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Properties

Label 18.0.30815486463...7376.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 7^{14}\cdot 13^{9}$
Root discriminant $93.67$
Ramified primes $2, 3, 7, 13$
Class number $59904$ (GRH)
Class group $[2, 2, 2, 4, 4, 468]$ (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![599486056, -713931504, 539250456, -169850772, 87864060, -4401372, 2610916, 410082, -55194, -297157, 161823, -97176, 42574, -10782, 3228, -518, 96, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 96*x^16 - 518*x^15 + 3228*x^14 - 10782*x^13 + 42574*x^12 - 97176*x^11 + 161823*x^10 - 297157*x^9 - 55194*x^8 + 410082*x^7 + 2610916*x^6 - 4401372*x^5 + 87864060*x^4 - 169850772*x^3 + 539250456*x^2 - 713931504*x + 599486056)
 
gp: K = bnfinit(x^18 - 9*x^17 + 96*x^16 - 518*x^15 + 3228*x^14 - 10782*x^13 + 42574*x^12 - 97176*x^11 + 161823*x^10 - 297157*x^9 - 55194*x^8 + 410082*x^7 + 2610916*x^6 - 4401372*x^5 + 87864060*x^4 - 169850772*x^3 + 539250456*x^2 - 713931504*x + 599486056, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 96 x^{16} - 518 x^{15} + 3228 x^{14} - 10782 x^{13} + 42574 x^{12} - 97176 x^{11} + 161823 x^{10} - 297157 x^{9} - 55194 x^{8} + 410082 x^{7} + 2610916 x^{6} - 4401372 x^{5} + 87864060 x^{4} - 169850772 x^{3} + 539250456 x^{2} - 713931504 x + 599486056 \)

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:  \(-308154864637927691969232913151717376=-\,2^{12}\cdot 3^{21}\cdot 7^{14}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{7}$, $\frac{1}{52} a^{16} - \frac{1}{52} a^{15} - \frac{3}{26} a^{14} - \frac{1}{13} a^{13} + \frac{3}{52} a^{12} - \frac{2}{13} a^{11} - \frac{9}{52} a^{10} - \frac{5}{26} a^{9} - \frac{1}{26} a^{8} + \frac{11}{52} a^{7} + \frac{7}{52} a^{6} - \frac{3}{26} a^{5} + \frac{9}{26} a^{4} + \frac{9}{26} a^{3} - \frac{4}{13} a^{2}$, $\frac{1}{20422596281701333819309725059530919519708843880960781548110232728220052} a^{17} - \frac{26804147324973632669381807474732720290243499063934147482163960438213}{20422596281701333819309725059530919519708843880960781548110232728220052} a^{16} - \frac{931876751034575480680338975921114486100624106990333511721102343327103}{20422596281701333819309725059530919519708843880960781548110232728220052} a^{15} - \frac{616756847293450974997662032690872595903397259262791111503230937051997}{5105649070425333454827431264882729879927210970240195387027558182055013} a^{14} - \frac{303489628730383009789370212420681990778670346089287007419944502392647}{5105649070425333454827431264882729879927210970240195387027558182055013} a^{13} + \frac{868610201345746417024633222196858342357402100136667734851594839154695}{10211298140850666909654862529765459759854421940480390774055116364110026} a^{12} + \frac{182613835005335201544759570244212743548430498246601215116082237928859}{5105649070425333454827431264882729879927210970240195387027558182055013} a^{11} + \frac{24571499759603797779353401410588431724666418198269455493070675552379}{176056864497425291545773491892507926894041757594489496104398558001897} a^{10} - \frac{405593860027457441481658915069087259588325419376854635248178017315243}{20422596281701333819309725059530919519708843880960781548110232728220052} a^{9} - \frac{1008041120680122177084804634427480542133384037824716607098204428924205}{20422596281701333819309725059530919519708843880960781548110232728220052} a^{8} + \frac{3415054663476180761327625466409596945897941781659126103637546849100593}{20422596281701333819309725059530919519708843880960781548110232728220052} a^{7} - \frac{1122537738132523952802751268529800709991249442405014338359715437928196}{5105649070425333454827431264882729879927210970240195387027558182055013} a^{6} + \frac{25276088576623318811813445555159025645176453906063665207580040987139}{352113728994850583091546983785015853788083515188978992208797116003794} a^{5} + \frac{948758512888714722846902385530043668443971158661615862447491011005999}{5105649070425333454827431264882729879927210970240195387027558182055013} a^{4} + \frac{136298094021705282044103369112811110314269251373393278352428521775459}{785484472373128223819604809981958443065724764652337751850393566470002} a^{3} + \frac{2111955198236513375072621550942826151957042861785424684523183191959236}{5105649070425333454827431264882729879927210970240195387027558182055013} a^{2} + \frac{162630834258721109476193414068599564591859481229747439011405356054270}{392742236186564111909802404990979221532862382326168875925196783235001} a - \frac{80908594656692457210965018875495269481813063885742304894747859181635}{392742236186564111909802404990979221532862382326168875925196783235001}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{468}$, which has order $59904$ (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

\(\Q(\sqrt{-39}) \), \(\Q(\zeta_{7})^+\), 3.3.756.1, 6.0.142424919.1, 6.0.3766993776.2, 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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}$
3Data not computed
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$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}$