x + 10835984384)
 
gp: K = bnfinit(x^18 - 93*x^16 - 48*x^15 + 4050*x^14 + 6240*x^13 - 91658*x^12 - 228384*x^11 + 1071381*x^10 + 4611968*x^9 - 823257*x^8 - 35960496*x^7 - 77344856*x^6 - 104708448*x^5 - 4174416*x^4 + 533722240*x^3 + 4177096704*x^2 + 12286132224*x + 10835984384, 1)
 

Normalized defining polynomial

\( x^{18} - 93 x^{16} - 48 x^{15} + 4050 x^{14} + 6240 x^{13} - 91658 x^{12} - 228384 x^{11} + 1071381 x^{10} + 4611968 x^{9} - 823257 x^{8} - 35960496 x^{7} - 77344856 x^{6} - 104708448 x^{5} - 4174416 x^{4} + 533722240 x^{3} + 4177096704 x^{2} + 12286132224 x + 10835984384 \)

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:  \(-2967726843159493858840932623319995101863=-\,3^{24}\cdot 7^{15}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.92$
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$$\chi_{1197}(1,·)$, $\chi_{1197}(1090,·)$, $\chi_{1197}(391,·)$, $\chi_{1197}(904,·)$, $\chi_{1197}(970,·)$, $\chi_{1197}(961,·)$, $\chi_{1197}(976,·)$, $\chi_{1197}(58,·)$, $\chi_{1197}(919,·)$, $\chi_{1197}(1018,·)$, $\chi_{1197}(862,·)$, $\chi_{1197}(676,·)$, $\chi_{1197}(229,·)$, $\chi_{1197}(1132,·)$, $\chi_{1197}(685,·)$, $\chi_{1197}(349,·)$, $\chi_{1197}(115,·)$, $\chi_{1197}(634,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{5} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{32} a^{6} - \frac{1}{32} a^{5} - \frac{1}{32} a^{4} - \frac{3}{32} a^{3} + \frac{1}{8} a$, $\frac{1}{32} a^{7} - \frac{5}{32} a^{3} + \frac{1}{8} a$, $\frac{1}{128} a^{8} - \frac{1}{64} a^{6} - \frac{1}{32} a^{5} + \frac{5}{128} a^{4} - \frac{3}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{512} a^{9} - \frac{1}{256} a^{8} - \frac{3}{256} a^{7} - \frac{1}{128} a^{6} + \frac{9}{512} a^{5} - \frac{1}{256} a^{4} + \frac{31}{128} a^{3} + \frac{1}{64} a^{2} - \frac{1}{4} a$, $\frac{1}{1024} a^{10} - \frac{1}{1024} a^{9} + \frac{3}{512} a^{7} - \frac{11}{1024} a^{6} + \frac{7}{1024} a^{5} + \frac{17}{512} a^{4} - \frac{51}{256} a^{3} - \frac{3}{128} a^{2} + \frac{3}{16} a$, $\frac{1}{2048} a^{11} - \frac{1}{2048} a^{10} + \frac{3}{1024} a^{8} + \frac{21}{2048} a^{7} - \frac{25}{2048} a^{6} + \frac{1}{1024} a^{5} - \frac{11}{512} a^{4} + \frac{61}{256} a^{3} + \frac{1}{32} a^{2} - \frac{1}{4} a$, $\frac{1}{2048} a^{12} - \frac{1}{2048} a^{10} - \frac{1}{1024} a^{9} - \frac{5}{2048} a^{8} - \frac{5}{512} a^{7} - \frac{23}{2048} a^{6} - \frac{25}{1024} a^{5} + \frac{23}{512} a^{4} - \frac{55}{256} a^{3} - \frac{1}{32} a^{2} + \frac{1}{4} a$, $\frac{1}{8192} a^{13} + \frac{1}{8192} a^{11} - \frac{1}{2048} a^{10} + \frac{3}{8192} a^{9} + \frac{1}{1024} a^{8} - \frac{93}{8192} a^{7} - \frac{1}{2048} a^{6} + \frac{13}{1024} a^{5} + \frac{3}{128} a^{4} - \frac{33}{512} a^{3} - \frac{3}{128} a^{2} + \frac{1}{16} a$, $\frac{1}{32768} a^{14} + \frac{1}{32768} a^{13} + \frac{5}{32768} a^{12} + \frac{5}{32768} a^{11} - \frac{5}{32768} a^{10} + \frac{27}{32768} a^{9} - \frac{121}{32768} a^{8} + \frac{359}{32768} a^{7} + \frac{13}{4096} a^{6} + \frac{57}{4096} a^{5} - \frac{103}{2048} a^{4} + \frac{203}{2048} a^{3} + \frac{13}{256} a^{2} - \frac{1}{8} a$, $\frac{1}{65536} a^{15} - \frac{1}{16384} a^{13} + \frac{7}{32768} a^{11} - \frac{3}{16384} a^{9} - \frac{5}{2048} a^{8} - \frac{823}{65536} a^{7} - \frac{9}{1024} a^{6} - \frac{115}{8192} a^{5} - \frac{29}{2048} a^{4} - \frac{19}{4096} a^{3} + \frac{13}{512} a^{2} + \frac{1}{32} a$, $\frac{1}{70254592} a^{16} + \frac{1}{548864} a^{15} + \frac{17}{17563648} a^{14} + \frac{1}{32768} a^{13} - \frac{693}{35127296} a^{12} + \frac{15}{68608} a^{11} - \frac{7661}{17563648} a^{10} - \frac{787}{1097728} a^{9} - \frac{20255}{70254592} a^{8} - \frac{185}{34304} a^{7} + \frac{13877}{8781824} a^{6} - \frac{35701}{2195456} a^{5} + \frac{225061}{4390912} a^{4} + \frac{115565}{548864} a^{3} + \frac{5125}{34304} a^{2} - \frac{217}{536} a - \frac{28}{67}$, $\frac{1}{96318032076834419753373117442593937948672} a^{17} + \frac{79074129927958754860214351324473}{96318032076834419753373117442593937948672} a^{16} - \frac{103268770506558139786543886355161339}{24079508019208604938343279360648484487168} a^{15} + \frac{283677637657430472749999616767774097}{24079508019208604938343279360648484487168} a^{14} - \frac{328009931299138993241788030729190405}{48159016038417209876686558721296968974336} a^{13} + \frac{8038734350061543638466654757116951059}{48159016038417209876686558721296968974336} a^{12} - \frac{66388900176592832200042292526050347}{304803898977324113143585814691752968192} a^{11} - \frac{5741703567314263668456851644115847013}{24079508019208604938343279360648484487168} a^{10} + \frac{179996314806453958611251886575193121}{96318032076834419753373117442593937948672} a^{9} - \frac{269488773765063595037657858557729896487}{96318032076834419753373117442593937948672} a^{8} + \frac{75684816541553193507083091081391113287}{12039754009604302469171639680324242243584} a^{7} - \frac{56488685186390898079097676315473355975}{12039754009604302469171639680324242243584} a^{6} - \frac{97668173799552377896977763841095313997}{6019877004802151234585819840162121121792} a^{5} - \frac{339013665742225107835338961944412571195}{6019877004802151234585819840162121121792} a^{4} - \frac{59647264869892322989388126785894837777}{752484625600268904323227480020265140224} a^{3} + \frac{801241995863131671436316504900685849}{23515144550008403260100858750633285632} a^{2} - \frac{672290136986640932786666741270742759}{2939393068751050407512607343829160704} a + \frac{12341744128075591863802710095176033}{45928016699235162617384489747330636}$

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

Class group and class number

$C_{4}\times C_{7372092}$, which has order $29488368$ (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:  \( 10681224266.072006 \) (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

\(\Q(\sqrt{-7}) \), 3.3.1432809.2, 3.3.29241.1, 3.3.3969.1, 3.3.17689.2, 6.0.14370591413367.6, 6.0.293277375783.2, 6.0.110270727.2, 6.0.2190305047.2, 9.9.2941473244627851129.8

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
19Data not computed