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Properties

Label 18.6.33637654972...8621.2
Degree $18$
Signature $[6, 6]$
Discriminant $3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 181$
Root discriminant $38.44$
Ramified primes $3, 7, 29, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11087, 14013, -21324, -29367, 12351, 28578, -5903, -16446, 5004, 5814, -2697, -1191, 975, 36, -207, 59, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 6*x^16 + 59*x^15 - 207*x^14 + 36*x^13 + 975*x^12 - 1191*x^11 - 2697*x^10 + 5814*x^9 + 5004*x^8 - 16446*x^7 - 5903*x^6 + 28578*x^5 + 12351*x^4 - 29367*x^3 - 21324*x^2 + 14013*x + 11087)
 
gp: K = bnfinit(x^18 - 6*x^17 + 6*x^16 + 59*x^15 - 207*x^14 + 36*x^13 + 975*x^12 - 1191*x^11 - 2697*x^10 + 5814*x^9 + 5004*x^8 - 16446*x^7 - 5903*x^6 + 28578*x^5 + 12351*x^4 - 29367*x^3 - 21324*x^2 + 14013*x + 11087, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 6 x^{16} + 59 x^{15} - 207 x^{14} + 36 x^{13} + 975 x^{12} - 1191 x^{11} - 2697 x^{10} + 5814 x^{9} + 5004 x^{8} - 16446 x^{7} - 5903 x^{6} + 28578 x^{5} + 12351 x^{4} - 29367 x^{3} - 21324 x^{2} + 14013 x + 11087 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33637654972350098104598658621=3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 181\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.44$
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:  $3, 7, 29, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{11} - \frac{1}{2} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{15} - \frac{1}{2} a^{10} - \frac{1}{3} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{5}{18} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{7}{18}$, $\frac{1}{18} a^{16} - \frac{1}{2} a^{11} - \frac{1}{3} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{5}{18} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{7}{18} a$, $\frac{1}{13894800532984846185357790945854} a^{17} - \frac{244325590220760790697512126795}{13894800532984846185357790945854} a^{16} - \frac{257930190740104451251508347199}{13894800532984846185357790945854} a^{15} - \frac{59304513161679610923695971861}{2315800088830807697559631824309} a^{14} + \frac{57714057555161456879777923658}{771933362943602565853210608103} a^{13} - \frac{208022586434971169276559670427}{4631600177661615395119263648618} a^{12} - \frac{691387365417017924550790301833}{4631600177661615395119263648618} a^{11} + \frac{899651746112540055915774999524}{2315800088830807697559631824309} a^{10} - \frac{352084003313705360436521785966}{2315800088830807697559631824309} a^{9} - \frac{123643823908164399114094401515}{1543866725887205131706421216206} a^{8} + \frac{681237560232068250351263143972}{2315800088830807697559631824309} a^{7} - \frac{558209376591696392629923131003}{1543866725887205131706421216206} a^{6} - \frac{49463500715995253308589874877}{479131052861546420184751411926} a^{5} + \frac{556702533724328305529454367345}{6947400266492423092678895472927} a^{4} + \frac{1114223389903699444610940784525}{13894800532984846185357790945854} a^{3} - \frac{3865990392947930162149145094161}{13894800532984846185357790945854} a^{2} + \frac{2390222394432197695327854243757}{6947400266492423092678895472927} a + \frac{4054198217169949640955672536119}{13894800532984846185357790945854}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 18642815.9973 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$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}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
181Data not computed