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Properties

Label 18.0.24227950659...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 5^{9}\cdot 17^{9}$
Root discriminant $33.22$
Ramified primes $3, 5, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14475, -49275, 111225, -156870, 178821, -174375, 159331, -111108, 62337, -30444, 15474, -7974, 4228, -1932, 744, -228, 54, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 54*x^16 - 228*x^15 + 744*x^14 - 1932*x^13 + 4228*x^12 - 7974*x^11 + 15474*x^10 - 30444*x^9 + 62337*x^8 - 111108*x^7 + 159331*x^6 - 174375*x^5 + 178821*x^4 - 156870*x^3 + 111225*x^2 - 49275*x + 14475)
 
gp: K = bnfinit(x^18 - 9*x^17 + 54*x^16 - 228*x^15 + 744*x^14 - 1932*x^13 + 4228*x^12 - 7974*x^11 + 15474*x^10 - 30444*x^9 + 62337*x^8 - 111108*x^7 + 159331*x^6 - 174375*x^5 + 178821*x^4 - 156870*x^3 + 111225*x^2 - 49275*x + 14475, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 54 x^{16} - 228 x^{15} + 744 x^{14} - 1932 x^{13} + 4228 x^{12} - 7974 x^{11} + 15474 x^{10} - 30444 x^{9} + 62337 x^{8} - 111108 x^{7} + 159331 x^{6} - 174375 x^{5} + 178821 x^{4} - 156870 x^{3} + 111225 x^{2} - 49275 x + 14475 \)

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:  \(-2422795065922582753693359375=-\,3^{21}\cdot 5^{9}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.22$
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:  $3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{115} a^{12} - \frac{6}{115} a^{11} - \frac{11}{115} a^{10} - \frac{1}{23} a^{9} + \frac{38}{115} a^{8} + \frac{42}{115} a^{7} + \frac{32}{115} a^{6} + \frac{11}{23} a^{5} - \frac{9}{115} a^{4} + \frac{29}{115} a^{3} - \frac{16}{115} a^{2} - \frac{7}{23} a - \frac{10}{23}$, $\frac{1}{115} a^{13} - \frac{1}{115} a^{11} - \frac{2}{115} a^{10} + \frac{8}{115} a^{9} + \frac{8}{23} a^{8} - \frac{38}{115} a^{7} - \frac{6}{115} a^{6} - \frac{24}{115} a^{5} - \frac{5}{23} a^{4} - \frac{26}{115} a^{3} + \frac{53}{115} a^{2} - \frac{6}{23} a + \frac{9}{23}$, $\frac{1}{171925} a^{14} - \frac{7}{171925} a^{13} - \frac{504}{171925} a^{12} + \frac{623}{34385} a^{11} + \frac{5923}{171925} a^{10} - \frac{58336}{171925} a^{9} - \frac{46227}{171925} a^{8} + \frac{2148}{6877} a^{7} - \frac{64769}{171925} a^{6} - \frac{15332}{171925} a^{5} + \frac{42051}{171925} a^{4} + \frac{453}{1495} a^{3} + \frac{9883}{34385} a^{2} - \frac{65}{529} a - \frac{2061}{6877}$, $\frac{1}{171925} a^{15} - \frac{553}{171925} a^{13} - \frac{413}{171925} a^{12} - \frac{6657}{171925} a^{11} - \frac{675}{6877} a^{10} + \frac{61196}{171925} a^{9} + \frac{73961}{171925} a^{8} + \frac{36051}{171925} a^{7} + \frac{724}{2645} a^{6} - \frac{5021}{13225} a^{5} + \frac{2602}{171925} a^{4} + \frac{7169}{34385} a^{3} - \frac{3814}{34385} a^{2} - \frac{1099}{6877} a - \frac{673}{6877}$, $\frac{1}{715822631875} a^{16} - \frac{8}{715822631875} a^{15} + \frac{819782}{715822631875} a^{14} - \frac{5738334}{715822631875} a^{13} - \frac{251739538}{715822631875} a^{12} + \frac{1585035206}{715822631875} a^{11} + \frac{28849248591}{715822631875} a^{10} - \frac{158912495017}{715822631875} a^{9} - \frac{216519625317}{715822631875} a^{8} - \frac{310244586773}{715822631875} a^{7} - \frac{173711905498}{715822631875} a^{6} - \frac{151856216659}{715822631875} a^{5} - \frac{74492389981}{715822631875} a^{4} + \frac{1776430774}{143164526375} a^{3} + \frac{13147632377}{28632905275} a^{2} + \frac{17311987}{5726581055} a - \frac{7690330514}{28632905275}$, $\frac{1}{1714395203340625} a^{17} + \frac{1189}{1714395203340625} a^{16} - \frac{3346704094}{1714395203340625} a^{15} - \frac{766677681}{342879040668125} a^{14} - \frac{2915181974811}{1714395203340625} a^{13} + \frac{968950717499}{342879040668125} a^{12} - \frac{151773346409352}{1714395203340625} a^{11} - \frac{17498986161443}{342879040668125} a^{10} + \frac{1503322754062}{3077908803125} a^{9} + \frac{11403336078086}{74538921884375} a^{8} + \frac{176358806714046}{1714395203340625} a^{7} - \frac{168969435745553}{342879040668125} a^{6} - \frac{45331895598358}{131876554103125} a^{5} + \frac{647348917873888}{1714395203340625} a^{4} + \frac{135521267947488}{342879040668125} a^{3} + \frac{5293240634789}{68575808133625} a^{2} + \frac{18072781510431}{68575808133625} a - \frac{21058028555833}{68575808133625}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 7766672.0712 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-255}) \), 3.1.255.1 x3, 6.0.16581375.1, 9.1.3082394705625.1 x9

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R R ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$