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Properties

Label 18.6.93265559882...3968.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 3^{33}$
Root discriminant $18.88$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_3\times S_3^2$ (as 18T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, -3, 270, 249, -336, 66, 240, -174, 52, -87, 120, -45, -24, 21, -12, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 12*x^16 - 12*x^15 + 21*x^14 - 24*x^13 - 45*x^12 + 120*x^11 - 87*x^10 + 52*x^9 - 174*x^8 + 240*x^7 + 66*x^6 - 336*x^5 + 249*x^4 + 270*x^3 - 3*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 12*x^16 - 12*x^15 + 21*x^14 - 24*x^13 - 45*x^12 + 120*x^11 - 87*x^10 + 52*x^9 - 174*x^8 + 240*x^7 + 66*x^6 - 336*x^5 + 249*x^4 + 270*x^3 - 3*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 12 x^{16} - 12 x^{15} + 21 x^{14} - 24 x^{13} - 45 x^{12} + 120 x^{11} - 87 x^{10} + 52 x^{9} - 174 x^{8} + 240 x^{7} + 66 x^{6} - 336 x^{5} + 249 x^{4} + 270 x^{3} - 3 x^{2} - 12 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(93265559882184385363968=2^{24}\cdot 3^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.88$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} + \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{2}{9} a^{6} + \frac{2}{9} a^{3} + \frac{2}{9}$, $\frac{1}{333} a^{16} - \frac{4}{111} a^{15} + \frac{4}{111} a^{14} + \frac{40}{333} a^{13} + \frac{3}{37} a^{12} + \frac{14}{111} a^{11} + \frac{16}{333} a^{10} - \frac{1}{111} a^{9} + \frac{83}{333} a^{7} - \frac{2}{111} a^{6} - \frac{13}{111} a^{5} - \frac{82}{333} a^{4} - \frac{16}{37} a^{3} - \frac{29}{111} a^{2} + \frac{23}{333} a + \frac{43}{111}$, $\frac{1}{30412907796519} a^{17} - \frac{11566524044}{30412907796519} a^{16} + \frac{530359463002}{10137635932173} a^{15} - \frac{1287846918256}{30412907796519} a^{14} + \frac{2417566497437}{30412907796519} a^{13} - \frac{2008167128015}{30412907796519} a^{12} + \frac{599343320666}{10137635932173} a^{11} - \frac{212757735467}{10137635932173} a^{10} - \frac{4882901714575}{30412907796519} a^{9} - \frac{2738318816152}{30412907796519} a^{8} - \frac{3283098540325}{30412907796519} a^{7} + \frac{3050934721582}{10137635932173} a^{6} - \frac{829511131970}{30412907796519} a^{5} - \frac{6870239108846}{30412907796519} a^{4} + \frac{9033782087723}{30412907796519} a^{3} + \frac{1111233338730}{3379211977391} a^{2} - \frac{3118919783666}{10137635932173} a - \frac{6112608516134}{30412907796519}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 44983.5370214 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), 6.2.5038848.1, \(\Q(\zeta_{36})^+\)

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

Sibling fields

Degree 12 sibling: 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed