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Properties

Label 18.0.13006915496...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 5^{9}\cdot 11^{9}$
Root discriminant $32.09$
Ramified primes $3, 5, 11$
Class number $28$ (GRH)
Class group $[28]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9784, -42972, 48636, 3137, 55563, 783, 823, -11433, 7326, 2370, -354, -1398, 78, 171, 57, -24, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 - 24*x^15 + 57*x^14 + 171*x^13 + 78*x^12 - 1398*x^11 - 354*x^10 + 2370*x^9 + 7326*x^8 - 11433*x^7 + 823*x^6 + 783*x^5 + 55563*x^4 + 3137*x^3 + 48636*x^2 - 42972*x + 9784)
 
gp: K = bnfinit(x^18 - 9*x^16 - 24*x^15 + 57*x^14 + 171*x^13 + 78*x^12 - 1398*x^11 - 354*x^10 + 2370*x^9 + 7326*x^8 - 11433*x^7 + 823*x^6 + 783*x^5 + 55563*x^4 + 3137*x^3 + 48636*x^2 - 42972*x + 9784, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{16} - 24 x^{15} + 57 x^{14} + 171 x^{13} + 78 x^{12} - 1398 x^{11} - 354 x^{10} + 2370 x^{9} + 7326 x^{8} - 11433 x^{7} + 823 x^{6} + 783 x^{5} + 55563 x^{4} + 3137 x^{3} + 48636 x^{2} - 42972 x + 9784 \)

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:  \(-1300691549639793389396484375=-\,3^{24}\cdot 5^{9}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.09$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{3}{10} a^{9} + \frac{2}{5} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} + \frac{2}{5} a^{5} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{1}{5} a^{8} - \frac{1}{2} a^{6} - \frac{1}{5} a^{5} + \frac{1}{10} a^{4} + \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{11} - \frac{1}{20} a^{10} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{3}{20} a^{7} - \frac{1}{20} a^{5} - \frac{1}{20} a^{4} - \frac{2}{5} a^{3} + \frac{3}{20} a^{2} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{100} a^{15} + \frac{1}{50} a^{14} - \frac{1}{50} a^{13} - \frac{1}{20} a^{12} + \frac{3}{20} a^{11} - \frac{2}{25} a^{10} - \frac{19}{50} a^{9} - \frac{3}{100} a^{8} - \frac{1}{2} a^{7} + \frac{3}{20} a^{6} - \frac{9}{20} a^{5} - \frac{12}{25} a^{4} - \frac{13}{100} a^{3} - \frac{3}{25} a^{2} + \frac{11}{25} a - \frac{8}{25}$, $\frac{1}{151300} a^{16} - \frac{263}{151300} a^{15} + \frac{3243}{151300} a^{14} + \frac{1031}{30260} a^{13} - \frac{83}{7565} a^{12} - \frac{13359}{75650} a^{11} + \frac{1087}{151300} a^{10} - \frac{37053}{151300} a^{9} - \frac{10243}{30260} a^{8} - \frac{1011}{15130} a^{7} + \frac{2043}{15130} a^{6} - \frac{681}{2225} a^{5} + \frac{29931}{75650} a^{4} + \frac{74863}{151300} a^{3} + \frac{50939}{151300} a^{2} - \frac{31921}{75650} a + \frac{1728}{7565}$, $\frac{1}{12117905985561708011349837035604700} a^{17} + \frac{2964401256092127648858591588}{3029476496390427002837459258901175} a^{16} - \frac{4011050660420380818394800775639}{3029476496390427002837459258901175} a^{15} - \frac{45529453571083149861845585301823}{12117905985561708011349837035604700} a^{14} - \frac{262372505564078277833758657961157}{12117905985561708011349837035604700} a^{13} - \frac{108014479301614186613304038149947}{3029476496390427002837459258901175} a^{12} + \frac{478967803220695277675133466947518}{3029476496390427002837459258901175} a^{11} + \frac{501852506305657195363667593385909}{12117905985561708011349837035604700} a^{10} - \frac{596215626460046664404474989930212}{3029476496390427002837459258901175} a^{9} - \frac{1507545430444263745858385764314343}{12117905985561708011349837035604700} a^{8} - \frac{237967178553596272629016200582825}{484716239422468320453993481424188} a^{7} - \frac{1415734670214206144380158575258627}{3029476496390427002837459258901175} a^{6} + \frac{2040955792159325448290217678837287}{12117905985561708011349837035604700} a^{5} + \frac{92035062265152285414291062758961}{1211790598556170801134983703560470} a^{4} - \frac{2908690863316192908258466157580627}{6058952992780854005674918517802350} a^{3} + \frac{1018695842849702698341782629010099}{3029476496390427002837459258901175} a^{2} - \frac{2798462191824512218927228016919633}{6058952992780854005674918517802350} a - \frac{834431398412387879025962877930318}{3029476496390427002837459258901175}$

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

Class group and class number

$C_{28}$, which has order $28$ (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:  \( 56262.000163134624 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-55}) \), \(\Q(\zeta_{9})^+\), 3.1.891.1, 6.0.1091586375.2, 6.0.1091586375.3, 9.3.707347971.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$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$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5_11.2t1.1c1$1$ $ 5 \cdot 11 $ $x^{2} - x + 14$ $C_2$ (as 2T1) $1$ $-1$
1.11.2t1.1c1$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.3e2_5.6t1.1c1$1$ $ 3^{2} \cdot 5 $ $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ $C_6$ (as 6T1) $0$ $1$
* 1.3e2_5_11.6t1.2c1$1$ $ 3^{2} \cdot 5 \cdot 11 $ $x^{6} - 3 x^{5} + 39 x^{4} - 71 x^{3} + 627 x^{2} - 681 x + 4049$ $C_6$ (as 6T1) $0$ $-1$
1.3e2_11.6t1.2c1$1$ $ 3^{2} \cdot 11 $ $x^{6} - 3 x^{5} + 6 x^{4} - 5 x^{3} + 33 x^{2} - 54 x + 111$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2_5_11.6t1.2c2$1$ $ 3^{2} \cdot 5 \cdot 11 $ $x^{6} - 3 x^{5} + 39 x^{4} - 71 x^{3} + 627 x^{2} - 681 x + 4049$ $C_6$ (as 6T1) $0$ $-1$
1.3e2_5.6t1.1c2$1$ $ 3^{2} \cdot 5 $ $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ $C_6$ (as 6T1) $0$ $1$
1.3e2_11.6t1.2c2$1$ $ 3^{2} \cdot 11 $ $x^{6} - 3 x^{5} + 6 x^{4} - 5 x^{3} + 33 x^{2} - 54 x + 111$ $C_6$ (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) $0$ $1$
* 2.3e4_11.3t2.1c1$2$ $ 3^{4} \cdot 11 $ $x^{3} + 6 x - 1$ $S_3$ (as 3T2) $1$ $0$
* 2.3e4_5e2_11.6t3.2c1$2$ $ 3^{4} \cdot 5^{2} \cdot 11 $ $x^{6} - 3 x^{5} + 12 x^{4} - 17 x^{3} + 51 x^{2} - 24 x - 64$ $D_{6}$ (as 6T3) $1$ $0$
* 2.3e2_11.6t5.1c1$2$ $ 3^{2} \cdot 11 $ $x^{6} - x^{4} - 2 x^{3} + 3 x^{2} + x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.3e2_5e2_11.12t18.2c1$2$ $ 3^{2} \cdot 5^{2} \cdot 11 $ $x^{18} - 9 x^{16} - 24 x^{15} + 57 x^{14} + 171 x^{13} + 78 x^{12} - 1398 x^{11} - 354 x^{10} + 2370 x^{9} + 7326 x^{8} - 11433 x^{7} + 823 x^{6} + 783 x^{5} + 55563 x^{4} + 3137 x^{3} + 48636 x^{2} - 42972 x + 9784$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.3e2_5e2_11.12t18.2c2$2$ $ 3^{2} \cdot 5^{2} \cdot 11 $ $x^{18} - 9 x^{16} - 24 x^{15} + 57 x^{14} + 171 x^{13} + 78 x^{12} - 1398 x^{11} - 354 x^{10} + 2370 x^{9} + 7326 x^{8} - 11433 x^{7} + 823 x^{6} + 783 x^{5} + 55563 x^{4} + 3137 x^{3} + 48636 x^{2} - 42972 x + 9784$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.3e2_11.6t5.1c2$2$ $ 3^{2} \cdot 11 $ $x^{6} - x^{4} - 2 x^{3} + 3 x^{2} + x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$

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 *.