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Properties

Label 18.0.50869871130...1703.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 23^{9}$
Root discriminant $20.75$
Ramified primes $3, 23$
Class number $3$
Class group $[3]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 45, -138, 405, -774, 956, -534, 468, -400, 450, -198, 254, -60, 45, -3, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 3*x^15 + 45*x^14 - 60*x^13 + 254*x^12 - 198*x^11 + 450*x^10 - 400*x^9 + 468*x^8 - 534*x^7 + 956*x^6 - 774*x^5 + 405*x^4 - 138*x^3 + 45*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^18 + 3*x^16 - 3*x^15 + 45*x^14 - 60*x^13 + 254*x^12 - 198*x^11 + 450*x^10 - 400*x^9 + 468*x^8 - 534*x^7 + 956*x^6 - 774*x^5 + 405*x^4 - 138*x^3 + 45*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} - 3 x^{15} + 45 x^{14} - 60 x^{13} + 254 x^{12} - 198 x^{11} + 450 x^{10} - 400 x^{9} + 468 x^{8} - 534 x^{7} + 956 x^{6} - 774 x^{5} + 405 x^{4} - 138 x^{3} + 45 x^{2} - 9 x + 1 \)

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:  \(-508698711308514601331703=-\,3^{24}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.75$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{26} a^{12} - \frac{2}{13} a^{11} + \frac{3}{26} a^{10} - \frac{1}{13} a^{9} - \frac{3}{26} a^{8} + \frac{3}{13} a^{7} + \frac{2}{13} a^{6} + \frac{7}{26} a^{5} - \frac{5}{13} a^{4} - \frac{5}{26} a^{3} + \frac{5}{13} a^{2} + \frac{7}{26} a - \frac{6}{13}$, $\frac{1}{104} a^{13} - \frac{1}{104} a^{12} + \frac{17}{104} a^{11} - \frac{19}{104} a^{10} + \frac{17}{104} a^{9} - \frac{3}{104} a^{8} + \frac{9}{104} a^{7} + \frac{19}{104} a^{6} + \frac{37}{104} a^{5} + \frac{43}{104} a^{4} - \frac{5}{104} a^{3} + \frac{11}{104} a^{2} + \frac{35}{104} a - \frac{49}{104}$, $\frac{1}{208} a^{14} - \frac{21}{104} a^{11} + \frac{1}{104} a^{10} + \frac{23}{104} a^{9} + \frac{1}{104} a^{8} + \frac{9}{52} a^{7} + \frac{6}{13} a^{6} - \frac{2}{13} a^{5} - \frac{5}{104} a^{4} + \frac{17}{104} a^{3} + \frac{47}{104} a^{2} - \frac{37}{104} a + \frac{3}{16}$, $\frac{1}{208} a^{15} - \frac{1}{104} a^{12} + \frac{25}{104} a^{11} - \frac{21}{104} a^{10} + \frac{1}{8} a^{9} + \frac{5}{52} a^{8} + \frac{3}{26} a^{7} - \frac{5}{13} a^{6} + \frac{31}{104} a^{5} + \frac{25}{104} a^{4} + \frac{51}{104} a^{3} + \frac{7}{104} a^{2} + \frac{7}{208} a + \frac{5}{26}$, $\frac{1}{191152} a^{16} + \frac{379}{191152} a^{15} + \frac{1}{11947} a^{14} + \frac{331}{95576} a^{13} - \frac{123}{47788} a^{12} + \frac{8009}{47788} a^{11} + \frac{7217}{47788} a^{10} + \frac{4661}{95576} a^{9} + \frac{6445}{47788} a^{8} + \frac{2463}{23894} a^{7} + \frac{10451}{95576} a^{6} - \frac{20411}{47788} a^{5} - \frac{19719}{47788} a^{4} - \frac{9869}{23894} a^{3} - \frac{81911}{191152} a^{2} - \frac{22235}{191152} a + \frac{6973}{23894}$, $\frac{1}{2780879296} a^{17} + \frac{3833}{2780879296} a^{16} + \frac{166905}{695219824} a^{15} - \frac{2071559}{2780879296} a^{14} - \frac{4605317}{1390439648} a^{13} + \frac{6298693}{1390439648} a^{12} + \frac{6180341}{347609912} a^{11} - \frac{252042831}{1390439648} a^{10} + \frac{159223221}{695219824} a^{9} - \frac{162112095}{695219824} a^{8} + \frac{64818553}{347609912} a^{7} - \frac{398590551}{1390439648} a^{6} - \frac{543304225}{1390439648} a^{5} + \frac{121440327}{347609912} a^{4} - \frac{44352703}{213913792} a^{3} - \frac{1364399973}{2780879296} a^{2} + \frac{15734112}{43451239} a - \frac{819081329}{2780879296}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 8598.44783956 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, \(\Q(\zeta_{9})^+\), 6.0.12167.1, 6.0.79827687.1, 6.0.79827687.2 x2, 9.3.6466042647.5 x3

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

Sibling fields

Degree 6 sibling: 6.0.79827687.2
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.23.2t1.1c1$1$ $ 23 $ $x^{2} - x + 6$ $C_2$ (as 2T1) $1$ $-1$
* 1.3e2_23.6t1.2c1$1$ $ 3^{2} \cdot 23 $ $x^{6} - 3 x^{5} + 15 x^{4} - 23 x^{3} + 123 x^{2} - 153 x + 489$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2_23.6t1.2c2$1$ $ 3^{2} \cdot 23 $ $x^{6} - 3 x^{5} + 15 x^{4} - 23 x^{3} + 123 x^{2} - 153 x + 489$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.23.3t2.1c1$2$ $ 23 $ $x^{3} - x^{2} + 1$ $S_3$ (as 3T2) $1$ $0$
*2 2.3e4_23.6t5.4c1$2$ $ 3^{4} \cdot 23 $ $x^{18} + 3 x^{16} - 3 x^{15} + 45 x^{14} - 60 x^{13} + 254 x^{12} - 198 x^{11} + 450 x^{10} - 400 x^{9} + 468 x^{8} - 534 x^{7} + 956 x^{6} - 774 x^{5} + 405 x^{4} - 138 x^{3} + 45 x^{2} - 9 x + 1$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.3e4_23.6t5.4c2$2$ $ 3^{4} \cdot 23 $ $x^{18} + 3 x^{16} - 3 x^{15} + 45 x^{14} - 60 x^{13} + 254 x^{12} - 198 x^{11} + 450 x^{10} - 400 x^{9} + 468 x^{8} - 534 x^{7} + 956 x^{6} - 774 x^{5} + 405 x^{4} - 138 x^{3} + 45 x^{2} - 9 x + 1$ $S_3 \times C_3$ (as 18T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.