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Properties

Label 18.0.23465261991...9951.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,31^{15}$
Root discriminant $17.49$
Ramified prime $31$
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, 4, 11, 8, -53, -76, 251, -30, -131, 148, -97, 20, 20, -34, 42, -34, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 19*x^16 - 34*x^15 + 42*x^14 - 34*x^13 + 20*x^12 + 20*x^11 - 97*x^10 + 148*x^9 - 131*x^8 - 30*x^7 + 251*x^6 - 76*x^5 - 53*x^4 + 8*x^3 + 11*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 19*x^16 - 34*x^15 + 42*x^14 - 34*x^13 + 20*x^12 + 20*x^11 - 97*x^10 + 148*x^9 - 131*x^8 - 30*x^7 + 251*x^6 - 76*x^5 - 53*x^4 + 8*x^3 + 11*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 19 x^{16} - 34 x^{15} + 42 x^{14} - 34 x^{13} + 20 x^{12} + 20 x^{11} - 97 x^{10} + 148 x^{9} - 131 x^{8} - 30 x^{7} + 251 x^{6} - 76 x^{5} - 53 x^{4} + 8 x^{3} + 11 x^{2} + 4 x + 1 \)

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:  \(-23465261991844685929951=-\,31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.49$
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:  $31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{3}{16} a^{4} + \frac{7}{16} a^{3} - \frac{5}{16} a + \frac{3}{16}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} + \frac{1}{32} a^{11} - \frac{1}{16} a^{10} - \frac{1}{32} a^{9} - \frac{1}{16} a^{7} + \frac{1}{32} a^{6} - \frac{3}{16} a^{5} - \frac{1}{8} a^{4} - \frac{15}{32} a^{3} + \frac{15}{32} a^{2} - \frac{1}{16} a + \frac{15}{32}$, $\frac{1}{64} a^{14} + \frac{3}{64} a^{11} + \frac{1}{64} a^{10} + \frac{3}{64} a^{9} + \frac{1}{32} a^{8} - \frac{13}{64} a^{7} + \frac{7}{64} a^{6} - \frac{1}{32} a^{5} + \frac{5}{64} a^{4} + \frac{1}{8} a^{3} + \frac{29}{64} a^{2} + \frac{1}{64} a + \frac{19}{64}$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{14} + \frac{3}{128} a^{12} + \frac{3}{64} a^{11} + \frac{1}{64} a^{10} + \frac{7}{128} a^{9} + \frac{1}{128} a^{8} - \frac{1}{32} a^{7} - \frac{25}{128} a^{6} - \frac{25}{128} a^{5} - \frac{29}{128} a^{4} - \frac{11}{128} a^{3} - \frac{3}{32} a^{2} - \frac{19}{64} a - \frac{3}{128}$, $\frac{1}{128} a^{16} - \frac{1}{128} a^{14} - \frac{1}{128} a^{13} - \frac{3}{128} a^{12} + \frac{1}{32} a^{11} + \frac{1}{128} a^{10} - \frac{1}{32} a^{9} - \frac{3}{128} a^{8} + \frac{11}{128} a^{7} + \frac{5}{64} a^{6} - \frac{7}{64} a^{5} + \frac{3}{16} a^{4} - \frac{43}{128} a^{3} - \frac{7}{64} a^{2} + \frac{15}{128} a + \frac{17}{128}$, $\frac{1}{482176} a^{17} - \frac{1767}{482176} a^{16} + \frac{41}{120544} a^{15} + \frac{1221}{482176} a^{14} - \frac{1679}{120544} a^{13} - \frac{567}{30136} a^{12} - \frac{7569}{482176} a^{11} - \frac{13685}{482176} a^{10} + \frac{341}{30136} a^{9} + \frac{31885}{482176} a^{8} - \frac{25083}{482176} a^{7} - \frac{60981}{482176} a^{6} + \frac{24525}{482176} a^{5} + \frac{6599}{60272} a^{4} - \frac{14187}{30136} a^{3} + \frac{155929}{482176} a^{2} - \frac{3397}{241088} a - \frac{75217}{241088}$

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:  \( 2615.01137266 \)
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

\(\Q(\sqrt{-31}) \), 3.1.31.1 x3, 3.3.961.1, 6.0.29791.1, 6.0.28629151.1, 6.0.28629151.2 x2, 9.3.27512614111.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.28629151.2
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$31$31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.31.2t1.1c1$1$ $ 31 $ $x^{2} - x + 8$ $C_2$ (as 2T1) $1$ $-1$
* 1.31.3t1.1c1$1$ $ 31 $ $x^{3} - x^{2} - 10 x + 8$ $C_3$ (as 3T1) $0$ $1$
* 1.31.6t1.1c1$1$ $ 31 $ $x^{6} - x^{5} + 3 x^{4} - 11 x^{3} + 44 x^{2} - 36 x + 32$ $C_6$ (as 6T1) $0$ $-1$
* 1.31.6t1.1c2$1$ $ 31 $ $x^{6} - x^{5} + 3 x^{4} - 11 x^{3} + 44 x^{2} - 36 x + 32$ $C_6$ (as 6T1) $0$ $-1$
* 1.31.3t1.1c2$1$ $ 31 $ $x^{3} - x^{2} - 10 x + 8$ $C_3$ (as 3T1) $0$ $1$
*2 2.31.3t2.1c1$2$ $ 31 $ $x^{3} + x - 1$ $S_3$ (as 3T2) $1$ $0$
*2 2.31e2.6t5.3c1$2$ $ 31^{2}$ $x^{18} - 6 x^{17} + 19 x^{16} - 34 x^{15} + 42 x^{14} - 34 x^{13} + 20 x^{12} + 20 x^{11} - 97 x^{10} + 148 x^{9} - 131 x^{8} - 30 x^{7} + 251 x^{6} - 76 x^{5} - 53 x^{4} + 8 x^{3} + 11 x^{2} + 4 x + 1$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.31e2.6t5.3c2$2$ $ 31^{2}$ $x^{18} - 6 x^{17} + 19 x^{16} - 34 x^{15} + 42 x^{14} - 34 x^{13} + 20 x^{12} + 20 x^{11} - 97 x^{10} + 148 x^{9} - 131 x^{8} - 30 x^{7} + 251 x^{6} - 76 x^{5} - 53 x^{4} + 8 x^{3} + 11 x^{2} + 4 x + 1$ $S_3 \times C_3$ (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 *.