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Properties

Label 18.0.11990701314...5739.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 59^{9}$
Root discriminant $28.11$
Ramified primes $7, 59$
Class number $9$
Class group $[3, 3]$
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![944, 1888, 3272, -2416, 3739, -7690, 7654, -6028, 7335, -3606, 3243, -1508, 1103, -426, 215, -80, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 32*x^16 - 80*x^15 + 215*x^14 - 426*x^13 + 1103*x^12 - 1508*x^11 + 3243*x^10 - 3606*x^9 + 7335*x^8 - 6028*x^7 + 7654*x^6 - 7690*x^5 + 3739*x^4 - 2416*x^3 + 3272*x^2 + 1888*x + 944)
 
gp: K = bnfinit(x^18 - 8*x^17 + 32*x^16 - 80*x^15 + 215*x^14 - 426*x^13 + 1103*x^12 - 1508*x^11 + 3243*x^10 - 3606*x^9 + 7335*x^8 - 6028*x^7 + 7654*x^6 - 7690*x^5 + 3739*x^4 - 2416*x^3 + 3272*x^2 + 1888*x + 944, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 32 x^{16} - 80 x^{15} + 215 x^{14} - 426 x^{13} + 1103 x^{12} - 1508 x^{11} + 3243 x^{10} - 3606 x^{9} + 7335 x^{8} - 6028 x^{7} + 7654 x^{6} - 7690 x^{5} + 3739 x^{4} - 2416 x^{3} + 3272 x^{2} + 1888 x + 944 \)

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:  \(-119907013147065124216135739=-\,7^{12}\cdot 59^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.11$
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:  $7, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{48} a^{10} + \frac{1}{48} a^{9} - \frac{1}{16} a^{8} - \frac{1}{24} a^{7} - \frac{1}{8} a^{6} - \frac{1}{24} a^{5} + \frac{3}{16} a^{4} - \frac{5}{16} a^{3} - \frac{19}{48} a^{2} + \frac{1}{3} a + \frac{1}{12}$, $\frac{1}{48} a^{11} - \frac{1}{12} a^{9} + \frac{1}{48} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{6} + \frac{11}{48} a^{5} - \frac{1}{12} a^{3} - \frac{13}{48} a^{2} + \frac{1}{4} a - \frac{1}{12}$, $\frac{1}{96} a^{12} - \frac{1}{96} a^{10} + \frac{1}{24} a^{9} - \frac{1}{96} a^{8} - \frac{1}{48} a^{7} + \frac{5}{96} a^{6} - \frac{3}{16} a^{5} + \frac{11}{96} a^{4} + \frac{1}{48} a^{3} + \frac{1}{32} a^{2} - \frac{5}{12} a - \frac{1}{8}$, $\frac{1}{384} a^{13} - \frac{1}{128} a^{11} + \frac{9}{128} a^{9} - \frac{5}{48} a^{8} - \frac{1}{128} a^{7} + \frac{23}{192} a^{6} - \frac{17}{128} a^{5} + \frac{7}{192} a^{4} + \frac{47}{384} a^{3} + \frac{79}{192} a^{2} - \frac{31}{96} a - \frac{7}{48}$, $\frac{1}{384} a^{14} + \frac{1}{384} a^{12} - \frac{1}{384} a^{10} - \frac{1}{8} a^{9} - \frac{31}{384} a^{8} - \frac{5}{192} a^{7} + \frac{17}{384} a^{6} + \frac{43}{192} a^{5} - \frac{29}{384} a^{4} - \frac{73}{192} a^{3} - \frac{5}{48} a^{2} + \frac{7}{16} a - \frac{3}{8}$, $\frac{1}{384} a^{15} + \frac{1}{192} a^{11} - \frac{5}{192} a^{9} - \frac{3}{64} a^{8} + \frac{5}{96} a^{7} + \frac{5}{48} a^{6} + \frac{11}{192} a^{5} - \frac{1}{24} a^{4} + \frac{19}{128} a^{3} + \frac{29}{192} a^{2} + \frac{43}{96} a - \frac{17}{48}$, $\frac{1}{3072} a^{16} + \frac{1}{1024} a^{15} + \frac{1}{1536} a^{14} + \frac{1}{3072} a^{13} - \frac{1}{768} a^{12} + \frac{11}{3072} a^{11} - \frac{1}{768} a^{10} - \frac{181}{3072} a^{9} - \frac{9}{128} a^{8} + \frac{349}{3072} a^{7} - \frac{85}{1536} a^{6} - \frac{749}{3072} a^{5} - \frac{233}{1024} a^{4} - \frac{5}{24} a^{3} + \frac{49}{384} a^{2} + \frac{27}{64} a - \frac{31}{192}$, $\frac{1}{41294861032928256} a^{17} + \frac{395694981319}{13764953677642752} a^{16} + \frac{267362546401}{397065971470464} a^{15} - \frac{20281716416587}{41294861032928256} a^{14} - \frac{1321429992983}{6882476838821376} a^{13} + \frac{4894167729147}{4588317892547584} a^{12} + \frac{145356325799989}{20647430516464128} a^{11} + \frac{281152194398003}{41294861032928256} a^{10} + \frac{2577824065925027}{20647430516464128} a^{9} + \frac{3881235871903709}{41294861032928256} a^{8} + \frac{347657606343785}{5161857629116032} a^{7} - \frac{328314966793061}{3176527771763712} a^{6} + \frac{367689592415521}{13764953677642752} a^{5} - \frac{3973026479875811}{20647430516464128} a^{4} - \frac{1206527885901937}{2580928814558016} a^{3} + \frac{261532532571893}{860309604852672} a^{2} + \frac{17669899645049}{2580928814558016} a - \frac{369233647150025}{1290464407279008}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 361929.852547 \)
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

\(\Q(\sqrt{-59}) \), 3.1.2891.1 x3, 3.1.2891.3 x3, 3.1.59.1 x3, 3.1.2891.2 x3, 6.0.493114979.3, 6.0.493114979.2, 6.0.205379.1, 6.0.493114979.1, 9.1.1425595404289.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ R

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$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$59$59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$