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Properties

Label 18.0.23236047581...9344.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 11^{12}$
Root discriminant $42.80$
Ramified primes $2, 3, 11$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
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![25088, -96768, 7968, 247136, 218880, -2592, -12422, 32706, 2787, -12424, 2037, 2148, -846, -216, 240, -6, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 21*x^16 - 6*x^15 + 240*x^14 - 216*x^13 - 846*x^12 + 2148*x^11 + 2037*x^10 - 12424*x^9 + 2787*x^8 + 32706*x^7 - 12422*x^6 - 2592*x^5 + 218880*x^4 + 247136*x^3 + 7968*x^2 - 96768*x + 25088)
 
gp: K = bnfinit(x^18 - 21*x^16 - 6*x^15 + 240*x^14 - 216*x^13 - 846*x^12 + 2148*x^11 + 2037*x^10 - 12424*x^9 + 2787*x^8 + 32706*x^7 - 12422*x^6 - 2592*x^5 + 218880*x^4 + 247136*x^3 + 7968*x^2 - 96768*x + 25088, 1)
 

Normalized defining polynomial

\( x^{18} - 21 x^{16} - 6 x^{15} + 240 x^{14} - 216 x^{13} - 846 x^{12} + 2148 x^{11} + 2037 x^{10} - 12424 x^{9} + 2787 x^{8} + 32706 x^{7} - 12422 x^{6} - 2592 x^{5} + 218880 x^{4} + 247136 x^{3} + 7968 x^{2} - 96768 x + 25088 \)

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:  \(-232360475811152994149020729344=-\,2^{18}\cdot 3^{24}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.80$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{16} a^{5} + \frac{1}{16} a^{4} - \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} + \frac{1}{16} a^{6} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{3}{32} a^{7} - \frac{1}{8} a^{6} + \frac{1}{32} a^{5} + \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{10} + \frac{1}{64} a^{8} + \frac{1}{16} a^{7} + \frac{1}{64} a^{6} + \frac{1}{8} a^{5} - \frac{5}{32} a^{4} - \frac{3}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{12} + \frac{1}{128} a^{11} - \frac{3}{128} a^{10} - \frac{1}{128} a^{9} + \frac{7}{128} a^{8} + \frac{7}{128} a^{7} - \frac{5}{128} a^{6} - \frac{1}{8} a^{5} - \frac{11}{64} a^{4} - \frac{3}{16} a^{3} - \frac{1}{16} a^{2}$, $\frac{1}{1024} a^{14} - \frac{1}{512} a^{13} - \frac{3}{512} a^{12} + \frac{3}{256} a^{11} + \frac{5}{512} a^{10} + \frac{3}{128} a^{9} + \frac{7}{128} a^{8} - \frac{23}{256} a^{7} - \frac{19}{1024} a^{6} + \frac{117}{512} a^{5} - \frac{25}{512} a^{4} + \frac{29}{64} a^{3} + \frac{1}{128} a^{2} - \frac{3}{8}$, $\frac{1}{3072} a^{15} + \frac{1}{512} a^{13} - \frac{1}{192} a^{12} + \frac{3}{512} a^{11} + \frac{5}{256} a^{10} - \frac{1}{384} a^{9} - \frac{5}{256} a^{8} + \frac{87}{1024} a^{7} + \frac{13}{768} a^{6} + \frac{43}{512} a^{5} - \frac{47}{256} a^{4} + \frac{63}{128} a^{3} + \frac{3}{64} a^{2} - \frac{3}{8} a + \frac{5}{12}$, $\frac{1}{71786496} a^{16} - \frac{9181}{71786496} a^{15} + \frac{725}{11964416} a^{14} + \frac{66169}{35893248} a^{13} + \frac{207209}{35893248} a^{12} + \frac{176803}{11964416} a^{11} - \frac{559817}{17946624} a^{10} - \frac{305029}{17946624} a^{9} + \frac{543067}{23928832} a^{8} - \frac{5253629}{71786496} a^{7} + \frac{2593243}{35893248} a^{6} - \frac{2281189}{11964416} a^{5} + \frac{142861}{5982208} a^{4} - \frac{557397}{2991104} a^{3} + \frac{526453}{1495552} a^{2} - \frac{63323}{560832} a - \frac{31781}{280416}$, $\frac{1}{3726405764849961178202112} a^{17} + \frac{94852965195405}{177447893564283865628672} a^{16} - \frac{1804351706794124633}{33271480043303224805376} a^{15} + \frac{702454677611579322091}{1863202882424980589101056} a^{14} + \frac{259802002409289199445}{621067627474993529700352} a^{13} - \frac{8023179555997456206905}{1863202882424980589101056} a^{12} + \frac{315972312304651583683}{20252205243749789011968} a^{11} - \frac{8003475331819971742777}{310533813737496764850176} a^{10} - \frac{5994303606082694623169}{532343680692851596886016} a^{9} - \frac{197530230799245903782159}{3726405764849961178202112} a^{8} - \frac{6067235463182335345241}{155266906868748382425088} a^{7} - \frac{45611679067901612799013}{1863202882424980589101056} a^{6} - \frac{17516337884597352960781}{155266906868748382425088} a^{5} + \frac{5540347585569748069029}{77633453434374191212544} a^{4} + \frac{17190188468173047723751}{38816726717187095606272} a^{3} - \frac{2104432036386343169045}{6128956850082172990464} a^{2} - \frac{1103412076872657624013}{2426045419824193475392} a + \frac{481169290458505282855}{1039733751353225775168}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( \frac{484198787}{15005512531968} a^{17} - \frac{48339481}{49303826890752} a^{16} - \frac{8308966417}{12325956722688} a^{15} - \frac{28303159253}{172563394117632} a^{14} + \frac{1320018632279}{172563394117632} a^{13} - \frac{1272056219425}{172563394117632} a^{12} - \frac{139000120829}{5392606066176} a^{11} + \frac{6107046710141}{86281697058816} a^{10} + \frac{2602428598955}{49303826890752} a^{9} - \frac{133373786940047}{345126788235264} a^{8} + \frac{11612755259687}{86281697058816} a^{7} + \frac{163220200521067}{172563394117632} a^{6} - \frac{1767534386351}{3595070710784} a^{5} + \frac{2722035821}{7076910848} a^{4} + \frac{1512704741145}{224691919424} a^{3} + \frac{4224112772533}{567642743808} a^{2} + \frac{74368467569}{42129734892} a - \frac{124168107439}{96296536896} \) (order $4$)
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:  \( 1166308755.75 \) (assuming GRH)
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

\(\Q(\sqrt{-1}) \), 3.1.39204.2 x3, 3.1.39204.1 x3, 3.1.324.1 x3, 3.1.484.1 x3, 6.0.6147814464.2, 6.0.6147814464.1, 6.0.419904.2, 6.0.937024.1, 9.1.241018918246656.6 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$3$3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$