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Properties

Label 18.0.24219057137...2064.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 113^{9}$
Root discriminant $29.23$
Ramified primes $2, 3, 113$
Class number $18$
Class group $[3, 6]$
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 1458, 2916, -486, 2916, 7614, -3861, -2916, 5508, -576, -1620, 312, 223, -74, 44, 22, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 4*x^16 + 22*x^15 + 44*x^14 - 74*x^13 + 223*x^12 + 312*x^11 - 1620*x^10 - 576*x^9 + 5508*x^8 - 2916*x^7 - 3861*x^6 + 7614*x^5 + 2916*x^4 - 486*x^3 + 2916*x^2 + 1458*x + 729)
 
gp: K = bnfinit(x^18 - 2*x^17 - 4*x^16 + 22*x^15 + 44*x^14 - 74*x^13 + 223*x^12 + 312*x^11 - 1620*x^10 - 576*x^9 + 5508*x^8 - 2916*x^7 - 3861*x^6 + 7614*x^5 + 2916*x^4 - 486*x^3 + 2916*x^2 + 1458*x + 729, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 4 x^{16} + 22 x^{15} + 44 x^{14} - 74 x^{13} + 223 x^{12} + 312 x^{11} - 1620 x^{10} - 576 x^{9} + 5508 x^{8} - 2916 x^{7} - 3861 x^{6} + 7614 x^{5} + 2916 x^{4} - 486 x^{3} + 2916 x^{2} + 1458 x + 729 \)

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:  \(-242190571378050467501912064=-\,2^{12}\cdot 3^{9}\cdot 113^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.23$
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:  $2, 3, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{18} a^{9} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{18} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{54} a^{10} + \frac{1}{54} a^{9} + \frac{1}{27} a^{8} - \frac{5}{54} a^{7} - \frac{1}{54} a^{6} - \frac{10}{27} a^{5} - \frac{11}{54} a^{4} + \frac{7}{18} a^{3} - \frac{1}{9} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{54} a^{11} + \frac{1}{54} a^{9} - \frac{1}{54} a^{8} - \frac{1}{27} a^{7} + \frac{5}{54} a^{6} + \frac{7}{18} a^{5} + \frac{10}{27} a^{4} + \frac{7}{18} a^{3} - \frac{7}{18} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{486} a^{12} + \frac{1}{486} a^{11} - \frac{1}{486} a^{10} + \frac{1}{486} a^{9} - \frac{25}{486} a^{8} + \frac{31}{486} a^{7} + \frac{55}{486} a^{6} + \frac{19}{54} a^{5} + \frac{7}{54} a^{4} - \frac{7}{18} a^{3} + \frac{1}{6} a^{2} - \frac{7}{18} a + \frac{4}{9}$, $\frac{1}{972} a^{13} + \frac{7}{972} a^{11} - \frac{7}{972} a^{10} + \frac{1}{972} a^{9} - \frac{25}{972} a^{8} - \frac{19}{324} a^{7} - \frac{127}{972} a^{6} + \frac{17}{108} a^{5} + \frac{11}{36} a^{4} + \frac{5}{36} a^{3} - \frac{5}{12} a^{2} - \frac{1}{3} a - \frac{5}{36}$, $\frac{1}{2916} a^{14} + \frac{1}{2916} a^{13} - \frac{1}{2916} a^{12} - \frac{13}{1458} a^{11} + \frac{1}{1458} a^{10} + \frac{1}{729} a^{9} - \frac{20}{729} a^{8} - \frac{5}{81} a^{7} - \frac{13}{162} a^{6} - \frac{13}{27} a^{5} + \frac{2}{9} a^{4} + \frac{19}{54} a^{3} - \frac{37}{108} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8748} a^{15} + \frac{1}{8748} a^{14} - \frac{1}{2187} a^{13} - \frac{1}{4374} a^{12} - \frac{49}{8748} a^{11} + \frac{55}{8748} a^{10} - \frac{221}{8748} a^{9} - \frac{73}{2916} a^{8} - \frac{259}{2916} a^{7} - \frac{43}{324} a^{6} - \frac{79}{324} a^{5} - \frac{61}{324} a^{4} - \frac{23}{162} a^{3} - \frac{25}{54} a^{2} - \frac{53}{108} a - \frac{5}{12}$, $\frac{1}{7479540} a^{16} + \frac{83}{3739770} a^{15} - \frac{101}{3739770} a^{14} - \frac{112}{373977} a^{13} + \frac{734}{1869885} a^{12} + \frac{37489}{7479540} a^{11} + \frac{64549}{7479540} a^{10} - \frac{8669}{498636} a^{9} + \frac{93007}{2493180} a^{8} - \frac{29729}{277020} a^{7} + \frac{104137}{831060} a^{6} - \frac{12727}{277020} a^{5} - \frac{5047}{69255} a^{4} - \frac{3799}{18468} a^{3} + \frac{38807}{92340} a^{2} - \frac{773}{2052} a - \frac{14197}{30780}$, $\frac{1}{23096108963700} a^{17} + \frac{1011097}{23096108963700} a^{16} - \frac{174325913}{11548054481850} a^{15} - \frac{240962477}{23096108963700} a^{14} + \frac{944156173}{11548054481850} a^{13} - \frac{1177493041}{1154805448185} a^{12} + \frac{76226785693}{23096108963700} a^{11} - \frac{53824542227}{7698702987900} a^{10} + \frac{170559783637}{7698702987900} a^{9} - \frac{22217510593}{2566234329300} a^{8} + \frac{422992709}{27012992940} a^{7} + \frac{126909456157}{855411443100} a^{6} - \frac{7917594305}{17108228862} a^{5} - \frac{74027753}{142568573850} a^{4} + \frac{9236862767}{285137147700} a^{3} - \frac{5252026529}{23761428975} a^{2} + \frac{3099029453}{95045715900} a + \frac{4111309651}{31681905300}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$

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:  \( 1281109.48365 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-339}) \), 3.1.1356.3 x3, 3.1.339.1 x3, 3.1.1356.2 x3, 3.1.1356.1 x3, 6.0.623331504.1, 6.0.38958219.1, 6.0.623331504.3, 6.0.623331504.2, 9.1.845237519424.1 x9

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

Sibling fields

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 R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$113$113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$