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Properties

Label 18.18.4180042727...3981.2
Degree $18$
Signature $[18, 0]$
Discriminant $29^{9}\cdot 257^{6}$
Root discriminant $34.24$
Ramified primes $29, 257$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3.S_3^2$ (as 18T57)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -125, -220, 2264, -117, -8480, 5568, 9114, -9129, -2796, 4953, -175, -1130, 187, 112, -25, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 25*x^16 + 112*x^15 + 187*x^14 - 1130*x^13 - 175*x^12 + 4953*x^11 - 2796*x^10 - 9129*x^9 + 9114*x^8 + 5568*x^7 - 8480*x^6 - 117*x^5 + 2264*x^4 - 220*x^3 - 125*x^2 + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 - 25*x^16 + 112*x^15 + 187*x^14 - 1130*x^13 - 175*x^12 + 4953*x^11 - 2796*x^10 - 9129*x^9 + 9114*x^8 + 5568*x^7 - 8480*x^6 - 117*x^5 + 2264*x^4 - 220*x^3 - 125*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 25 x^{16} + 112 x^{15} + 187 x^{14} - 1130 x^{13} - 175 x^{12} + 4953 x^{11} - 2796 x^{10} - 9129 x^{9} + 9114 x^{8} + 5568 x^{7} - 8480 x^{6} - 117 x^{5} + 2264 x^{4} - 220 x^{3} - 125 x^{2} + 1 \)

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:  \(4180042727650388025593873981=29^{9}\cdot 257^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.24$
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:  $29, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{412207706894954637} a^{17} + \frac{17299200728365683}{137402568964984879} a^{16} + \frac{55822733496544567}{137402568964984879} a^{15} + \frac{24052274847058828}{412207706894954637} a^{14} - \frac{3960965712156628}{412207706894954637} a^{13} - \frac{150500796128876792}{412207706894954637} a^{12} + \frac{181211604651045941}{412207706894954637} a^{11} + \frac{6643957251082537}{137402568964984879} a^{10} + \frac{174704801023028300}{412207706894954637} a^{9} - \frac{93823387556602363}{412207706894954637} a^{8} + \frac{130051983619647310}{412207706894954637} a^{7} + \frac{16453453905963779}{137402568964984879} a^{6} - \frac{165511241466377104}{412207706894954637} a^{5} - \frac{37989769752263737}{412207706894954637} a^{4} + \frac{64533964117128557}{137402568964984879} a^{3} - \frac{114759458016415406}{412207706894954637} a^{2} + \frac{31716795552295619}{412207706894954637} a - \frac{57734584068886974}{137402568964984879}$

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:  \( 95471510.8169 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3.S_3^2$ (as 18T57):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 11 conjugacy class representatives for $C_3.S_3^2$
Character table for $C_3.S_3^2$

Intermediate fields

\(\Q(\sqrt{29}) \), 3.3.257.1, 6.6.1610869061.2, 9.9.413993348677.1 x3

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

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$29$29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
257Data not computed