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Properties

Label 18.0.12451904740...0000.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{31}\cdot 5^{9}\cdot 13^{14}$
Root discriminant $218.08$
Ramified primes $2, 3, 5, 13$
Class number $38558688$ (GRH)
Class group $[14, 2754192]$ (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![1721055744, 242697600, 625363200, 167694048, 128208960, 36778968, 20836656, 5088618, 816822, 301551, 210087, -19494, -22314, 1116, 1944, 60, -60, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 60*x^16 + 60*x^15 + 1944*x^14 + 1116*x^13 - 22314*x^12 - 19494*x^11 + 210087*x^10 + 301551*x^9 + 816822*x^8 + 5088618*x^7 + 20836656*x^6 + 36778968*x^5 + 128208960*x^4 + 167694048*x^3 + 625363200*x^2 + 242697600*x + 1721055744)
 
gp: K = bnfinit(x^18 - 3*x^17 - 60*x^16 + 60*x^15 + 1944*x^14 + 1116*x^13 - 22314*x^12 - 19494*x^11 + 210087*x^10 + 301551*x^9 + 816822*x^8 + 5088618*x^7 + 20836656*x^6 + 36778968*x^5 + 128208960*x^4 + 167694048*x^3 + 625363200*x^2 + 242697600*x + 1721055744, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 60 x^{16} + 60 x^{15} + 1944 x^{14} + 1116 x^{13} - 22314 x^{12} - 19494 x^{11} + 210087 x^{10} + 301551 x^{9} + 816822 x^{8} + 5088618 x^{7} + 20836656 x^{6} + 36778968 x^{5} + 128208960 x^{4} + 167694048 x^{3} + 625363200 x^{2} + 242697600 x + 1721055744 \)

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:  \(-1245190474036783044610867207005696000000000=-\,2^{18}\cdot 3^{31}\cdot 5^{9}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $218.08$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{72} a^{9} + \frac{1}{24} a^{8} + \frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{12} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{288} a^{10} + \frac{1}{288} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{1}{96} a^{6} - \frac{1}{32} a^{5} + \frac{5}{48} a^{4} - \frac{13}{48} a^{3} - \frac{1}{2} a^{2} - \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{288} a^{11} + \frac{1}{288} a^{9} - \frac{1}{24} a^{8} - \frac{1}{96} a^{7} + \frac{1}{24} a^{6} + \frac{1}{96} a^{5} + \frac{17}{48} a^{3} - \frac{1}{3} a^{2} - \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{576} a^{12} - \frac{1}{576} a^{10} + \frac{1}{288} a^{9} + \frac{1}{64} a^{8} + \frac{5}{64} a^{6} - \frac{7}{32} a^{5} - \frac{5}{96} a^{4} - \frac{5}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{1152} a^{13} - \frac{1}{1152} a^{12} - \frac{1}{1152} a^{11} - \frac{1}{1152} a^{10} + \frac{1}{384} a^{9} + \frac{7}{128} a^{8} + \frac{13}{128} a^{7} + \frac{35}{384} a^{6} - \frac{13}{96} a^{5} + \frac{1}{64} a^{4} - \frac{25}{96} a^{3} - \frac{5}{48} a^{2} - \frac{1}{24} a - \frac{1}{2}$, $\frac{1}{2304} a^{14} - \frac{1}{2304} a^{13} - \frac{1}{2304} a^{12} + \frac{1}{768} a^{11} - \frac{1}{2304} a^{10} + \frac{5}{768} a^{9} - \frac{1}{768} a^{8} - \frac{41}{768} a^{7} + \frac{7}{96} a^{6} + \frac{83}{384} a^{5} - \frac{23}{192} a^{4} - \frac{11}{96} a^{3} + \frac{7}{16} a^{2} - \frac{1}{3} a$, $\frac{1}{502272} a^{15} + \frac{43}{251136} a^{14} + \frac{17}{251136} a^{13} + \frac{137}{251136} a^{12} - \frac{271}{251136} a^{11} - \frac{397}{251136} a^{10} - \frac{133}{41856} a^{9} - \frac{4487}{83712} a^{8} + \frac{12503}{167424} a^{7} - \frac{1783}{41856} a^{6} + \frac{2961}{27904} a^{5} + \frac{575}{10464} a^{4} - \frac{1423}{5232} a^{3} - \frac{1981}{5232} a^{2} + \frac{611}{5232} a + \frac{115}{436}$, $\frac{1}{139631616} a^{16} - \frac{3}{3878656} a^{15} - \frac{511}{34907904} a^{14} - \frac{103}{320256} a^{13} - \frac{5801}{11635968} a^{12} + \frac{21737}{17453952} a^{11} - \frac{83465}{69815808} a^{10} - \frac{209509}{34907904} a^{9} + \frac{615515}{15514624} a^{8} + \frac{246263}{3878656} a^{7} - \frac{1570655}{23271936} a^{6} - \frac{426607}{1939328} a^{5} - \frac{148213}{1454496} a^{4} + \frac{104695}{1454496} a^{3} - \frac{492307}{1454496} a^{2} + \frac{6401}{45453} a - \frac{149}{654}$, $\frac{1}{8447214997753954044045953848014848054243328} a^{17} - \frac{9865435786758405644550225041900279}{2815738332584651348015317949338282684747776} a^{16} - \frac{678162489890450532468283507013995255}{1407869166292325674007658974669141342373888} a^{15} + \frac{2023288174481570682625642165523862661}{21997955723317588656369671479205333474592} a^{14} + \frac{41263097928963679041639701822341971031}{117322430524360472833971581222428445197824} a^{13} + \frac{17715596987329727838162474816924562073}{78214953682906981889314387481618963465216} a^{12} - \frac{2406426816687378375305076229771636021339}{1407869166292325674007658974669141342373888} a^{11} + \frac{79280648000872565345908983076924395287}{469289722097441891335886324889713780791296} a^{10} - \frac{8532337510549397848291868671498499331895}{2815738332584651348015317949338282684747776} a^{9} + \frac{30775400101972967042124609488663625046579}{2815738332584651348015317949338282684747776} a^{8} - \frac{28975273927567137630247836608937341128217}{234644861048720945667943162444856890395648} a^{7} + \frac{25541685069000323311227741343467219005561}{469289722097441891335886324889713780791296} a^{6} - \frac{12000557418584285860884628569150874069841}{234644861048720945667943162444856890395648} a^{5} + \frac{300687485511136066399059889922239052969}{4888434605181686368082149217601185216576} a^{4} + \frac{811706854072477512472996699608482587619}{14665303815545059104246447652803555649728} a^{3} - \frac{693496311421565430778381254954348988199}{9776869210363372736164298435202370433152} a^{2} + \frac{534861834887976424945166021051358753903}{4888434605181686368082149217601185216576} a - \frac{6309410269178102427000406234784520877}{26376445711411976806198646857560351888}$

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

Class group and class number

$C_{14}\times C_{2754192}$, which has order $38558688$ (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:  \( 3533133948.6916637 \) (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

\(\Q(\sqrt{-15}) \), 3.3.2808.1, 3.3.13689.1, 6.0.70270770375.5, 6.0.2956824000.11, 9.9.460990789028310528.1

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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.12.10.2$x^{12} + 39 x^{6} + 676$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$