↑

Properties

Label 18.0.44241302708...1936.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 37^{14}\cdot 79^{9}$
Root discriminant $234.00$
Ramified primes $2, 37, 79$
Class number $107911440$ (GRH)
Class group $[6, 6, 2997540]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1602731119741, -567101046533, 597897722246, -162811998196, 95605863343, -20860733787, 8881121966, -1583274847, 540532502, -79178651, 22786278, -2722651, 676338, -64155, 13779, -962, 174, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 174*x^16 - 962*x^15 + 13779*x^14 - 64155*x^13 + 676338*x^12 - 2722651*x^11 + 22786278*x^10 - 79178651*x^9 + 540532502*x^8 - 1583274847*x^7 + 8881121966*x^6 - 20860733787*x^5 + 95605863343*x^4 - 162811998196*x^3 + 597897722246*x^2 - 567101046533*x + 1602731119741)
 
gp: K = bnfinit(x^18 - 7*x^17 + 174*x^16 - 962*x^15 + 13779*x^14 - 64155*x^13 + 676338*x^12 - 2722651*x^11 + 22786278*x^10 - 79178651*x^9 + 540532502*x^8 - 1583274847*x^7 + 8881121966*x^6 - 20860733787*x^5 + 95605863343*x^4 - 162811998196*x^3 + 597897722246*x^2 - 567101046533*x + 1602731119741, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 174 x^{16} - 962 x^{15} + 13779 x^{14} - 64155 x^{13} + 676338 x^{12} - 2722651 x^{11} + 22786278 x^{10} - 79178651 x^{9} + 540532502 x^{8} - 1583274847 x^{7} + 8881121966 x^{6} - 20860733787 x^{5} + 95605863343 x^{4} - 162811998196 x^{3} + 597897722246 x^{2} - 567101046533 x + 1602731119741 \)

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:  \(-4424130270896887308908712516757313137831936=-\,2^{12}\cdot 37^{14}\cdot 79^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $234.00$
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:  $2, 37, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{17} - \frac{1458280310509708520142766575559052850054818821654099931027003293548878}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{16} + \frac{1492987330158135323489143142205624829799277728229294078294265717888867}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{15} - \frac{409829051798248383405642533709867778136854416500975980543726471920}{6144572201358873121311270255285376974730114563959257233782878438609} a^{14} + \frac{1361730991291646460737205027947283268650130849570979419150496998288891}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{13} + \frac{833789445781052207411747619685915859842797051302076757254299293768745}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{12} + \frac{1277107921346783609358996690591981311586010998340880875203508929318613}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{11} - \frac{1258212712265965924462798932352904915303187514799467516661402385357983}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{10} + \frac{1389708065180441766931002325299126413792767215920824856954281508243935}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{9} + \frac{1325815653781864949311678673173619139128547864901998211092011693161577}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{8} + \frac{114485979622359586070510422168199207516176953371474676082063005151601}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{7} + \frac{741178161570522377776216291597436302159362001446161250411031124384978}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{6} + \frac{118089084473789295272621065112908905690436533627837378222542737424899}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{5} + \frac{1082372986331197269234314281067678064406921381759248900988969357794181}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{4} - \frac{311012739108002245497732733258904933527225847646987208958224861256297}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{3} - \frac{642973447307365522089658560108645726712919960133900882159837228744167}{3361080994143303597357264829641101205177372666485713706879234505919123} a^{2} - \frac{125165110508846621133555553248321247175588510743005053524403059103955}{3361080994143303597357264829641101205177372666485713706879234505919123} a + \frac{36107094632566290079390407393043308935994678776164019526160765440303}{3361080994143303597357264829641101205177372666485713706879234505919123}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{2997540}$, which has order $107911440$ (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:  \( 615797.1340659427 \) (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

\(\Q(\sqrt{-79}) \), 3.3.1369.1, 3.3.148.1, 6.0.10799526256.4, 6.0.924034465279.1, 9.9.6075640136512.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$79$79.6.3.2$x^{6} - 6241 x^{2} + 1972156$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
79.12.6.1$x^{12} + 5916468 x^{6} - 3077056399 x^{2} + 8751148398756$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$