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Properties

Label 18.0.54381578892...8931.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 13^{12}\cdot 17^{9}$
Root discriminant $39.48$
Ramified primes $3, 13, 17$
Class number $162$ (GRH)
Class group $[3, 3, 18]$ (GRH)
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, -1, 28, -14, 464, -147, 2650, 2413, 4817, 1662, 3756, 793, 1229, -804, 291, -81, 27, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 27*x^16 - 81*x^15 + 291*x^14 - 804*x^13 + 1229*x^12 + 793*x^11 + 3756*x^10 + 1662*x^9 + 4817*x^8 + 2413*x^7 + 2650*x^6 - 147*x^5 + 464*x^4 - 14*x^3 + 28*x^2 - x + 1)
 
gp: K = bnfinit(x^18 - 8*x^17 + 27*x^16 - 81*x^15 + 291*x^14 - 804*x^13 + 1229*x^12 + 793*x^11 + 3756*x^10 + 1662*x^9 + 4817*x^8 + 2413*x^7 + 2650*x^6 - 147*x^5 + 464*x^4 - 14*x^3 + 28*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 27 x^{16} - 81 x^{15} + 291 x^{14} - 804 x^{13} + 1229 x^{12} + 793 x^{11} + 3756 x^{10} + 1662 x^{9} + 4817 x^{8} + 2413 x^{7} + 2650 x^{6} - 147 x^{5} + 464 x^{4} - 14 x^{3} + 28 x^{2} - x + 1 \)

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:  \(-54381578892591924651626528931=-\,3^{9}\cdot 13^{12}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.48$
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:  $3, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{785} a^{16} + \frac{343}{785} a^{15} - \frac{39}{785} a^{14} - \frac{172}{785} a^{13} + \frac{8}{157} a^{12} + \frac{334}{785} a^{11} - \frac{102}{785} a^{10} - \frac{34}{157} a^{9} - \frac{181}{785} a^{8} - \frac{119}{785} a^{7} - \frac{353}{785} a^{6} - \frac{79}{785} a^{5} + \frac{188}{785} a^{4} + \frac{392}{785} a^{3} + \frac{69}{785} a^{2} + \frac{47}{785} a - \frac{51}{785}$, $\frac{1}{289541481089685835098330575} a^{17} + \frac{11031230986549814281009}{57908296217937167019666115} a^{16} + \frac{75944995363110044579450212}{289541481089685835098330575} a^{15} + \frac{28825885097415939451919591}{57908296217937167019666115} a^{14} - \frac{139658053468408388478316019}{289541481089685835098330575} a^{13} - \frac{19598277467796657987757286}{289541481089685835098330575} a^{12} - \frac{132064499220707219046854}{289541481089685835098330575} a^{11} - \frac{113281992663038375758324644}{289541481089685835098330575} a^{10} - \frac{38396885066890741027592976}{289541481089685835098330575} a^{9} - \frac{131327630960676359412265916}{289541481089685835098330575} a^{8} + \frac{32382659896786778373024344}{289541481089685835098330575} a^{7} + \frac{1609737622300021893137844}{57908296217937167019666115} a^{6} + \frac{16944569253787254178313567}{57908296217937167019666115} a^{5} - \frac{71364448902218129520427892}{289541481089685835098330575} a^{4} + \frac{64342454015472693791903313}{289541481089685835098330575} a^{3} - \frac{357355646527217790278895}{11581659243587433403933223} a^{2} + \frac{94445971004526075958596478}{289541481089685835098330575} a + \frac{89742120639804952004961358}{289541481089685835098330575}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{18}$, which has order $162$ (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:  \( 48887.8339977 \) (assuming GRH)
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

\(\Q(\sqrt{-51}) \), 3.1.8619.1 x3, 3.3.169.1, 6.0.3788645211.1, 6.0.22418019.1 x2, 6.0.3788645211.3, 9.3.640281040659.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.22418019.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$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$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3_17.2t1.1c1$1$ $ 3 \cdot 17 $ $x^{2} - x + 13$ $C_2$ (as 2T1) $1$ $-1$
* 1.3_13_17.6t1.1c1$1$ $ 3 \cdot 13 \cdot 17 $ $x^{6} - x^{5} + 30 x^{4} - 15 x^{3} + 509 x^{2} - 374 x + 3775$ $C_6$ (as 6T1) $0$ $-1$
* 1.3_13_17.6t1.1c2$1$ $ 3 \cdot 13 \cdot 17 $ $x^{6} - x^{5} + 30 x^{4} - 15 x^{3} + 509 x^{2} - 374 x + 3775$ $C_6$ (as 6T1) $0$ $-1$
* 1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.3_13e2_17.3t2.1c1$2$ $ 3 \cdot 13^{2} \cdot 17 $ $x^{3} - x^{2} + 9 x + 12$ $S_3$ (as 3T2) $1$ $0$
*2 2.3_13_17.6t5.3c1$2$ $ 3 \cdot 13 \cdot 17 $ $x^{18} - 8 x^{17} + 27 x^{16} - 81 x^{15} + 291 x^{14} - 804 x^{13} + 1229 x^{12} + 793 x^{11} + 3756 x^{10} + 1662 x^{9} + 4817 x^{8} + 2413 x^{7} + 2650 x^{6} - 147 x^{5} + 464 x^{4} - 14 x^{3} + 28 x^{2} - x + 1$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.3_13_17.6t5.3c2$2$ $ 3 \cdot 13 \cdot 17 $ $x^{18} - 8 x^{17} + 27 x^{16} - 81 x^{15} + 291 x^{14} - 804 x^{13} + 1229 x^{12} + 793 x^{11} + 3756 x^{10} + 1662 x^{9} + 4817 x^{8} + 2413 x^{7} + 2650 x^{6} - 147 x^{5} + 464 x^{4} - 14 x^{3} + 28 x^{2} - x + 1$ $S_3 \times C_3$ (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 *.