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Properties

Label 18.0.34744128540...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{8}\cdot 3^{33}\cdot 5^{12}$
Root discriminant $29.82$
Ramified primes $2, 3, 5$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6412, 26412, 37548, 20930, 5868, 9540, 8909, -2421, -510, 2877, -732, -600, 525, -99, -111, 55, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 3*x^16 + 55*x^15 - 111*x^14 - 99*x^13 + 525*x^12 - 600*x^11 - 732*x^10 + 2877*x^9 - 510*x^8 - 2421*x^7 + 8909*x^6 + 9540*x^5 + 5868*x^4 + 20930*x^3 + 37548*x^2 + 26412*x + 6412)
 
gp: K = bnfinit(x^18 - 6*x^17 + 3*x^16 + 55*x^15 - 111*x^14 - 99*x^13 + 525*x^12 - 600*x^11 - 732*x^10 + 2877*x^9 - 510*x^8 - 2421*x^7 + 8909*x^6 + 9540*x^5 + 5868*x^4 + 20930*x^3 + 37548*x^2 + 26412*x + 6412, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 3 x^{16} + 55 x^{15} - 111 x^{14} - 99 x^{13} + 525 x^{12} - 600 x^{11} - 732 x^{10} + 2877 x^{9} - 510 x^{8} - 2421 x^{7} + 8909 x^{6} + 9540 x^{5} + 5868 x^{4} + 20930 x^{3} + 37548 x^{2} + 26412 x + 6412 \)

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:  \(-347441285409720187500000000=-\,2^{8}\cdot 3^{33}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.82$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10} a^{15} - \frac{2}{5} a^{14} - \frac{3}{10} a^{13} - \frac{1}{2} a^{12} - \frac{3}{10} a^{11} + \frac{3}{10} a^{10} - \frac{1}{2} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} + \frac{2}{5} a^{5} + \frac{1}{10} a^{4} + \frac{3}{10} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{110} a^{16} - \frac{2}{55} a^{15} - \frac{43}{110} a^{14} - \frac{1}{2} a^{13} + \frac{27}{110} a^{12} + \frac{43}{110} a^{11} + \frac{5}{22} a^{10} - \frac{3}{55} a^{9} - \frac{1}{5} a^{8} - \frac{19}{110} a^{7} + \frac{12}{55} a^{6} + \frac{21}{110} a^{5} + \frac{53}{110} a^{4} + \frac{2}{55} a^{3} - \frac{27}{55} a^{2} + \frac{6}{55} a - \frac{2}{11}$, $\frac{1}{43936500400696524405080739427630030} a^{17} + \frac{79732797925144928317182346670479}{21968250200348262202540369713815015} a^{16} - \frac{34200552690301801845355600475997}{828990573598047630284542253351510} a^{15} - \frac{2723577741929621764140107156919583}{6276642914385217772154391346804290} a^{14} + \frac{15663929134497662970108436517425027}{43936500400696524405080739427630030} a^{13} - \frac{5706873803150426216670147317728433}{43936500400696524405080739427630030} a^{12} + \frac{15436039382602339115168408079071481}{43936500400696524405080739427630030} a^{11} + \frac{2465827170674187594048066058017752}{21968250200348262202540369713815015} a^{10} - \frac{1346770206798524287928177820184186}{3138321457192608886077195673402145} a^{9} + \frac{2586157390460335344282776752456061}{6276642914385217772154391346804290} a^{8} + \frac{529764511551650746630041764547598}{21968250200348262202540369713815015} a^{7} - \frac{26687206956512095437489775223491}{3994227309154229491370976311602730} a^{6} - \frac{4306535001507927583376847068629729}{8787300080139304881016147885526006} a^{5} + \frac{6860257702284071431718494177305}{4393650040069652440508073942763003} a^{4} - \frac{3922469322157022108603014920979793}{21968250200348262202540369713815015} a^{3} - \frac{13075748998210991180050781812226}{47447624622782423763586111692905} a^{2} + \frac{4159195684146670700990017138877347}{21968250200348262202540369713815015} a + \frac{201354900332502481544856951379627}{627664291438521777215439134680429}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -\frac{235403268965567448}{88672878506237415565} a^{17} + \frac{312155967721532700}{17734575701247483113} a^{16} - \frac{63622824581561081}{3346146358725940210} a^{15} - \frac{340600955506956054}{2533510814463926159} a^{14} + \frac{67493868125401093809}{177345757012474831130} a^{13} + \frac{4263753416308651479}{177345757012474831130} a^{12} - \frac{252054652920248807781}{177345757012474831130} a^{11} + \frac{88911304120885300863}{35469151402494966226} a^{10} + \frac{9827630880993184873}{25335108144639261590} a^{9} - \frac{101114838003981953349}{12667554072319630795} a^{8} + \frac{572813303129619715173}{88672878506237415565} a^{7} + \frac{441681896277411590271}{177345757012474831130} a^{6} - \frac{453270569357945815146}{17734575701247483113} a^{5} - \frac{1612793907017531308113}{177345757012474831130} a^{4} - \frac{1649253164175097744563}{177345757012474831130} a^{3} - \frac{9595977353706646194}{191518096125782755} a^{2} - \frac{1210321047248291524536}{17734575701247483113} a - \frac{328384289569839115451}{12667554072319630795} \) (order $6$)
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:  \( 1517059.6926500278 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.6075.2 x3, 6.0.110716875.2, 9.3.3587226750000.2

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

Sibling fields

Degree 9 siblings: 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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$