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Properties

Label 18.2.74140900471...6601.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{13}\cdot 83^{5}\cdot 181^{5}$
Root discriminant $58.96$
Ramified primes $7, 83, 181$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-79183, 196056, -427833, 582461, -689064, 647554, -551651, 385625, -225634, 113360, -48163, 19528, -5870, 1711, -324, 74, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 6*x^16 + 74*x^15 - 324*x^14 + 1711*x^13 - 5870*x^12 + 19528*x^11 - 48163*x^10 + 113360*x^9 - 225634*x^8 + 385625*x^7 - 551651*x^6 + 647554*x^5 - 689064*x^4 + 582461*x^3 - 427833*x^2 + 196056*x - 79183)
 
gp: K = bnfinit(x^18 - x^17 + 6*x^16 + 74*x^15 - 324*x^14 + 1711*x^13 - 5870*x^12 + 19528*x^11 - 48163*x^10 + 113360*x^9 - 225634*x^8 + 385625*x^7 - 551651*x^6 + 647554*x^5 - 689064*x^4 + 582461*x^3 - 427833*x^2 + 196056*x - 79183, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 6 x^{16} + 74 x^{15} - 324 x^{14} + 1711 x^{13} - 5870 x^{12} + 19528 x^{11} - 48163 x^{10} + 113360 x^{9} - 225634 x^{8} + 385625 x^{7} - 551651 x^{6} + 647554 x^{5} - 689064 x^{4} + 582461 x^{3} - 427833 x^{2} + 196056 x - 79183 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(74140900471891896178682056546601=7^{13}\cdot 83^{5}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.96$
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:  $7, 83, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{181} a^{16} + \frac{76}{181} a^{15} + \frac{77}{181} a^{14} - \frac{39}{181} a^{13} + \frac{54}{181} a^{12} + \frac{10}{181} a^{11} + \frac{19}{181} a^{10} - \frac{76}{181} a^{9} - \frac{49}{181} a^{8} - \frac{30}{181} a^{7} - \frac{61}{181} a^{6} - \frac{45}{181} a^{5} + \frac{64}{181} a^{4} + \frac{25}{181} a^{3} - \frac{83}{181} a^{2} + \frac{41}{181} a - \frac{58}{181}$, $\frac{1}{28734650518972349935817253364898902061815052723} a^{17} - \frac{47502142924587408185267028373740347236780148}{28734650518972349935817253364898902061815052723} a^{16} - \frac{5655876620748425504344571897908640044169670820}{28734650518972349935817253364898902061815052723} a^{15} - \frac{7310259609604230463838019021320898059832290289}{28734650518972349935817253364898902061815052723} a^{14} + \frac{10788247594917307327811159913056051121572588268}{28734650518972349935817253364898902061815052723} a^{13} - \frac{189726308300408679465580104518986324107892889}{28734650518972349935817253364898902061815052723} a^{12} - \frac{9691289256044378037334280110109574613784787346}{28734650518972349935817253364898902061815052723} a^{11} - \frac{9550496094881088801297458420012980946050462098}{28734650518972349935817253364898902061815052723} a^{10} + \frac{11160429904512446461467665495219579285779967045}{28734650518972349935817253364898902061815052723} a^{9} + \frac{2124314854235856826895713228214388400319750435}{28734650518972349935817253364898902061815052723} a^{8} - \frac{3785292810268986048033824132729831769335146914}{28734650518972349935817253364898902061815052723} a^{7} - \frac{4042872212881680668901575568998000606415018861}{28734650518972349935817253364898902061815052723} a^{6} + \frac{6364640644283171626903470799513505512009492486}{28734650518972349935817253364898902061815052723} a^{5} + \frac{4103296085486180715606325895732939281480703696}{28734650518972349935817253364898902061815052723} a^{4} - \frac{13467590532696119478292619623354169069000798737}{28734650518972349935817253364898902061815052723} a^{3} - \frac{2526627191541990711966093647469289739286307114}{28734650518972349935817253364898902061815052723} a^{2} - \frac{4735444776221950956996336476355191951243840251}{28734650518972349935817253364898902061815052723} a - \frac{13412538909903553129508170754388391000093058925}{28734650518972349935817253364898902061815052723}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $9$
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:  \( 59862015.9899 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
83Data not computed
181Data not computed