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Properties

Label 18.6.80722386956...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 5^{9}\cdot 7^{12}$
Root discriminant $18.73$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (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![-1, -5, 28, 11, -75, 69, -65, -162, 295, 131, -304, -50, 44, 25, 30, -4, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 - 4*x^15 + 30*x^14 + 25*x^13 + 44*x^12 - 50*x^11 - 304*x^10 + 131*x^9 + 295*x^8 - 162*x^7 - 65*x^6 + 69*x^5 - 75*x^4 + 11*x^3 + 28*x^2 - 5*x - 1)
 
gp: K = bnfinit(x^18 - 9*x^16 - 4*x^15 + 30*x^14 + 25*x^13 + 44*x^12 - 50*x^11 - 304*x^10 + 131*x^9 + 295*x^8 - 162*x^7 - 65*x^6 + 69*x^5 - 75*x^4 + 11*x^3 + 28*x^2 - 5*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{16} - 4 x^{15} + 30 x^{14} + 25 x^{13} + 44 x^{12} - 50 x^{11} - 304 x^{10} + 131 x^{9} + 295 x^{8} - 162 x^{7} - 65 x^{6} + 69 x^{5} - 75 x^{4} + 11 x^{3} + 28 x^{2} - 5 x - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(80722386956232000000000=2^{12}\cdot 3^{6}\cdot 5^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.73$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{251} a^{16} - \frac{109}{251} a^{15} - \frac{92}{251} a^{14} + \frac{115}{251} a^{13} + \frac{98}{251} a^{12} + \frac{7}{251} a^{11} - \frac{17}{251} a^{10} - \frac{119}{251} a^{9} - \frac{56}{251} a^{8} + \frac{4}{251} a^{7} - \frac{76}{251} a^{6} - \frac{76}{251} a^{5} + \frac{78}{251} a^{4} - \frac{8}{251} a^{3} + \frac{70}{251} a^{2} - \frac{8}{251} a + \frac{3}{251}$, $\frac{1}{6832826114954482217} a^{17} - \frac{3625612921324399}{6832826114954482217} a^{16} + \frac{722329000027568295}{6832826114954482217} a^{15} - \frac{482805184884482658}{6832826114954482217} a^{14} + \frac{2641728517857068046}{6832826114954482217} a^{13} - \frac{2658292027045742631}{6832826114954482217} a^{12} + \frac{3252532761338959297}{6832826114954482217} a^{11} + \frac{1431586054900295169}{6832826114954482217} a^{10} + \frac{1562382547266327060}{6832826114954482217} a^{9} + \frac{319004128624873638}{6832826114954482217} a^{8} - \frac{474814844572608187}{6832826114954482217} a^{7} + \frac{2644251958448350595}{6832826114954482217} a^{6} - \frac{1575131762797618053}{6832826114954482217} a^{5} + \frac{3154660602575436626}{6832826114954482217} a^{4} - \frac{1965667048283050273}{6832826114954482217} a^{3} + \frac{1401142149463669852}{6832826114954482217} a^{2} + \frac{1614118709768711085}{6832826114954482217} a + \frac{2963659863169465768}{6832826114954482217}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 26434.0630612897 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 3.1.588.1, 6.2.43218000.1, 6.6.300125.1, 9.3.203297472.1

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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$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}$
7Data not computed

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_5.2t1.1c1$1$ $ 3 \cdot 5 $ $x^{2} - x + 4$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.3_7.6t1.2c1$1$ $ 3 \cdot 7 $ $x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.5_7.6t1.1c1$1$ $ 5 \cdot 7 $ $x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$ $C_6$ (as 6T1) $0$ $1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.3_5_7.6t1.2c1$1$ $ 3 \cdot 5 \cdot 7 $ $x^{6} - x^{5} + 7 x^{4} - 5 x^{3} + 49 x^{2} - 9 x + 139$ $C_6$ (as 6T1) $0$ $-1$
1.3_7.6t1.2c2$1$ $ 3 \cdot 7 $ $x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$ $C_6$ (as 6T1) $0$ $-1$
1.3_5_7.6t1.2c2$1$ $ 3 \cdot 5 \cdot 7 $ $x^{6} - x^{5} + 7 x^{4} - 5 x^{3} + 49 x^{2} - 9 x + 139$ $C_6$ (as 6T1) $0$ $-1$
* 1.5_7.6t1.1c2$1$ $ 5 \cdot 7 $ $x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$ $C_6$ (as 6T1) $0$ $1$
* 2.2e2_3_7e2.3t2.1c1$2$ $ 2^{2} \cdot 3 \cdot 7^{2}$ $x^{3} - x^{2} + 5 x + 1$ $S_3$ (as 3T2) $1$ $0$
* 2.2e2_3_5e2_7e2.6t3.7c1$2$ $ 2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ $x^{6} - 3 x^{5} + 5 x^{4} + 23 x^{3} - 62 x^{2} + 8 x + 244$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e2_3_7.6t5.1c1$2$ $ 2^{2} \cdot 3 \cdot 7 $ $x^{6} - 3 x^{5} + 4 x^{4} - x^{3} - 2 x^{2} + x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e2_3_5e2_7.12t18.1c1$2$ $ 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $ $x^{18} - 9 x^{16} - 4 x^{15} + 30 x^{14} + 25 x^{13} + 44 x^{12} - 50 x^{11} - 304 x^{10} + 131 x^{9} + 295 x^{8} - 162 x^{7} - 65 x^{6} + 69 x^{5} - 75 x^{4} + 11 x^{3} + 28 x^{2} - 5 x - 1$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.2e2_3_7.6t5.1c2$2$ $ 2^{2} \cdot 3 \cdot 7 $ $x^{6} - 3 x^{5} + 4 x^{4} - x^{3} - 2 x^{2} + x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e2_3_5e2_7.12t18.1c2$2$ $ 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $ $x^{18} - 9 x^{16} - 4 x^{15} + 30 x^{14} + 25 x^{13} + 44 x^{12} - 50 x^{11} - 304 x^{10} + 131 x^{9} + 295 x^{8} - 162 x^{7} - 65 x^{6} + 69 x^{5} - 75 x^{4} + 11 x^{3} + 28 x^{2} - 5 x - 1$ $S_3 \times C_6$ (as 18T6) $0$ $0$

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 *.