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Properties

Label 18.18.1016032598...3125.1
Degree $18$
Signature $[18, 0]$
Discriminant $5^{9}\cdot 139^{12}$
Root discriminant $60.00$
Ramified primes $5, 139$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![-739, -12897, -46372, 25097, 151141, -28892, -191308, 34914, 122098, -29674, -41286, 13412, 6882, -2982, -406, 292, -11, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 11*x^16 + 292*x^15 - 406*x^14 - 2982*x^13 + 6882*x^12 + 13412*x^11 - 41286*x^10 - 29674*x^9 + 122098*x^8 + 34914*x^7 - 191308*x^6 - 28892*x^5 + 151141*x^4 + 25097*x^3 - 46372*x^2 - 12897*x - 739)
 
gp: K = bnfinit(x^18 - 9*x^17 - 11*x^16 + 292*x^15 - 406*x^14 - 2982*x^13 + 6882*x^12 + 13412*x^11 - 41286*x^10 - 29674*x^9 + 122098*x^8 + 34914*x^7 - 191308*x^6 - 28892*x^5 + 151141*x^4 + 25097*x^3 - 46372*x^2 - 12897*x - 739, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 11 x^{16} + 292 x^{15} - 406 x^{14} - 2982 x^{13} + 6882 x^{12} + 13412 x^{11} - 41286 x^{10} - 29674 x^{9} + 122098 x^{8} + 34914 x^{7} - 191308 x^{6} - 28892 x^{5} + 151141 x^{4} + 25097 x^{3} - 46372 x^{2} - 12897 x - 739 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(101603259838455245078148283203125=5^{9}\cdot 139^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.00$
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:  $5, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{10330} a^{14} - \frac{7}{10330} a^{13} + \frac{711}{10330} a^{12} + \frac{99}{1033} a^{11} - \frac{2177}{10330} a^{10} + \frac{1252}{5165} a^{9} - \frac{1163}{10330} a^{8} - \frac{4361}{10330} a^{7} - \frac{1991}{5165} a^{6} - \frac{161}{5165} a^{5} + \frac{4527}{10330} a^{4} + \frac{5061}{10330} a^{3} - \frac{1933}{10330} a^{2} + \frac{151}{10330} a + \frac{4011}{10330}$, $\frac{1}{10330} a^{15} + \frac{331}{5165} a^{13} + \frac{401}{5165} a^{12} - \frac{206}{5165} a^{11} - \frac{481}{2066} a^{10} + \frac{87}{1033} a^{9} - \frac{1086}{5165} a^{8} - \frac{3519}{10330} a^{7} + \frac{1397}{5165} a^{6} - \frac{1446}{5165} a^{5} - \frac{457}{1033} a^{4} - \frac{2661}{10330} a^{3} + \frac{423}{2066} a^{2} - \frac{97}{10330} a - \frac{2913}{10330}$, $\frac{1}{106708900} a^{16} - \frac{2}{26677225} a^{15} + \frac{733}{26677225} a^{14} - \frac{5096}{26677225} a^{13} + \frac{4315771}{53354450} a^{12} + \frac{914913}{53354450} a^{11} + \frac{234445}{1067089} a^{10} - \frac{9439731}{53354450} a^{9} + \frac{4607888}{26677225} a^{8} + \frac{1788829}{26677225} a^{7} + \frac{7650704}{26677225} a^{6} + \frac{8806523}{53354450} a^{5} - \frac{688024}{26677225} a^{4} + \frac{3041729}{53354450} a^{3} + \frac{48030153}{106708900} a^{2} - \frac{8366073}{26677225} a - \frac{5948361}{106708900}$, $\frac{1}{474534478300} a^{17} + \frac{443}{94906895660} a^{16} - \frac{3329973}{118633619575} a^{15} + \frac{3602558}{118633619575} a^{14} - \frac{5489154393}{47453447830} a^{13} + \frac{37884740971}{237267239150} a^{12} - \frac{4892039368}{118633619575} a^{11} + \frac{6807096707}{118633619575} a^{10} + \frac{27267932073}{237267239150} a^{9} + \frac{13995321888}{118633619575} a^{8} + \frac{16692896341}{118633619575} a^{7} + \frac{106658022897}{237267239150} a^{6} + \frac{39245097608}{118633619575} a^{5} + \frac{21290847021}{47453447830} a^{4} + \frac{46659415007}{474534478300} a^{3} + \frac{103371194437}{474534478300} a^{2} + \frac{172345021283}{474534478300} a + \frac{130261423197}{474534478300}$

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:  $17$
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:  \( 9517414523.29 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 3.3.96605.1 x3, 3.3.19321.1, 6.6.46662630125.2, 6.6.46662630125.1, 6.6.2415125.1 x2, 9.9.901568676645125.1 x3

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

Sibling fields

Degree 6 sibling: 6.6.2415125.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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$$[\ ]_{2}^{3}$
$139$139.9.6.1$x^{9} + 2085 x^{6} + 1429754 x^{3} + 335702375$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
139.9.6.1$x^{9} + 2085 x^{6} + 1429754 x^{3} + 335702375$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$