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Properties

Label 18.10.8581450218...7312.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{6}\cdot 3^{24}\cdot 7^{15}$
Root discriminant $27.59$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T459

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, -243, -351, 567, -405, -801, 3216, -963, -3090, 2633, 438, -1461, 514, 177, -171, 34, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 9*x^16 + 34*x^15 - 171*x^14 + 177*x^13 + 514*x^12 - 1461*x^11 + 438*x^10 + 2633*x^9 - 3090*x^8 - 963*x^7 + 3216*x^6 - 801*x^5 - 405*x^4 + 567*x^3 - 351*x^2 - 243*x - 27)
 
gp: K = bnfinit(x^18 - 6*x^17 + 9*x^16 + 34*x^15 - 171*x^14 + 177*x^13 + 514*x^12 - 1461*x^11 + 438*x^10 + 2633*x^9 - 3090*x^8 - 963*x^7 + 3216*x^6 - 801*x^5 - 405*x^4 + 567*x^3 - 351*x^2 - 243*x - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 9 x^{16} + 34 x^{15} - 171 x^{14} + 177 x^{13} + 514 x^{12} - 1461 x^{11} + 438 x^{10} + 2633 x^{9} - 3090 x^{8} - 963 x^{7} + 3216 x^{6} - 801 x^{5} - 405 x^{4} + 567 x^{3} - 351 x^{2} - 243 x - 27 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(85814502186767229614757312=2^{6}\cdot 3^{24}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.59$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{90} a^{15} + \frac{7}{90} a^{12} - \frac{13}{30} a^{11} - \frac{1}{3} a^{10} - \frac{22}{45} a^{9} - \frac{7}{15} a^{8} + \frac{7}{15} a^{7} + \frac{41}{90} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{11}{30} a^{3} - \frac{3}{10} a^{2} + \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{90} a^{16} + \frac{7}{90} a^{13} - \frac{1}{10} a^{12} - \frac{1}{3} a^{11} - \frac{22}{45} a^{10} - \frac{2}{15} a^{9} + \frac{7}{15} a^{8} + \frac{41}{90} a^{7} + \frac{2}{15} a^{6} + \frac{2}{5} a^{5} - \frac{11}{30} a^{4} + \frac{11}{30} a^{3} + \frac{1}{5} a^{2} + \frac{1}{10} a$, $\frac{1}{476154329917630640130} a^{17} + \frac{815815444826289601}{158718109972543546710} a^{16} - \frac{1022604373121778733}{238077164958815320065} a^{15} + \frac{22084442049490931857}{476154329917630640130} a^{14} + \frac{12299072768076760772}{79359054986271773355} a^{13} - \frac{4684603241230493849}{476154329917630640130} a^{12} - \frac{11822482245439546718}{47615432991763064013} a^{11} + \frac{35333073563621014171}{79359054986271773355} a^{10} + \frac{2852127698998615546}{47615432991763064013} a^{9} - \frac{191613026772032494891}{476154329917630640130} a^{8} - \frac{46516733109082647049}{158718109972543546710} a^{7} + \frac{4167646487817841738}{238077164958815320065} a^{6} + \frac{66238989752791964861}{158718109972543546710} a^{5} + \frac{38568766972252686178}{79359054986271773355} a^{4} - \frac{7195377356545687133}{31743621994508709342} a^{3} + \frac{703989376142865573}{10581207331502903114} a^{2} + \frac{4305978734647329011}{52906036657514515570} a + \frac{1192738370917482107}{26453018328757257785}$

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:  $13$
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:  \( 2716594.14709 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T459:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 96 conjugacy class representatives for t18n459 are not computed
Character table for t18n459 is not computed

Intermediate fields

3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 9.9.62523502209.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed