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Properties

Label 18.0.11590726342...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{12}\cdot 7^{15}$
Root discriminant $14.80$
Ramified primes $5, 7$
Class number $1$
Class group Trivial
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, -5, 16, -34, 63, -91, 92, -7, -128, 193, -128, -7, 92, -91, 63, -34, 16, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 16*x^16 - 34*x^15 + 63*x^14 - 91*x^13 + 92*x^12 - 7*x^11 - 128*x^10 + 193*x^9 - 128*x^8 - 7*x^7 + 92*x^6 - 91*x^5 + 63*x^4 - 34*x^3 + 16*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^18 - 5*x^17 + 16*x^16 - 34*x^15 + 63*x^14 - 91*x^13 + 92*x^12 - 7*x^11 - 128*x^10 + 193*x^9 - 128*x^8 - 7*x^7 + 92*x^6 - 91*x^5 + 63*x^4 - 34*x^3 + 16*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 16 x^{16} - 34 x^{15} + 63 x^{14} - 91 x^{13} + 92 x^{12} - 7 x^{11} - 128 x^{10} + 193 x^{9} - 128 x^{8} - 7 x^{7} + 92 x^{6} - 91 x^{5} + 63 x^{4} - 34 x^{3} + 16 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1159072634263427734375=-\,5^{12}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.80$
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:  $5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{1688} a^{16} + \frac{15}{1688} a^{14} + \frac{41}{1688} a^{13} + \frac{21}{844} a^{12} + \frac{289}{1688} a^{11} - \frac{101}{422} a^{10} + \frac{5}{1688} a^{9} + \frac{45}{844} a^{8} + \frac{5}{1688} a^{7} - \frac{101}{422} a^{6} + \frac{711}{1688} a^{5} + \frac{58}{211} a^{4} + \frac{41}{1688} a^{3} - \frac{829}{1688} a^{2} - \frac{3}{8} a + \frac{1}{1688}$, $\frac{1}{21944} a^{17} - \frac{3}{10972} a^{16} - \frac{407}{21944} a^{15} + \frac{373}{21944} a^{14} - \frac{1259}{21944} a^{13} + \frac{37}{21944} a^{12} - \frac{1505}{21944} a^{11} + \frac{3695}{21944} a^{10} - \frac{573}{21944} a^{9} + \frac{4107}{21944} a^{8} - \frac{4443}{21944} a^{7} - \frac{3195}{21944} a^{6} - \frac{4435}{21944} a^{5} + \frac{5}{104} a^{4} - \frac{4019}{10972} a^{3} - \frac{4099}{21944} a^{2} - \frac{3903}{10972} a + \frac{1318}{2743}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{27109}{10972} a^{17} - \frac{261283}{21944} a^{16} + \frac{812637}{21944} a^{15} - \frac{831379}{10972} a^{14} + \frac{376703}{2743} a^{13} - \frac{1048051}{5486} a^{12} + \frac{484435}{2743} a^{11} + \frac{98517}{2743} a^{10} - \frac{3593581}{10972} a^{9} + \frac{2235179}{5486} a^{8} - \frac{2214161}{10972} a^{7} - \frac{1056147}{10972} a^{6} + \frac{2480535}{10972} a^{5} - \frac{1844107}{10972} a^{4} + \frac{553597}{5486} a^{3} - \frac{1098597}{21944} a^{2} + \frac{444357}{21944} a - \frac{45925}{10972} \) (order $14$)
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:  \( 13425.8394152 \)
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

\(\Q(\sqrt{-7}) \), 3.1.175.1 x3, \(\Q(\zeta_{7})^+\), 6.0.214375.1, 6.0.10504375.1 x2, \(\Q(\zeta_{7})\), 9.3.12867859375.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.10504375.1
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
5Data not computed
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.6t1.1c1$1$ $ 7 $ $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.6t1.1c2$1$ $ 7 $ $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.5e2_7.3t2.1c1$2$ $ 5^{2} \cdot 7 $ $x^{3} - x^{2} + 2 x - 3$ $S_3$ (as 3T2) $1$ $0$
*2 2.5e2_7e2.6t5.3c1$2$ $ 5^{2} \cdot 7^{2}$ $x^{18} - 5 x^{17} + 16 x^{16} - 34 x^{15} + 63 x^{14} - 91 x^{13} + 92 x^{12} - 7 x^{11} - 128 x^{10} + 193 x^{9} - 128 x^{8} - 7 x^{7} + 92 x^{6} - 91 x^{5} + 63 x^{4} - 34 x^{3} + 16 x^{2} - 5 x + 1$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.5e2_7e2.6t5.3c2$2$ $ 5^{2} \cdot 7^{2}$ $x^{18} - 5 x^{17} + 16 x^{16} - 34 x^{15} + 63 x^{14} - 91 x^{13} + 92 x^{12} - 7 x^{11} - 128 x^{10} + 193 x^{9} - 128 x^{8} - 7 x^{7} + 92 x^{6} - 91 x^{5} + 63 x^{4} - 34 x^{3} + 16 x^{2} - 5 x + 1$ $S_3 \times C_3$ (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 *.