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Properties

Label 18.0.10522666778...5647.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 53^{6}$
Root discriminant $19.01$
Ramified primes $7, 53$
Class number $2$
Class group $[2]$
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![1, 3, 21, 45, 183, 144, 387, -346, -166, 240, 94, -180, 214, -98, 40, -26, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 17*x^16 - 26*x^15 + 40*x^14 - 98*x^13 + 214*x^12 - 180*x^11 + 94*x^10 + 240*x^9 - 166*x^8 - 346*x^7 + 387*x^6 + 144*x^5 + 183*x^4 + 45*x^3 + 21*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 17*x^16 - 26*x^15 + 40*x^14 - 98*x^13 + 214*x^12 - 180*x^11 + 94*x^10 + 240*x^9 - 166*x^8 - 346*x^7 + 387*x^6 + 144*x^5 + 183*x^4 + 45*x^3 + 21*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 17 x^{16} - 26 x^{15} + 40 x^{14} - 98 x^{13} + 214 x^{12} - 180 x^{11} + 94 x^{10} + 240 x^{9} - 166 x^{8} - 346 x^{7} + 387 x^{6} + 144 x^{5} + 183 x^{4} + 45 x^{3} + 21 x^{2} + 3 x + 1 \)

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:  \(-105226667788517176205647=-\,7^{15}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.01$
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:  $7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5}$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{98745078826041104260} a^{17} - \frac{4714061555271744181}{98745078826041104260} a^{16} - \frac{127785655527003547}{4937253941302055213} a^{15} - \frac{2474074754225539561}{49372539413020552130} a^{14} - \frac{3296612258676563361}{49372539413020552130} a^{13} + \frac{1944239369822935947}{24686269706510276065} a^{12} + \frac{489210846464088147}{49372539413020552130} a^{11} + \frac{3326471407070951801}{49372539413020552130} a^{10} - \frac{3828602743051459337}{24686269706510276065} a^{9} - \frac{2275348515375749139}{24686269706510276065} a^{8} - \frac{18808514879415822777}{49372539413020552130} a^{7} + \frac{1991559125182078147}{24686269706510276065} a^{6} + \frac{31505165916110647471}{98745078826041104260} a^{5} - \frac{11378737627714114441}{98745078826041104260} a^{4} - \frac{14859142428626506911}{49372539413020552130} a^{3} - \frac{17116920831631833661}{98745078826041104260} a^{2} - \frac{484896488630393913}{24686269706510276065} a + \frac{8178400699570292551}{19749015765208220852}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{1079221961855783189}{24686269706510276065} a^{17} + \frac{9559957616114906433}{49372539413020552130} a^{16} - \frac{7862821012445822079}{24686269706510276065} a^{15} - \frac{2698281314406409891}{24686269706510276065} a^{14} + \frac{1334614879615159058}{4937253941302055213} a^{13} + \frac{28353905884660856891}{24686269706510276065} a^{12} - \frac{49707137782911400077}{24686269706510276065} a^{11} - \frac{40555873751918850873}{4937253941302055213} a^{10} + \frac{278279393436072230658}{24686269706510276065} a^{9} - \frac{493227439224437268034}{24686269706510276065} a^{8} - \frac{176264853243586564903}{24686269706510276065} a^{7} + \frac{144019997988855549745}{4937253941302055213} a^{6} + \frac{96652775618478636262}{24686269706510276065} a^{5} - \frac{1774626336418581316847}{49372539413020552130} a^{4} - \frac{301758603604306121189}{24686269706510276065} a^{3} - \frac{391478876759448471347}{24686269706510276065} a^{2} - \frac{11110056191179123083}{9874507882604110426} a - \frac{3586954569388599281}{9874507882604110426} \) (order $14$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23621.7133529 \)
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

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), 3.3.2597.1, \(\Q(\zeta_{7})\), 6.0.47210863.1, 9.9.17515230173.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
7Data not computed
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$