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Properties

Label 18.4.61536947391...7883.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,7^{13}\cdot 83^{6}\cdot 181^{5}$
Root discriminant $75.37$
Ramified primes $7, 83, 181$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 18T705

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-862709, 7576254, -17157215, 18580467, -8647295, -2574322, 7643106, -6574343, 3726139, -1580866, 553361, -165051, 42845, -9567, 1843, -360, 47, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 47*x^16 - 360*x^15 + 1843*x^14 - 9567*x^13 + 42845*x^12 - 165051*x^11 + 553361*x^10 - 1580866*x^9 + 3726139*x^8 - 6574343*x^7 + 7643106*x^6 - 2574322*x^5 - 8647295*x^4 + 18580467*x^3 - 17157215*x^2 + 7576254*x - 862709)
 
gp: K = bnfinit(x^18 - 6*x^17 + 47*x^16 - 360*x^15 + 1843*x^14 - 9567*x^13 + 42845*x^12 - 165051*x^11 + 553361*x^10 - 1580866*x^9 + 3726139*x^8 - 6574343*x^7 + 7643106*x^6 - 2574322*x^5 - 8647295*x^4 + 18580467*x^3 - 17157215*x^2 + 7576254*x - 862709, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 47 x^{16} - 360 x^{15} + 1843 x^{14} - 9567 x^{13} + 42845 x^{12} - 165051 x^{11} + 553361 x^{10} - 1580866 x^{9} + 3726139 x^{8} - 6574343 x^{7} + 7643106 x^{6} - 2574322 x^{5} - 8647295 x^{4} + 18580467 x^{3} - 17157215 x^{2} + 7576254 x - 862709 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6153694739167027382830610693367883=-\,7^{13}\cdot 83^{6}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{4} + \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{16} - \frac{3}{7} a^{14} + \frac{1}{7} a^{13} + \frac{2}{7} a^{12} + \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{2409764413071130427658510862976842345781997646434571003883} a^{17} - \frac{93059547681987096262474512771639977219000537060609355849}{2409764413071130427658510862976842345781997646434571003883} a^{16} + \frac{136853411843499216615984887585392128997052751579873290700}{2409764413071130427658510862976842345781997646434571003883} a^{15} + \frac{17439150137564942439487117283552174974296688966089378684}{344252059010161489665501551853834620825999663776367286269} a^{14} + \frac{1040232736196026985484837458821092465374453830456541734560}{2409764413071130427658510862976842345781997646434571003883} a^{13} + \frac{248136216313649363510284661300266950159270333800180203771}{2409764413071130427658510862976842345781997646434571003883} a^{12} + \frac{858988104462101010544438369475926859396411296142471011033}{2409764413071130427658510862976842345781997646434571003883} a^{11} + \frac{141157678612090337582245539370937865248097437196491582543}{344252059010161489665501551853834620825999663776367286269} a^{10} + \frac{1123645086856589642384449002127424253781471852011003184995}{2409764413071130427658510862976842345781997646434571003883} a^{9} + \frac{212739799712595823748511337274660563797504544400891997513}{2409764413071130427658510862976842345781997646434571003883} a^{8} - \frac{409397188570812700230784247737087246521092484937225069305}{2409764413071130427658510862976842345781997646434571003883} a^{7} + \frac{677644245952925467567082395988733027184105383951411987056}{2409764413071130427658510862976842345781997646434571003883} a^{6} - \frac{242182709758948004583055942075876304931490100830412747309}{2409764413071130427658510862976842345781997646434571003883} a^{5} - \frac{94109237640132410261075809492310219432099835329089896925}{344252059010161489665501551853834620825999663776367286269} a^{4} + \frac{53667286681086341500294863535056127556309998286683560255}{344252059010161489665501551853834620825999663776367286269} a^{3} + \frac{141684216057733814579619708070854719843842568600018086}{1901945077404207125223765479855439894066296484952305449} a^{2} + \frac{1199303481205032337068147067852507769049454437694483594687}{2409764413071130427658510862976842345781997646434571003883} a + \frac{825587474953934297568770965114716855897075270048776898882}{2409764413071130427658510862976842345781997646434571003883}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $10$
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:  \( 2502547216.88 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T705:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 64 conjugacy class representatives for t18n705 are not computed
Character table for t18n705 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 $18$ $18$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ $18$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
83Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$