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Properties

Label 18.0.19309844779...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 5^{9}\cdot 7^{9}\cdot 163^{14}$
Root discriminant $621.81$
Ramified primes $2, 5, 7, 163$
Class number $3991494528$ (GRH)
Class group $[2, 2, 2, 28, 17819172]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![193437757489, 4718555316, 45599080719, -2439482531, 5750426701, -138548203, 468059929, -5102579, 22067877, 1462242, 1006988, 63753, 15397, -175, 1775, -2, -37, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 37*x^16 - 2*x^15 + 1775*x^14 - 175*x^13 + 15397*x^12 + 63753*x^11 + 1006988*x^10 + 1462242*x^9 + 22067877*x^8 - 5102579*x^7 + 468059929*x^6 - 138548203*x^5 + 5750426701*x^4 - 2439482531*x^3 + 45599080719*x^2 + 4718555316*x + 193437757489)
 
gp: K = bnfinit(x^18 - x^17 - 37*x^16 - 2*x^15 + 1775*x^14 - 175*x^13 + 15397*x^12 + 63753*x^11 + 1006988*x^10 + 1462242*x^9 + 22067877*x^8 - 5102579*x^7 + 468059929*x^6 - 138548203*x^5 + 5750426701*x^4 - 2439482531*x^3 + 45599080719*x^2 + 4718555316*x + 193437757489, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 37 x^{16} - 2 x^{15} + 1775 x^{14} - 175 x^{13} + 15397 x^{12} + 63753 x^{11} + 1006988 x^{10} + 1462242 x^{9} + 22067877 x^{8} - 5102579 x^{7} + 468059929 x^{6} - 138548203 x^{5} + 5750426701 x^{4} - 2439482531 x^{3} + 45599080719 x^{2} + 4718555316 x + 193437757489 \)

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:  \(-193098447796943280994699574357059823115776000000000=-\,2^{18}\cdot 5^{9}\cdot 7^{9}\cdot 163^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $621.81$
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:  $2, 5, 7, 163$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{74} a^{14} - \frac{7}{74} a^{13} - \frac{1}{37} a^{12} - \frac{9}{74} a^{11} + \frac{11}{74} a^{10} - \frac{9}{37} a^{9} - \frac{27}{74} a^{8} + \frac{1}{74} a^{7} + \frac{14}{37} a^{6} - \frac{9}{74} a^{5} - \frac{13}{74} a^{4} + \frac{8}{37} a^{3} - \frac{31}{74} a^{2} - \frac{27}{74} a + \frac{23}{74}$, $\frac{1}{1258} a^{15} + \frac{4}{629} a^{14} + \frac{189}{1258} a^{13} - \frac{39}{1258} a^{12} + \frac{308}{629} a^{11} - \frac{593}{1258} a^{10} - \frac{297}{1258} a^{9} - \frac{202}{629} a^{8} - \frac{105}{1258} a^{7} + \frac{485}{1258} a^{6} - \frac{8}{17} a^{5} + \frac{43}{1258} a^{4} + \frac{61}{1258} a^{3} - \frac{209}{629} a^{2} + \frac{290}{629} a + \frac{29}{74}$, $\frac{1}{68929594} a^{16} - \frac{22395}{68929594} a^{15} - \frac{13434}{34464797} a^{14} - \frac{3414866}{34464797} a^{13} - \frac{1674021}{34464797} a^{12} + \frac{22879753}{68929594} a^{11} + \frac{6328504}{34464797} a^{10} - \frac{23438413}{68929594} a^{9} + \frac{2499404}{34464797} a^{8} + \frac{2432}{2027341} a^{7} - \frac{10109409}{34464797} a^{6} - \frac{4609957}{68929594} a^{5} - \frac{7586172}{34464797} a^{4} + \frac{12769587}{68929594} a^{3} + \frac{3994955}{68929594} a^{2} + \frac{25717419}{68929594} a + \frac{1871535}{4054682}$, $\frac{1}{11141702217060193393962579897795769253284781868584432912964754307597902} a^{17} - \frac{80349864927591907613078245083636113664311459007820460399642209}{11141702217060193393962579897795769253284781868584432912964754307597902} a^{16} - \frac{3424105885068401873049991393395733443892666748943379877597536824995}{11141702217060193393962579897795769253284781868584432912964754307597902} a^{15} - \frac{11227398902984222269131890099935944378596530521329990713074240818365}{11141702217060193393962579897795769253284781868584432912964754307597902} a^{14} + \frac{80838155747126966282476433216053558837559243578650529183068252207199}{327697124031182158645958232288110860390728878487777438616610420811703} a^{13} - \frac{2031214797695335717307526950784379196488912250082609396585281012888137}{11141702217060193393962579897795769253284781868584432912964754307597902} a^{12} + \frac{1566014976585555247566270309993832382996141629478190686478777404054387}{11141702217060193393962579897795769253284781868584432912964754307597902} a^{11} + \frac{833301900029426800109682507164537532170932888403036058047785753162665}{5570851108530096696981289948897884626642390934292216456482377153798951} a^{10} + \frac{2583773613788426472397350284895081318368516080855527717967422824501587}{11141702217060193393962579897795769253284781868584432912964754307597902} a^{9} - \frac{1721032067553287045595574233758520936079247508006955414500245864220851}{11141702217060193393962579897795769253284781868584432912964754307597902} a^{8} + \frac{967351118412372074960652696678633016354910736100587551547654366608411}{5570851108530096696981289948897884626642390934292216456482377153798951} a^{7} + \frac{89354075204955520585950608074654350806803037443717226380261520302555}{301127086947572794431421078318804574413102212664444132782831197502646} a^{6} - \frac{3352099263417613879514395132880996147779168677307579114796913117953255}{11141702217060193393962579897795769253284781868584432912964754307597902} a^{5} - \frac{254714169223860243389849101824495492823337032527522338754379148795517}{5570851108530096696981289948897884626642390934292216456482377153798951} a^{4} - \frac{2783313031056795931785729159836703447970975821768029503940225512526344}{5570851108530096696981289948897884626642390934292216456482377153798951} a^{3} + \frac{233804383148616103669759177159842995395249798474675033479350630544409}{5570851108530096696981289948897884626642390934292216456482377153798951} a^{2} - \frac{1773687304380332008564685589412211101164889028302808454428606610678312}{5570851108530096696981289948897884626642390934292216456482377153798951} a - \frac{1959199850641066605592568721789094988958676267084297875705626093499}{38552602827197901017171556739777748281262220998562051601954167154318}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{28}\times C_{17819172}$, which has order $3991494528$ (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:  \( 87986390.20265311 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-35}) \), 3.3.26569.1, 3.3.1304.1, 6.0.30265966752875.1, 6.0.72905336000.1, 9.9.1565248123502319104.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$163$163.6.4.1$x^{6} + 5216 x^{3} + 35363339$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
163.12.10.1$x^{12} + 266994 x^{6} + 47068604209$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$