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Properties

Label 18.18.1401934994...6409.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{27}\cdot 107^{9}$
Root discriminant $53.75$
Ramified primes $3, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times D_9$ (as 18T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -216, -1728, 10989, -10521, -20790, 31560, 13491, -31995, -3117, 16005, 0, -4331, 69, 621, -4, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 42*x^16 - 4*x^15 + 621*x^14 + 69*x^13 - 4331*x^12 + 16005*x^10 - 3117*x^9 - 31995*x^8 + 13491*x^7 + 31560*x^6 - 20790*x^5 - 10521*x^4 + 10989*x^3 - 1728*x^2 - 216*x + 9)
 
gp: K = bnfinit(x^18 - 42*x^16 - 4*x^15 + 621*x^14 + 69*x^13 - 4331*x^12 + 16005*x^10 - 3117*x^9 - 31995*x^8 + 13491*x^7 + 31560*x^6 - 20790*x^5 - 10521*x^4 + 10989*x^3 - 1728*x^2 - 216*x + 9, 1)
 

Normalized defining polynomial

\( x^{18} - 42 x^{16} - 4 x^{15} + 621 x^{14} + 69 x^{13} - 4331 x^{12} + 16005 x^{10} - 3117 x^{9} - 31995 x^{8} + 13491 x^{7} + 31560 x^{6} - 20790 x^{5} - 10521 x^{4} + 10989 x^{3} - 1728 x^{2} - 216 x + 9 \)

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:  \(14019349946482311002129152886409=3^{27}\cdot 107^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.75$
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:  $3, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{9} a^{12} - \frac{1}{3} a^{10} - \frac{1}{9} a^{9} + \frac{1}{3} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{9} a^{13} - \frac{1}{3} a^{11} - \frac{1}{9} a^{10} + \frac{1}{3} a^{8} + \frac{1}{9} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} + \frac{1}{9} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{4}{9} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{9} a^{16} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{4}{9} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{142114943571259941} a^{17} - \frac{1888613000633797}{47371647857086647} a^{16} - \frac{112906940142943}{15790549285695549} a^{15} - \frac{7680405118121335}{142114943571259941} a^{14} - \frac{214642632258779}{15790549285695549} a^{13} + \frac{153639105527971}{47371647857086647} a^{12} - \frac{652423885436654}{142114943571259941} a^{11} + \frac{2625658398984637}{15790549285695549} a^{10} + \frac{11771664340911818}{47371647857086647} a^{9} - \frac{1965001876362053}{47371647857086647} a^{8} + \frac{12663781110873356}{47371647857086647} a^{7} + \frac{17957971948577812}{47371647857086647} a^{6} + \frac{21325915165151422}{47371647857086647} a^{5} + \frac{2585165183980821}{5263516428565183} a^{4} + \frac{4746295971330428}{15790549285695549} a^{3} + \frac{1398568892808119}{5263516428565183} a^{2} + \frac{5917367204052668}{15790549285695549} a - \frac{946120595562866}{15790549285695549}$

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:  \( 14865346201.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_9$ (as 18T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times D_9$
Character table for $C_3\times D_9$

Intermediate fields

\(\Q(\sqrt{321}) \), 3.3.321.1 x3, 6.6.33076161.2

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

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{9}$

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$3.6.9.15$x^{6} + 6 x^{4} + 6 x^{3} + 12$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$
3.6.9.15$x^{6} + 6 x^{4} + 6 x^{3} + 12$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$
3.6.9.15$x^{6} + 6 x^{4} + 6 x^{3} + 12$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$
$107$107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.1$x^{2} - 107$$2$$1$$1$$C_2$$[\ ]_{2}$