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Properties

Label 18.0.52667711830...7559.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{39}\cdot 37^{9}$
Root discriminant $65.75$
Ramified primes $3, 37$
Class number $1568$ (GRH)
Class group $[2, 14, 56]$ (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![39304, 52020, 93636, 65865, 143037, 44955, 96036, 17055, 43740, -1415, 16344, -2025, 3642, -315, 486, -15, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 36*x^16 - 15*x^15 + 486*x^14 - 315*x^13 + 3642*x^12 - 2025*x^11 + 16344*x^10 - 1415*x^9 + 43740*x^8 + 17055*x^7 + 96036*x^6 + 44955*x^5 + 143037*x^4 + 65865*x^3 + 93636*x^2 + 52020*x + 39304)
 
gp: K = bnfinit(x^18 + 36*x^16 - 15*x^15 + 486*x^14 - 315*x^13 + 3642*x^12 - 2025*x^11 + 16344*x^10 - 1415*x^9 + 43740*x^8 + 17055*x^7 + 96036*x^6 + 44955*x^5 + 143037*x^4 + 65865*x^3 + 93636*x^2 + 52020*x + 39304, 1)
 

Normalized defining polynomial

\( x^{18} + 36 x^{16} - 15 x^{15} + 486 x^{14} - 315 x^{13} + 3642 x^{12} - 2025 x^{11} + 16344 x^{10} - 1415 x^{9} + 43740 x^{8} + 17055 x^{7} + 96036 x^{6} + 44955 x^{5} + 143037 x^{4} + 65865 x^{3} + 93636 x^{2} + 52020 x + 39304 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-526677118301850649055916906437559=-\,3^{39}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.75$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{3} + \frac{1}{3} a$, $\frac{1}{36} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{5}{12} a^{5} - \frac{1}{3} a^{4} - \frac{1}{12} a^{3} + \frac{5}{36} a^{2} - \frac{5}{18} a + \frac{2}{9}$, $\frac{1}{21420} a^{12} - \frac{5}{612} a^{11} + \frac{421}{10710} a^{10} - \frac{61}{4284} a^{9} + \frac{11}{238} a^{8} + \frac{29}{714} a^{7} + \frac{286}{1785} a^{6} + \frac{115}{476} a^{5} - \frac{1399}{7140} a^{4} - \frac{19}{63} a^{3} + \frac{311}{1260} a^{2} - \frac{5}{306} a + \frac{22}{315}$, $\frac{1}{21420} a^{13} - \frac{11}{7140} a^{11} - \frac{341}{4284} a^{10} - \frac{27}{476} a^{9} - \frac{9}{238} a^{8} - \frac{233}{3570} a^{7} - \frac{25}{476} a^{6} + \frac{248}{595} a^{5} - \frac{1817}{4284} a^{4} - \frac{51}{140} a^{3} + \frac{25}{204} a^{2} - \frac{956}{5355} a$, $\frac{1}{21420} a^{14} + \frac{1}{84} a^{11} + \frac{23}{1260} a^{10} - \frac{271}{4284} a^{9} + \frac{103}{2380} a^{8} + \frac{9}{238} a^{7} + \frac{863}{7140} a^{6} + \frac{565}{4284} a^{5} - \frac{393}{1190} a^{4} + \frac{31}{357} a^{3} + \frac{1567}{3570} a^{2} - \frac{97}{306} a - \frac{149}{315}$, $\frac{1}{599760} a^{15} + \frac{1}{119952} a^{13} + \frac{1}{74970} a^{12} - \frac{6529}{599760} a^{11} + \frac{106}{1785} a^{10} - \frac{11923}{599760} a^{9} + \frac{113}{2499} a^{8} - \frac{1359}{66640} a^{7} + \frac{10499}{149940} a^{6} + \frac{421}{28560} a^{5} + \frac{85997}{299880} a^{4} + \frac{48757}{599760} a^{3} + \frac{66919}{149940} a^{2} - \frac{16951}{49980} a - \frac{1457}{4410}$, $\frac{1}{1799280} a^{16} - \frac{1}{1799280} a^{15} + \frac{11}{599760} a^{14} + \frac{31}{1799280} a^{13} - \frac{13}{1799280} a^{12} + \frac{7627}{599760} a^{11} + \frac{77677}{1799280} a^{10} + \frac{43103}{1799280} a^{9} - \frac{163}{2352} a^{8} - \frac{34057}{1799280} a^{7} - \frac{28555}{359856} a^{6} - \frac{260681}{599760} a^{5} - \frac{567877}{1799280} a^{4} - \frac{151813}{359856} a^{3} + \frac{37913}{149940} a^{2} - \frac{35743}{449820} a + \frac{311}{2646}$, $\frac{1}{406831003079760} a^{17} - \frac{342611}{1994269622940} a^{16} - \frac{10603909}{14529678681420} a^{15} - \frac{272265359}{14529678681420} a^{14} + \frac{254926417}{22601722393320} a^{13} + \frac{258137087}{25426937692485} a^{12} - \frac{527292691349}{40683100307976} a^{11} - \frac{846117531433}{11300861196660} a^{10} - \frac{237155209988}{5085387538497} a^{9} + \frac{1581723383143}{20341550153988} a^{8} - \frac{1972910137391}{33902583589980} a^{7} - \frac{3164900280821}{40683100307976} a^{6} + \frac{2563608004009}{20341550153988} a^{5} - \frac{4186563627859}{22601722393320} a^{4} - \frac{27061556607341}{406831003079760} a^{3} - \frac{3909890286373}{14529678681420} a^{2} - \frac{217474969363}{664756540980} a - \frac{10833991415}{35192993346}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{56}$, which has order $1568$ (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:  $8$
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:  \( 2657736.68779 \) (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

\(\Q(\sqrt{-111}) \), 3.1.8991.3 x3, \(\Q(\zeta_{9})^+\), 6.0.8973026991.3, 6.0.8973026991.5 x2, 6.0.997002999.1, 9.3.58872030087951.2 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$37$37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$