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Properties

Label 18.18.8224668627...2224.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 7^{12}\cdot 29^{9}$
Root discriminant $31.28$
Ramified primes $2, 7, 29$
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![29, -986, 4189, 2338, -21852, -8833, 35618, 14876, -27282, -11183, 11347, 4210, -2727, -819, 385, 79, -30, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 30*x^16 + 79*x^15 + 385*x^14 - 819*x^13 - 2727*x^12 + 4210*x^11 + 11347*x^10 - 11183*x^9 - 27282*x^8 + 14876*x^7 + 35618*x^6 - 8833*x^5 - 21852*x^4 + 2338*x^3 + 4189*x^2 - 986*x + 29)
 
gp: K = bnfinit(x^18 - 3*x^17 - 30*x^16 + 79*x^15 + 385*x^14 - 819*x^13 - 2727*x^12 + 4210*x^11 + 11347*x^10 - 11183*x^9 - 27282*x^8 + 14876*x^7 + 35618*x^6 - 8833*x^5 - 21852*x^4 + 2338*x^3 + 4189*x^2 - 986*x + 29, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 30 x^{16} + 79 x^{15} + 385 x^{14} - 819 x^{13} - 2727 x^{12} + 4210 x^{11} + 11347 x^{10} - 11183 x^{9} - 27282 x^{8} + 14876 x^{7} + 35618 x^{6} - 8833 x^{5} - 21852 x^{4} + 2338 x^{3} + 4189 x^{2} - 986 x + 29 \)

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:  \(822466862771545065687732224=2^{12}\cdot 7^{12}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.28$
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:  $2, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{29} a^{16} - \frac{10}{29} a^{15} + \frac{6}{29} a^{14} + \frac{7}{29} a^{12} + \frac{2}{29} a^{11} + \frac{8}{29} a^{10} - \frac{3}{29} a^{9} - \frac{11}{29} a^{8} - \frac{13}{29} a^{7} + \frac{8}{29} a^{6} + \frac{8}{29} a^{5} - \frac{3}{29} a^{4} - \frac{7}{29} a^{3} - \frac{9}{29} a^{2}$, $\frac{1}{34238316485044211} a^{17} + \frac{459514369827680}{34238316485044211} a^{16} + \frac{16137686494299430}{34238316485044211} a^{15} + \frac{13954046519715360}{34238316485044211} a^{14} - \frac{2451614332163051}{34238316485044211} a^{13} - \frac{2297905833551588}{34238316485044211} a^{12} + \frac{16919717882777141}{34238316485044211} a^{11} - \frac{11566865479473943}{34238316485044211} a^{10} - \frac{16886109617211549}{34238316485044211} a^{9} - \frac{8497795294102}{352972334897363} a^{8} + \frac{11922261501316360}{34238316485044211} a^{7} - \frac{3665055314360452}{34238316485044211} a^{6} - \frac{2425656714367153}{34238316485044211} a^{5} - \frac{13716866252549663}{34238316485044211} a^{4} - \frac{4812572924704803}{34238316485044211} a^{3} + \frac{5482911013936556}{34238316485044211} a^{2} + \frac{583190926591941}{1180631602932559} a - \frac{204166753344311}{1180631602932559}$

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:  \( 36105703.1937 \) (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

\(\Q(\sqrt{29}) \), 3.3.5684.1 x3, \(\Q(\zeta_{7})^+\), 6.6.936927824.1, 6.6.19120976.1 x2, 6.6.58557989.1, 9.9.183637853504.1 x3

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

Sibling fields

Degree 6 sibling: 6.6.19120976.1
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$