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Properties

Label 18.0.72089101682...0304.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 127^{15}$
Root discriminant $588.68$
Ramified primes $2, 3, 127$
Class number $2713518080$ (GRH)
Class group $[2, 2, 2, 2, 8, 8, 728, 3640]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![226534772736, 0, 339802159104, 0, 184059502848, 0, 45949843584, 0, 5811085152, 0, 378579888, 0, 12079605, 0, 163449, 0, 762, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 762*x^16 + 163449*x^14 + 12079605*x^12 + 378579888*x^10 + 5811085152*x^8 + 45949843584*x^6 + 184059502848*x^4 + 339802159104*x^2 + 226534772736)
 
gp: K = bnfinit(x^18 + 762*x^16 + 163449*x^14 + 12079605*x^12 + 378579888*x^10 + 5811085152*x^8 + 45949843584*x^6 + 184059502848*x^4 + 339802159104*x^2 + 226534772736, 1)
 

Normalized defining polynomial

\( x^{18} + 762 x^{16} + 163449 x^{14} + 12079605 x^{12} + 378579888 x^{10} + 5811085152 x^{8} + 45949843584 x^{6} + 184059502848 x^{4} + 339802159104 x^{2} + 226534772736 \)

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:  \(-72089101682458965160031671674946384142833324130304=-\,2^{18}\cdot 3^{27}\cdot 127^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $588.68$
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:  $2, 3, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4572=2^{2}\cdot 3^{2}\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{4572}(1,·)$, $\chi_{4572}(4211,·)$, $\chi_{4572}(2305,·)$, $\chi_{4572}(1163,·)$, $\chi_{4572}(781,·)$, $\chi_{4572}(3791,·)$, $\chi_{4572}(3829,·)$, $\chi_{4572}(3409,·)$, $\chi_{4572}(4571,·)$, $\chi_{4572}(1885,·)$, $\chi_{4572}(2267,·)$, $\chi_{4572}(743,·)$, $\chi_{4572}(3049,·)$, $\chi_{4572}(3047,·)$, $\chi_{4572}(1523,·)$, $\chi_{4572}(1525,·)$, $\chi_{4572}(361,·)$, $\chi_{4572}(2687,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{381} a^{6}$, $\frac{1}{762} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{1524} a^{8} - \frac{1}{762} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{3048} a^{9} - \frac{1}{1524} a^{7} - \frac{3}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{6096} a^{10} - \frac{1}{3048} a^{8} + \frac{1}{6096} a^{6} - \frac{3}{16} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{12192} a^{11} - \frac{1}{6096} a^{9} + \frac{1}{12192} a^{7} + \frac{13}{32} a^{5} - \frac{3}{8} a^{3}$, $\frac{1}{371612160} a^{12} - \frac{7}{97536} a^{10} - \frac{307}{975360} a^{8} - \frac{1}{65024} a^{6} + \frac{15}{64} a^{4} - \frac{11}{32} a^{2} + \frac{1}{40}$, $\frac{1}{743224320} a^{13} - \frac{7}{195072} a^{11} - \frac{307}{1950720} a^{9} - \frac{1}{130048} a^{7} + \frac{15}{128} a^{5} - \frac{11}{64} a^{3} + \frac{1}{80} a$, $\frac{1}{135266826240} a^{14} + \frac{1}{13526682624} a^{12} + \frac{1353}{16906240} a^{10} + \frac{2549}{71006208} a^{8} + \frac{1661}{1479296} a^{6} + \frac{167}{1664} a^{4} - \frac{3849}{14560} a^{2} - \frac{17}{364}$, $\frac{1}{270533652480} a^{15} + \frac{1}{27053365248} a^{13} + \frac{1353}{33812480} a^{11} + \frac{2549}{142012416} a^{9} + \frac{1661}{2958592} a^{7} - \frac{1497}{3328} a^{5} - \frac{3849}{29120} a^{3} + \frac{347}{728} a$, $\frac{1}{293824462201606103040} a^{16} - \frac{346905821}{146912231100803051520} a^{14} + \frac{294310633457}{293824462201606103040} a^{12} + \frac{2821696908469}{771192814177443840} a^{10} - \frac{3087926270419}{64266067848120320} a^{8} + \frac{1403241649605}{1606651696203008} a^{6} + \frac{457220259637}{3953375236720} a^{4} - \frac{73352465009}{247085952295} a^{2} - \frac{791794132821}{1976687618360}$, $\frac{1}{587648924403212206080} a^{17} - \frac{346905821}{293824462201606103040} a^{15} + \frac{294310633457}{587648924403212206080} a^{13} + \frac{2821696908469}{1542385628354887680} a^{11} - \frac{3087926270419}{128532135696240640} a^{9} + \frac{1403241649605}{3213303392406016} a^{7} - \frac{3496154977083}{7906750473440} a^{5} + \frac{86866743643}{247085952295} a^{3} + \frac{1184893485539}{3953375236720} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{8}\times C_{728}\times C_{3640}$, which has order $2713518080$ (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:  \( 139246964.12762704 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-381}) \), \(\Q(\zeta_{9})^+\), 3.3.1306449.2, 3.3.1306449.1, 3.3.16129.1, 6.0.2580372645696.7, 6.0.41618830402430784.1, 6.0.41618830402430784.2, 6.0.57090302335296.2, 9.9.2229858897655236849.1

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

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$127$127.6.5.1$x^{6} - 127$$6$$1$$5$$C_6$$[\ ]_{6}$
127.6.5.1$x^{6} - 127$$6$$1$$5$$C_6$$[\ ]_{6}$
127.6.5.1$x^{6} - 127$$6$$1$$5$$C_6$$[\ ]_{6}$