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Properties

Label 18.0.74037208411...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}\cdot 5^{9}$
Root discriminant $27.36$
Ramified primes $2, 3, 5$
Class number $14$ (GRH)
Class group $[14]$ (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![15931, -54078, 101529, -128200, 124314, -97902, 66838, -40962, 22581, -11958, 5586, -2514, 996, -342, 147, -24, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 - 24*x^15 + 147*x^14 - 342*x^13 + 996*x^12 - 2514*x^11 + 5586*x^10 - 11958*x^9 + 22581*x^8 - 40962*x^7 + 66838*x^6 - 97902*x^5 + 124314*x^4 - 128200*x^3 + 101529*x^2 - 54078*x + 15931)
 
gp: K = bnfinit(x^18 + 18*x^16 - 24*x^15 + 147*x^14 - 342*x^13 + 996*x^12 - 2514*x^11 + 5586*x^10 - 11958*x^9 + 22581*x^8 - 40962*x^7 + 66838*x^6 - 97902*x^5 + 124314*x^4 - 128200*x^3 + 101529*x^2 - 54078*x + 15931, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} - 24 x^{15} + 147 x^{14} - 342 x^{13} + 996 x^{12} - 2514 x^{11} + 5586 x^{10} - 11958 x^{9} + 22581 x^{8} - 40962 x^{7} + 66838 x^{6} - 97902 x^{5} + 124314 x^{4} - 128200 x^{3} + 101529 x^{2} - 54078 x + 15931 \)

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:  \(-74037208411275264000000000=-\,2^{27}\cdot 3^{24}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.36$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{89} a^{15} - \frac{24}{89} a^{14} + \frac{1}{89} a^{13} - \frac{21}{89} a^{12} + \frac{19}{89} a^{11} + \frac{31}{89} a^{10} - \frac{2}{89} a^{9} + \frac{19}{89} a^{8} + \frac{14}{89} a^{7} + \frac{43}{89} a^{6} + \frac{28}{89} a^{5} + \frac{18}{89} a^{4} + \frac{43}{89} a^{3} + \frac{41}{89} a^{2} + \frac{27}{89} a$, $\frac{1}{89} a^{16} - \frac{41}{89} a^{14} + \frac{3}{89} a^{13} - \frac{40}{89} a^{12} + \frac{42}{89} a^{11} + \frac{30}{89} a^{10} - \frac{29}{89} a^{9} + \frac{25}{89} a^{8} + \frac{23}{89} a^{7} - \frac{8}{89} a^{6} - \frac{22}{89} a^{5} + \frac{30}{89} a^{4} + \frac{5}{89} a^{3} + \frac{32}{89} a^{2} + \frac{25}{89} a$, $\frac{1}{129437747179102706621319986935447} a^{17} + \frac{525016406814843271359583372230}{129437747179102706621319986935447} a^{16} + \frac{675800082994899903280098733165}{129437747179102706621319986935447} a^{15} + \frac{4286731991021986245603864409563}{129437747179102706621319986935447} a^{14} - \frac{32900435922211943353563036956395}{129437747179102706621319986935447} a^{13} - \frac{19001293212048667777204980444361}{129437747179102706621319986935447} a^{12} - \frac{8344126639182351020389458086536}{129437747179102706621319986935447} a^{11} + \frac{54690244317339487934679739722559}{129437747179102706621319986935447} a^{10} + \frac{63828747761785578114745900451947}{129437747179102706621319986935447} a^{9} - \frac{15196567396685929106311910554959}{129437747179102706621319986935447} a^{8} + \frac{44648440423939747417838576978682}{129437747179102706621319986935447} a^{7} + \frac{61933210450767848879677985755773}{129437747179102706621319986935447} a^{6} + \frac{51248945856209130586240189099606}{129437747179102706621319986935447} a^{5} - \frac{2509984562783937226349748706232}{129437747179102706621319986935447} a^{4} + \frac{55022857254382930076454224012362}{129437747179102706621319986935447} a^{3} + \frac{35752188196901289686409900164569}{129437747179102706621319986935447} a^{2} - \frac{33840330289857636233590815944148}{129437747179102706621319986935447} a - \frac{2744762415288312275139862226}{8124897820545019560687966037}$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( 17326.71661590147 \) (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

\(\Q(\sqrt{-10}) \), \(\Q(\zeta_{9})^+\), 3.1.648.1, 6.0.419904000.3, 6.0.419904000.2, 9.3.272097792.1

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 R R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e3_5.2t1.2c1$1$ $ 2^{3} \cdot 5 $ $x^{2} + 10$ $C_2$ (as 2T1) $1$ $-1$
1.2e3.2t1.2c1$1$ $ 2^{3}$ $x^{2} + 2$ $C_2$ (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) $0$ $1$
* 1.2e3_3e2_5.6t1.4c1$1$ $ 2^{3} \cdot 3^{2} \cdot 5 $ $x^{6} + 24 x^{4} - 2 x^{3} + 309 x^{2} + 66 x + 1691$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.2e3_3e2.6t1.3c1$1$ $ 2^{3} \cdot 3^{2}$ $x^{6} + 12 x^{4} + 36 x^{2} + 8$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.2e3_3e2.6t1.3c2$1$ $ 2^{3} \cdot 3^{2}$ $x^{6} + 12 x^{4} + 36 x^{2} + 8$ $C_6$ (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) $0$ $1$
* 1.2e3_3e2_5.6t1.4c2$1$ $ 2^{3} \cdot 3^{2} \cdot 5 $ $x^{6} + 24 x^{4} - 2 x^{3} + 309 x^{2} + 66 x + 1691$ $C_6$ (as 6T1) $0$ $-1$
* 2.2e3_3e4.3t2.1c1$2$ $ 2^{3} \cdot 3^{4}$ $x^{3} - 3 x - 10$ $S_3$ (as 3T2) $1$ $0$
* 2.2e3_3e4_5e2.6t3.10c1$2$ $ 2^{3} \cdot 3^{4} \cdot 5^{2}$ $x^{6} - 3 x^{5} - 6 x^{4} - 3 x^{3} + 30 x^{2} - 39 x + 109$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e3_3e2.6t5.1c1$2$ $ 2^{3} \cdot 3^{2}$ $x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} + 2 x^{2} + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e3_3e2_5e2.12t18.2c1$2$ $ 2^{3} \cdot 3^{2} \cdot 5^{2}$ $x^{18} + 18 x^{16} - 24 x^{15} + 147 x^{14} - 342 x^{13} + 996 x^{12} - 2514 x^{11} + 5586 x^{10} - 11958 x^{9} + 22581 x^{8} - 40962 x^{7} + 66838 x^{6} - 97902 x^{5} + 124314 x^{4} - 128200 x^{3} + 101529 x^{2} - 54078 x + 15931$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.2e3_3e2_5e2.12t18.2c2$2$ $ 2^{3} \cdot 3^{2} \cdot 5^{2}$ $x^{18} + 18 x^{16} - 24 x^{15} + 147 x^{14} - 342 x^{13} + 996 x^{12} - 2514 x^{11} + 5586 x^{10} - 11958 x^{9} + 22581 x^{8} - 40962 x^{7} + 66838 x^{6} - 97902 x^{5} + 124314 x^{4} - 128200 x^{3} + 101529 x^{2} - 54078 x + 15931$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.2e3_3e2.6t5.1c2$2$ $ 2^{3} \cdot 3^{2}$ $x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} + 2 x^{2} + 1$ $S_3\times C_3$ (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 *.