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Properties

Label 18.18.1114205477...8352.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{12}\cdot 13^{15}$
Root discriminant $27.99$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2\times S_3$ (as 18T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79, 381, -319, -4367, -6240, 4714, 15963, 4448, -12791, -7794, 4484, 3876, -833, -925, 112, 114, -14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 14*x^16 + 114*x^15 + 112*x^14 - 925*x^13 - 833*x^12 + 3876*x^11 + 4484*x^10 - 7794*x^9 - 12791*x^8 + 4448*x^7 + 15963*x^6 + 4714*x^5 - 6240*x^4 - 4367*x^3 - 319*x^2 + 381*x + 79)
 
gp: K = bnfinit(x^18 - 6*x^17 - 14*x^16 + 114*x^15 + 112*x^14 - 925*x^13 - 833*x^12 + 3876*x^11 + 4484*x^10 - 7794*x^9 - 12791*x^8 + 4448*x^7 + 15963*x^6 + 4714*x^5 - 6240*x^4 - 4367*x^3 - 319*x^2 + 381*x + 79, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 14 x^{16} + 114 x^{15} + 112 x^{14} - 925 x^{13} - 833 x^{12} + 3876 x^{11} + 4484 x^{10} - 7794 x^{9} - 12791 x^{8} + 4448 x^{7} + 15963 x^{6} + 4714 x^{5} - 6240 x^{4} - 4367 x^{3} - 319 x^{2} + 381 x + 79 \)

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:  \(111420547765458558938468352=2^{12}\cdot 3^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.99$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{21} a^{15} + \frac{2}{21} a^{14} + \frac{1}{7} a^{13} - \frac{2}{21} a^{12} - \frac{1}{7} a^{11} + \frac{2}{21} a^{10} + \frac{1}{7} a^{9} + \frac{2}{21} a^{8} - \frac{2}{21} a^{7} - \frac{3}{7} a^{6} - \frac{10}{21} a^{5} - \frac{5}{21} a^{4} - \frac{8}{21} a^{3} + \frac{10}{21} a^{2} - \frac{5}{21} a - \frac{2}{7}$, $\frac{1}{21} a^{16} - \frac{1}{21} a^{14} - \frac{1}{21} a^{13} + \frac{1}{21} a^{12} + \frac{8}{21} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{5}{21} a^{7} + \frac{1}{21} a^{6} - \frac{2}{7} a^{5} + \frac{2}{21} a^{4} - \frac{2}{21} a^{3} - \frac{4}{21} a^{2} - \frac{10}{21} a + \frac{5}{21}$, $\frac{1}{59249757} a^{17} + \frac{164018}{19749919} a^{16} + \frac{191371}{59249757} a^{15} - \frac{4420835}{59249757} a^{14} - \frac{741571}{59249757} a^{13} - \frac{248386}{19749919} a^{12} - \frac{5200078}{59249757} a^{11} - \frac{27049423}{59249757} a^{10} + \frac{3583597}{8464251} a^{9} + \frac{24151816}{59249757} a^{8} - \frac{2409096}{19749919} a^{7} - \frac{15556850}{59249757} a^{6} + \frac{26759930}{59249757} a^{5} + \frac{8495147}{19749919} a^{4} - \frac{154765}{59249757} a^{3} + \frac{16124482}{59249757} a^{2} - \frac{6989779}{59249757} a - \frac{474977}{2821417}$

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:  \( 11924892.4742 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2\times S_3$ (as 18T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_3^2\times S_3$
Character table for $C_3^2\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 6.6.481195728.1, 6.6.481195728.2, 6.6.2847312.1, \(\Q(\zeta_{13})^+\)

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

Sibling fields

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
13Data not computed