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Properties

Label 18.0.79892001943...5616.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{18}\cdot 7^{15}\cdot 13^{9}$
Root discriminant $86.90$
Ramified primes $2, 3, 7, 13$
Class number $29952$ (GRH)
Class group $[2, 2, 2, 2, 4, 468]$ (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![2682840203, -1117528098, 462427464, -125897574, 134273652, -45616323, 20679591, -5768394, 2781420, -744566, 246777, -47466, 17387, -3216, 972, -91, 33, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 33*x^16 - 91*x^15 + 972*x^14 - 3216*x^13 + 17387*x^12 - 47466*x^11 + 246777*x^10 - 744566*x^9 + 2781420*x^8 - 5768394*x^7 + 20679591*x^6 - 45616323*x^5 + 134273652*x^4 - 125897574*x^3 + 462427464*x^2 - 1117528098*x + 2682840203)
 
gp: K = bnfinit(x^18 - 3*x^17 + 33*x^16 - 91*x^15 + 972*x^14 - 3216*x^13 + 17387*x^12 - 47466*x^11 + 246777*x^10 - 744566*x^9 + 2781420*x^8 - 5768394*x^7 + 20679591*x^6 - 45616323*x^5 + 134273652*x^4 - 125897574*x^3 + 462427464*x^2 - 1117528098*x + 2682840203, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 33 x^{16} - 91 x^{15} + 972 x^{14} - 3216 x^{13} + 17387 x^{12} - 47466 x^{11} + 246777 x^{10} - 744566 x^{9} + 2781420 x^{8} - 5768394 x^{7} + 20679591 x^{6} - 45616323 x^{5} + 134273652 x^{4} - 125897574 x^{3} + 462427464 x^{2} - 1117528098 x + 2682840203 \)

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:  \(-79892001943166438658690014520815616=-\,2^{12}\cdot 3^{18}\cdot 7^{15}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.90$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{6} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7}$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{7} + \frac{1}{7}$, $\frac{1}{35} a^{15} - \frac{2}{35} a^{14} + \frac{1}{35} a^{13} + \frac{2}{35} a^{12} + \frac{3}{7} a^{11} + \frac{6}{35} a^{10} + \frac{12}{35} a^{9} - \frac{9}{35} a^{8} + \frac{2}{7} a^{7} + \frac{8}{35} a^{6} - \frac{3}{7} a^{5} - \frac{9}{35} a^{4} + \frac{13}{35} a^{3} + \frac{3}{7} a^{2} - \frac{11}{35} a + \frac{6}{35}$, $\frac{1}{96005} a^{16} - \frac{1143}{96005} a^{15} - \frac{3932}{96005} a^{14} - \frac{249}{96005} a^{13} + \frac{151}{7385} a^{12} - \frac{5662}{13715} a^{11} + \frac{43516}{96005} a^{10} - \frac{25176}{96005} a^{9} - \frac{33451}{96005} a^{8} - \frac{33767}{96005} a^{7} - \frac{6444}{13715} a^{6} - \frac{3629}{96005} a^{5} + \frac{13787}{96005} a^{4} - \frac{5334}{13715} a^{3} + \frac{45574}{96005} a^{2} - \frac{17373}{96005} a + \frac{11909}{96005}$, $\frac{1}{17959693180299501978789128110232854850154997071667541560776455395} a^{17} - \frac{8946776112730087523804571846676012624135363945684652311763}{2565670454328500282684161158604693550022142438809648794396636485} a^{16} - \frac{100369983452753179743015651839985755322434789733003201378520388}{17959693180299501978789128110232854850154997071667541560776455395} a^{15} + \frac{983556412756225921878222037722892295393167796935109806770362047}{17959693180299501978789128110232854850154997071667541560776455395} a^{14} + \frac{29609277044276640930610381815013709666873195414226006898836895}{513134090865700056536832231720938710004428487761929758879327297} a^{13} + \frac{588010230880403503718038327863056356781644310356838939341719472}{17959693180299501978789128110232854850154997071667541560776455395} a^{12} - \frac{400837917839205760078064512218629959921678530907701659391134884}{1381514860023038613753009854633296526934999774743657043136650415} a^{11} - \frac{1904484205016302645621384457065375168021487552309919657188327519}{17959693180299501978789128110232854850154997071667541560776455395} a^{10} - \frac{7991174968977164489208205513125115124955761187703725110577516133}{17959693180299501978789128110232854850154997071667541560776455395} a^{9} + \frac{4626062852690087627038828196578621711303732150795323383640563661}{17959693180299501978789128110232854850154997071667541560776455395} a^{8} + \frac{5079377386966585480917068193805277654320425434209876288600047228}{17959693180299501978789128110232854850154997071667541560776455395} a^{7} + \frac{1257421411642703584613894396304908685860041504157868402989715097}{3591938636059900395757825622046570970030999414333508312155291079} a^{6} - \frac{236339117978330524289744021862858735017162825123901162852222718}{2565670454328500282684161158604693550022142438809648794396636485} a^{5} + \frac{8194703733279411929049221413621902112753724365688566545216963406}{17959693180299501978789128110232854850154997071667541560776455395} a^{4} + \frac{788205203502196844645088533486101817347807095543881767272774579}{2565670454328500282684161158604693550022142438809648794396636485} a^{3} + \frac{170173887596789733628068876488250218679967709157070806450799880}{513134090865700056536832231720938710004428487761929758879327297} a^{2} - \frac{6591521945660548929051005690937558309952470566093684703289705837}{17959693180299501978789128110232854850154997071667541560776455395} a - \frac{3587616032195639123138490669362389153303747656227609200103511002}{17959693180299501978789128110232854850154997071667541560776455395}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{468}$, which has order $29952$ (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:  \( 296124.35954857944 \) (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

\(\Q(\sqrt{-91}) \), \(\Q(\zeta_{7})^+\), 3.3.756.1, 6.0.36924979.1, 6.0.8789652144.11, 9.9.1037426999616.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
3Data not computed
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$