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Properties

Label 18.0.79343716147...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{18}\cdot 5^{13}$
Root discriminant $24.17$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![344, 288, 1512, -616, -264, -3264, 2020, 240, 1128, -620, -252, -264, 224, 48, 12, -28, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 28*x^15 + 12*x^14 + 48*x^13 + 224*x^12 - 264*x^11 - 252*x^10 - 620*x^9 + 1128*x^8 + 240*x^7 + 2020*x^6 - 3264*x^5 - 264*x^4 - 616*x^3 + 1512*x^2 + 288*x + 344)
 
gp: K = bnfinit(x^18 - 28*x^15 + 12*x^14 + 48*x^13 + 224*x^12 - 264*x^11 - 252*x^10 - 620*x^9 + 1128*x^8 + 240*x^7 + 2020*x^6 - 3264*x^5 - 264*x^4 - 616*x^3 + 1512*x^2 + 288*x + 344, 1)
 

Normalized defining polynomial

\( x^{18} - 28 x^{15} + 12 x^{14} + 48 x^{13} + 224 x^{12} - 264 x^{11} - 252 x^{10} - 620 x^{9} + 1128 x^{8} + 240 x^{7} + 2020 x^{6} - 3264 x^{5} - 264 x^{4} - 616 x^{3} + 1512 x^{2} + 288 x + 344 \)

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:  \(-7934371614720000000000000=-\,2^{24}\cdot 3^{18}\cdot 5^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{4} a^{14}$, $\frac{1}{92} a^{15} - \frac{1}{23} a^{14} + \frac{3}{92} a^{13} - \frac{1}{23} a^{12} - \frac{4}{23} a^{11} - \frac{11}{46} a^{10} - \frac{7}{46} a^{9} + \frac{3}{23} a^{8} + \frac{1}{23} a^{7} - \frac{4}{23} a^{6} + \frac{7}{23} a^{5} - \frac{1}{23} a^{4} - \frac{1}{23} a^{3} + \frac{9}{23} a^{2} - \frac{4}{23} a - \frac{9}{23}$, $\frac{1}{2116} a^{16} + \frac{5}{2116} a^{15} + \frac{151}{2116} a^{14} - \frac{5}{46} a^{13} - \frac{49}{1058} a^{12} + \frac{101}{1058} a^{11} + \frac{131}{529} a^{10} + \frac{75}{529} a^{9} - \frac{110}{529} a^{8} + \frac{33}{1058} a^{7} - \frac{6}{529} a^{6} - \frac{7}{529} a^{5} + \frac{243}{529} a^{4} + \frac{2}{23} a^{3} + \frac{100}{529} a^{2} - \frac{206}{529} a - \frac{219}{529}$, $\frac{1}{102271664246835911427956} a^{17} - \frac{16772566540594998795}{102271664246835911427956} a^{16} - \frac{165180583708049811045}{102271664246835911427956} a^{15} + \frac{287153300127115851655}{51135832123417955713978} a^{14} + \frac{1310935726338373826791}{51135832123417955713978} a^{13} - \frac{5756863630886724787709}{102271664246835911427956} a^{12} + \frac{133549817486641356062}{25567916061708977856989} a^{11} + \frac{4667224055810216048296}{25567916061708977856989} a^{10} - \frac{3073093577411347217642}{25567916061708977856989} a^{9} + \frac{7805473689045844030099}{51135832123417955713978} a^{8} - \frac{2766181357547741524696}{25567916061708977856989} a^{7} - \frac{4668287049244756840893}{25567916061708977856989} a^{6} + \frac{4782961299896253260580}{25567916061708977856989} a^{5} - \frac{1831139849968600568424}{25567916061708977856989} a^{4} + \frac{7116519014316493310171}{25567916061708977856989} a^{3} + \frac{8412969580791006906077}{25567916061708977856989} a^{2} - \frac{8258889444873828532482}{25567916061708977856989} a - \frac{10833420155983143854167}{25567916061708977856989}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 56479.65794539926 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-5}) \), 3.1.108.1, 6.0.23328000.1, 9.3.787320000.1

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

Sibling fields

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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.10.2$x^{12} + 15 x^{6} + 100$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$