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Properties

Label 18.0.20375763784...4672.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 19^{15}$
Root discriminant $32.90$
Ramified primes $2, 19$
Class number $6$
Class group $[6]$
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![1432, 7392, 13300, 8536, -1276, -4340, -1637, 2310, 637, -1136, 695, -470, 129, -142, 155, -60, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 17*x^16 - 60*x^15 + 155*x^14 - 142*x^13 + 129*x^12 - 470*x^11 + 695*x^10 - 1136*x^9 + 637*x^8 + 2310*x^7 - 1637*x^6 - 4340*x^5 - 1276*x^4 + 8536*x^3 + 13300*x^2 + 7392*x + 1432)
 
gp: K = bnfinit(x^18 - 6*x^17 + 17*x^16 - 60*x^15 + 155*x^14 - 142*x^13 + 129*x^12 - 470*x^11 + 695*x^10 - 1136*x^9 + 637*x^8 + 2310*x^7 - 1637*x^6 - 4340*x^5 - 1276*x^4 + 8536*x^3 + 13300*x^2 + 7392*x + 1432, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 17 x^{16} - 60 x^{15} + 155 x^{14} - 142 x^{13} + 129 x^{12} - 470 x^{11} + 695 x^{10} - 1136 x^{9} + 637 x^{8} + 2310 x^{7} - 1637 x^{6} - 4340 x^{5} - 1276 x^{4} + 8536 x^{3} + 13300 x^{2} + 7392 x + 1432 \)

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:  \(-2037576378429183552150044672=-\,2^{27}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.90$
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:  $2, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{74} a^{14} - \frac{16}{37} a^{13} - \frac{9}{74} a^{12} + \frac{9}{37} a^{11} + \frac{5}{74} a^{10} - \frac{2}{37} a^{9} + \frac{11}{74} a^{8} - \frac{12}{37} a^{7} - \frac{5}{74} a^{6} - \frac{11}{37} a^{5} + \frac{29}{74} a^{4} - \frac{5}{37} a^{3} + \frac{1}{74} a^{2} + \frac{13}{37} a + \frac{15}{37}$, $\frac{1}{222} a^{15} + \frac{1}{222} a^{14} + \frac{15}{74} a^{13} + \frac{17}{222} a^{12} + \frac{27}{74} a^{11} - \frac{61}{222} a^{10} + \frac{9}{74} a^{9} + \frac{43}{222} a^{8} + \frac{91}{222} a^{7} + \frac{35}{222} a^{6} - \frac{31}{222} a^{5} - \frac{89}{222} a^{4} - \frac{11}{74} a^{3} + \frac{59}{222} a^{2} + \frac{14}{111}$, $\frac{1}{4884} a^{16} + \frac{1}{814} a^{15} - \frac{31}{4884} a^{14} - \frac{13}{1221} a^{13} - \frac{1325}{4884} a^{12} + \frac{775}{2442} a^{11} - \frac{1571}{4884} a^{10} + \frac{695}{2442} a^{9} - \frac{639}{1628} a^{8} - \frac{446}{1221} a^{7} - \frac{409}{1628} a^{6} - \frac{785}{2442} a^{5} - \frac{1717}{4884} a^{4} + \frac{65}{1221} a^{3} - \frac{224}{1221} a^{2} + \frac{424}{1221} a + \frac{38}{1221}$, $\frac{1}{52711633148971909754678916} a^{17} - \frac{73291838749202575033}{52711633148971909754678916} a^{16} + \frac{9097525611749692389731}{52711633148971909754678916} a^{15} + \frac{93974321178573358885789}{17570544382990636584892972} a^{14} + \frac{21004392260699981983557299}{52711633148971909754678916} a^{13} + \frac{10877242848619482481246567}{52711633148971909754678916} a^{12} + \frac{14824075074752753092484279}{52711633148971909754678916} a^{11} - \frac{3839988775300464527204957}{17570544382990636584892972} a^{10} - \frac{20888937417821735355412351}{52711633148971909754678916} a^{9} - \frac{9492104917984622817221387}{52711633148971909754678916} a^{8} - \frac{3255866025096391947374659}{52711633148971909754678916} a^{7} + \frac{23087309042173206229831385}{52711633148971909754678916} a^{6} + \frac{8298729587890497933261455}{17570544382990636584892972} a^{5} + \frac{4210542949831080641599339}{17570544382990636584892972} a^{4} - \frac{5087722175274480513003853}{13177908287242977438669729} a^{3} + \frac{3546932638473251838546609}{8785272191495318292446486} a^{2} - \frac{165724121440135428524359}{1197991662476634312606339} a - \frac{262433537119934307430760}{13177908287242977438669729}$

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

Class group and class number

$C_{6}$, which has order $6$

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:  \( 399403.722263 \)
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

\(\Q(\sqrt{-38}) \), 3.1.152.1 x3, 3.3.361.1, 6.0.3511808.1, 6.0.1267762688.3 x2, 6.0.1267762688.1, 9.3.457662330368.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.1267762688.3
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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$19$19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e3_19.2t1.2c1$1$ $ 2^{3} \cdot 19 $ $x^{2} + 38$ $C_2$ (as 2T1) $1$ $-1$
* 1.19.3t1.1c1$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
* 1.2e3_19.6t1.4c1$1$ $ 2^{3} \cdot 19 $ $x^{6} + 38 x^{4} + 152 x^{2} + 152$ $C_6$ (as 6T1) $0$ $-1$
* 1.2e3_19.6t1.4c2$1$ $ 2^{3} \cdot 19 $ $x^{6} + 38 x^{4} + 152 x^{2} + 152$ $C_6$ (as 6T1) $0$ $-1$
* 1.19.3t1.1c2$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
*2 2.2e3_19.3t2.1c1$2$ $ 2^{3} \cdot 19 $ $x^{3} - x^{2} - 2 x - 2$ $S_3$ (as 3T2) $1$ $0$
*2 2.2e3_19e2.6t5.3c1$2$ $ 2^{3} \cdot 19^{2}$ $x^{18} - 6 x^{17} + 17 x^{16} - 60 x^{15} + 155 x^{14} - 142 x^{13} + 129 x^{12} - 470 x^{11} + 695 x^{10} - 1136 x^{9} + 637 x^{8} + 2310 x^{7} - 1637 x^{6} - 4340 x^{5} - 1276 x^{4} + 8536 x^{3} + 13300 x^{2} + 7392 x + 1432$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.2e3_19e2.6t5.3c2$2$ $ 2^{3} \cdot 19^{2}$ $x^{18} - 6 x^{17} + 17 x^{16} - 60 x^{15} + 155 x^{14} - 142 x^{13} + 129 x^{12} - 470 x^{11} + 695 x^{10} - 1136 x^{9} + 637 x^{8} + 2310 x^{7} - 1637 x^{6} - 4340 x^{5} - 1276 x^{4} + 8536 x^{3} + 13300 x^{2} + 7392 x + 1432$ $S_3 \times C_3$ (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 *.