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Properties

Label 18.0.65501171436...5231.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 13^{9}$
Root discriminant $23.92$
Ramified primes $3, 13$
Class number $4$
Class group $[4]$
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![4096, -7680, 6528, -120, -3480, 5526, -4305, 1350, 2157, -3110, 2649, -1398, 465, -12, -27, 6, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 9*x^16 + 6*x^15 - 27*x^14 - 12*x^13 + 465*x^12 - 1398*x^11 + 2649*x^10 - 3110*x^9 + 2157*x^8 + 1350*x^7 - 4305*x^6 + 5526*x^5 - 3480*x^4 - 120*x^3 + 6528*x^2 - 7680*x + 4096)
 
gp: K = bnfinit(x^18 - 6*x^17 + 9*x^16 + 6*x^15 - 27*x^14 - 12*x^13 + 465*x^12 - 1398*x^11 + 2649*x^10 - 3110*x^9 + 2157*x^8 + 1350*x^7 - 4305*x^6 + 5526*x^5 - 3480*x^4 - 120*x^3 + 6528*x^2 - 7680*x + 4096, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 9 x^{16} + 6 x^{15} - 27 x^{14} - 12 x^{13} + 465 x^{12} - 1398 x^{11} + 2649 x^{10} - 3110 x^{9} + 2157 x^{8} + 1350 x^{7} - 4305 x^{6} + 5526 x^{5} - 3480 x^{4} - 120 x^{3} + 6528 x^{2} - 7680 x + 4096 \)

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:  \(-6550117143611896491465231=-\,3^{31}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.92$
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:  $3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{68} a^{14} + \frac{4}{17} a^{13} - \frac{13}{68} a^{12} + \frac{7}{68} a^{10} - \frac{1}{17} a^{9} + \frac{1}{68} a^{8} + \frac{3}{34} a^{7} - \frac{1}{4} a^{6} + \frac{7}{17} a^{5} + \frac{7}{68} a^{4} - \frac{3}{17} a^{3} + \frac{7}{68} a^{2} + \frac{5}{34} a + \frac{8}{17}$, $\frac{1}{2584} a^{15} - \frac{1}{323} a^{14} - \frac{91}{2584} a^{13} + \frac{5}{323} a^{12} - \frac{5}{136} a^{11} + \frac{101}{1292} a^{10} - \frac{39}{2584} a^{9} - \frac{13}{646} a^{8} - \frac{467}{2584} a^{7} - \frac{39}{323} a^{6} - \frac{563}{2584} a^{5} + \frac{295}{646} a^{4} - \frac{997}{2584} a^{3} + \frac{112}{323} a^{2} - \frac{9}{323} a + \frac{125}{323}$, $\frac{1}{764864} a^{16} - \frac{7}{382432} a^{15} - \frac{423}{764864} a^{14} + \frac{53455}{382432} a^{13} - \frac{65163}{764864} a^{12} - \frac{453}{191216} a^{11} - \frac{135695}{764864} a^{10} - \frac{78607}{382432} a^{9} + \frac{102825}{764864} a^{8} - \frac{36071}{382432} a^{7} + \frac{15277}{44992} a^{6} - \frac{18545}{382432} a^{5} - \frac{346657}{764864} a^{4} - \frac{9899}{20128} a^{3} + \frac{8515}{95608} a^{2} - \frac{619}{95608} a - \frac{4725}{11951}$, $\frac{1}{73680431697112228688384} a^{17} + \frac{978550546347997}{2167071520503300843776} a^{16} - \frac{14144631039093611127}{73680431697112228688384} a^{15} + \frac{27691646557106617715}{36840215848556114344192} a^{14} - \frac{4726018307117479240219}{73680431697112228688384} a^{13} - \frac{2139488982513767674731}{18420107924278057172096} a^{12} - \frac{9758375204008644064239}{73680431697112228688384} a^{11} + \frac{3793987477927413101685}{36840215848556114344192} a^{10} + \frac{2820751898679129006297}{73680431697112228688384} a^{9} - \frac{4242836501059472713379}{36840215848556114344192} a^{8} + \frac{124442807575467226623}{3877917457742748878336} a^{7} + \frac{7819569012910705752019}{36840215848556114344192} a^{6} - \frac{184278735436328930109}{1991363018840871045632} a^{5} - \frac{13127559606765618320165}{36840215848556114344192} a^{4} + \frac{122074428312677765103}{484739682217843609792} a^{3} + \frac{2281130207455030888097}{9210053962139028586048} a^{2} + \frac{118944765750073359013}{575628372633689286628} a - \frac{71917827451109428584}{143907093158422321657}$

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

Class group and class number

$C_{4}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 29831.8168774 \)
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

\(\Q(\sqrt{-39}) \), 3.1.351.1 x3, \(\Q(\zeta_{9})^+\), 6.0.4804839.1, 6.0.43243551.1, 6.0.43243551.2 x2, 9.3.31524548679.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.43243551.2
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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$