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Properties

Label 18.0.60036344830...1024.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 73^{12}$
Root discriminant $34.93$
Ramified primes $2, 73$
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![576, 0, 784, 0, 1304, 0, 9513, 0, 11626, 0, 6721, 0, 2158, 0, 378, 0, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 32*x^16 + 378*x^14 + 2158*x^12 + 6721*x^10 + 11626*x^8 + 9513*x^6 + 1304*x^4 + 784*x^2 + 576)
 
gp: K = bnfinit(x^18 + 32*x^16 + 378*x^14 + 2158*x^12 + 6721*x^10 + 11626*x^8 + 9513*x^6 + 1304*x^4 + 784*x^2 + 576, 1)
 

Normalized defining polynomial

\( x^{18} + 32 x^{16} + 378 x^{14} + 2158 x^{12} + 6721 x^{10} + 11626 x^{8} + 9513 x^{6} + 1304 x^{4} + 784 x^{2} + 576 \)

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:  \(-6003634483099142379672961024=-\,2^{18}\cdot 73^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.93$
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:  $2, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{12} - \frac{1}{4} a^{6} - \frac{1}{12} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{12} a^{13} - \frac{1}{4} a^{7} - \frac{1}{12} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{10} + \frac{7}{18} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{5}{36} a^{2} - \frac{1}{2} a$, $\frac{1}{72} a^{15} + \frac{1}{36} a^{11} + \frac{5}{72} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{17}{72} a^{3} - \frac{1}{2} a$, $\frac{1}{181235756016} a^{16} + \frac{575320267}{45308939004} a^{14} - \frac{814155617}{90617878008} a^{12} + \frac{1004765599}{90617878008} a^{10} + \frac{635249057}{181235756016} a^{8} + \frac{39803667121}{90617878008} a^{6} - \frac{80975653427}{181235756016} a^{4} - \frac{1}{2} a^{3} - \frac{5666524525}{22654469502} a^{2} + \frac{357880945}{1258581639}$, $\frac{1}{362471512032} a^{17} + \frac{575320267}{90617878008} a^{15} + \frac{6737334217}{181235756016} a^{13} + \frac{1004765599}{181235756016} a^{11} + \frac{635249057}{362471512032} a^{9} + \frac{17149197619}{181235756016} a^{7} - \frac{1}{2} a^{6} - \frac{96078633095}{362471512032} a^{5} + \frac{45303124705}{90617878008} a^{3} - \frac{1}{2} a^{2} + \frac{357880945}{2517163278} a$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{23141}{38758716} a^{17} + \frac{1455973}{77517432} a^{15} + \frac{4162307}{19379358} a^{13} + \frac{44559805}{38758716} a^{11} + \frac{30486490}{9689679} a^{9} + \frac{312528041}{77517432} a^{7} + \frac{4420007}{9689679} a^{5} - \frac{276404783}{77517432} a^{3} + \frac{151223}{2153262} a \) (order $4$)
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:  \( 12079603.9433 \) (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

\(\Q(\sqrt{-1}) \), 3.1.21316.1 x3, 3.3.5329.1, 6.0.1817487424.2, 6.0.341056.1 x2, 6.0.1817487424.1, 9.3.9685390482496.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.341056.1
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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$73$73.9.6.1$x^{9} + 3066 x^{6} + 3128123 x^{3} + 1067462648$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
73.9.6.1$x^{9} + 3066 x^{6} + 3128123 x^{3} + 1067462648$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$