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Properties

Label 18.0.56270551408...1696.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 11^{9}\cdot 17^{12}$
Root discriminant $34.81$
Ramified primes $2, 11, 17$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![271568, -494560, 703168, -171840, 243424, 113584, 29584, 22376, 60140, -39464, 38972, -18332, 9543, -3090, 1059, -224, 53, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 53*x^16 - 224*x^15 + 1059*x^14 - 3090*x^13 + 9543*x^12 - 18332*x^11 + 38972*x^10 - 39464*x^9 + 60140*x^8 + 22376*x^7 + 29584*x^6 + 113584*x^5 + 243424*x^4 - 171840*x^3 + 703168*x^2 - 494560*x + 271568)
 
gp: K = bnfinit(x^18 - 6*x^17 + 53*x^16 - 224*x^15 + 1059*x^14 - 3090*x^13 + 9543*x^12 - 18332*x^11 + 38972*x^10 - 39464*x^9 + 60140*x^8 + 22376*x^7 + 29584*x^6 + 113584*x^5 + 243424*x^4 - 171840*x^3 + 703168*x^2 - 494560*x + 271568, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 53 x^{16} - 224 x^{15} + 1059 x^{14} - 3090 x^{13} + 9543 x^{12} - 18332 x^{11} + 38972 x^{10} - 39464 x^{9} + 60140 x^{8} + 22376 x^{7} + 29584 x^{6} + 113584 x^{5} + 243424 x^{4} - 171840 x^{3} + 703168 x^{2} - 494560 x + 271568 \)

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:  \(-5627055140868788987624861696=-\,2^{12}\cdot 11^{9}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.81$
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:  $2, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{4} a^{7} - \frac{5}{12} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{120} a^{12} - \frac{1}{30} a^{11} - \frac{1}{40} a^{10} + \frac{1}{30} a^{9} + \frac{7}{40} a^{8} - \frac{1}{5} a^{7} - \frac{29}{120} a^{6} - \frac{1}{3} a^{5} + \frac{11}{60} a^{4} + \frac{1}{15} a^{3} - \frac{4}{15} a^{2} + \frac{13}{30} a - \frac{7}{30}$, $\frac{1}{120} a^{13} + \frac{1}{120} a^{11} + \frac{1}{60} a^{10} + \frac{7}{120} a^{9} - \frac{1}{24} a^{7} - \frac{1}{20} a^{6} - \frac{7}{30} a^{5} + \frac{2}{15} a^{4} + \frac{1}{6} a^{3} - \frac{7}{15} a^{2} + \frac{1}{6} a + \frac{1}{15}$, $\frac{1}{120} a^{14} - \frac{1}{30} a^{11} - \frac{1}{30} a^{9} - \frac{13}{60} a^{8} - \frac{1}{10} a^{7} - \frac{29}{120} a^{6} - \frac{11}{30} a^{5} - \frac{7}{20} a^{4} - \frac{11}{30} a^{3} + \frac{13}{30} a^{2} - \frac{1}{30} a + \frac{7}{30}$, $\frac{1}{720} a^{15} - \frac{1}{360} a^{14} - \frac{1}{720} a^{13} - \frac{1}{360} a^{12} - \frac{11}{720} a^{11} + \frac{7}{180} a^{10} - \frac{7}{720} a^{9} + \frac{11}{360} a^{8} + \frac{17}{120} a^{7} + \frac{71}{360} a^{6} + \frac{43}{90} a^{5} + \frac{2}{5} a^{4} - \frac{41}{180} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{5}{18}$, $\frac{1}{4601520} a^{16} - \frac{439}{1533840} a^{15} + \frac{10111}{4601520} a^{14} - \frac{25}{83664} a^{13} - \frac{4891}{1533840} a^{12} + \frac{547}{306768} a^{11} - \frac{4547}{4601520} a^{10} - \frac{10181}{920304} a^{9} - \frac{249313}{1150380} a^{8} + \frac{148279}{1150380} a^{7} - \frac{153287}{1150380} a^{6} + \frac{33821}{104580} a^{5} - \frac{22102}{57519} a^{4} - \frac{1369}{2772} a^{3} + \frac{27029}{191730} a^{2} - \frac{12349}{52290} a - \frac{2302}{26145}$, $\frac{1}{334624598441205997320758688480} a^{17} - \frac{825936008702678420131}{16731229922060299866037934424} a^{16} + \frac{37467167555031253830798397}{334624598441205997320758688480} a^{15} - \frac{808840239972668419840659}{1859025546895588874004214936} a^{14} + \frac{954282197927247458591372447}{334624598441205997320758688480} a^{13} - \frac{491679017496985389017407}{5070069673351606020011495280} a^{12} + \frac{473625343495358831303247731}{66924919688241199464151737696} a^{11} - \frac{38650273579510106629818273}{1327875390639706338574439240} a^{10} - \frac{511232252454164968366169377}{7605104510027409030017242920} a^{9} - \frac{881417125772580728833361851}{7967252343838238031446635440} a^{8} + \frac{19094661114604555506896422913}{83656149610301499330189672120} a^{7} - \frac{3965726876590123157906200397}{16731229922060299866037934424} a^{6} + \frac{40475426535728203507762226189}{83656149610301499330189672120} a^{5} + \frac{89141432818788937244691589}{1991813085959559507861658860} a^{4} - \frac{1396498933081875990881277787}{2987719628939339261792488290} a^{3} + \frac{16428655242906601708088448511}{41828074805150749665094836060} a^{2} + \frac{58415262301573742838539317}{126751741833790150500287382} a - \frac{147947282384701172026345897}{1901276127506852257504310730}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 3932278.1508 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.12716.2 x3, 3.1.44.1 x3, 3.1.12716.1 x3, 3.1.3179.1 x3, 6.0.1778663216.2, 6.0.21296.1, 6.0.1778663216.1, 6.0.111166451.1, 9.1.22617481454656.1 x9

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$