↑

Properties

Label 18.0.23634501652...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{45}\cdot 5^{9}$
Root discriminant $55.33$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![38400, -138240, 331776, -25920, 20736, 4320, -71352, -2592, 2592, 2079, 5796, 1296, 117, 0, -126, -27, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 27*x^15 - 126*x^14 + 117*x^12 + 1296*x^11 + 5796*x^10 + 2079*x^9 + 2592*x^8 - 2592*x^7 - 71352*x^6 + 4320*x^5 + 20736*x^4 - 25920*x^3 + 331776*x^2 - 138240*x + 38400)
 
gp: K = bnfinit(x^18 - 27*x^15 - 126*x^14 + 117*x^12 + 1296*x^11 + 5796*x^10 + 2079*x^9 + 2592*x^8 - 2592*x^7 - 71352*x^6 + 4320*x^5 + 20736*x^4 - 25920*x^3 + 331776*x^2 - 138240*x + 38400, 1)
 

Normalized defining polynomial

\( x^{18} - 27 x^{15} - 126 x^{14} + 117 x^{12} + 1296 x^{11} + 5796 x^{10} + 2079 x^{9} + 2592 x^{8} - 2592 x^{7} - 71352 x^{6} + 4320 x^{5} + 20736 x^{4} - 25920 x^{3} + 331776 x^{2} - 138240 x + 38400 \)

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:  \(-23634501652406669589144000000000=-\,2^{12}\cdot 3^{45}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.33$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{3}{8} a^{6} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{3}{16} a^{7} + \frac{1}{4} a^{5} - \frac{5}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{14} - \frac{3}{32} a^{11} + \frac{1}{16} a^{10} - \frac{3}{32} a^{8} - \frac{3}{8} a^{6} + \frac{7}{32} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{15} - \frac{3}{64} a^{12} + \frac{1}{32} a^{11} + \frac{13}{64} a^{9} + \frac{1}{16} a^{7} - \frac{9}{64} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{8390912} a^{16} + \frac{3117}{1048864} a^{15} - \frac{3429}{2097728} a^{14} + \frac{92485}{8390912} a^{13} - \frac{242835}{4195456} a^{12} + \frac{4667}{2097728} a^{11} - \frac{156947}{8390912} a^{10} - \frac{204483}{1048864} a^{9} + \frac{10721}{262216} a^{8} + \frac{1984543}{8390912} a^{7} - \frac{23889}{1048864} a^{6} + \frac{282717}{2097728} a^{5} - \frac{370531}{1048864} a^{4} + \frac{22109}{524432} a^{3} - \frac{1769}{3592} a^{2} - \frac{152}{449} a + \frac{9965}{65554}$, $\frac{1}{708084545982364388724035428695040} a^{17} + \frac{1932508309536611761906573}{70808454598236438872403542869504} a^{16} - \frac{138471389969815944302334558723}{35404227299118219436201771434752} a^{15} - \frac{8825432931188039579749290508147}{708084545982364388724035428695040} a^{14} + \frac{2934005615457687072930246756551}{177021136495591097181008857173760} a^{13} - \frac{185508723850731861713212917133}{17702113649559109718100885717376} a^{12} - \frac{61304868723083636220535129232763}{708084545982364388724035428695040} a^{11} - \frac{14514985054246705153347155753107}{354042272991182194362017714347520} a^{10} - \frac{1666842822660096642747317586333}{88510568247795548590504428586880} a^{9} - \frac{155296684161824108929336841341201}{708084545982364388724035428695040} a^{8} - \frac{17330176004170949677416321393089}{354042272991182194362017714347520} a^{7} + \frac{56421357308442073583672699332967}{177021136495591097181008857173760} a^{6} + \frac{10324104563303831443134361674019}{22127642061948887147626107146720} a^{5} + \frac{687846614968657425988799036561}{4425528412389777429525221429344} a^{4} - \frac{4600235447377445747971578574011}{11063821030974443573813053573360} a^{3} + \frac{7783115483085391883991360689}{30311838441025872804967270064} a^{2} - \frac{1846465310316636121366068602983}{5531910515487221786906526786680} a + \frac{257432270814742919332863809401}{553191051548722178690652678668}$

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:  \( 787576192.9529331 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

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 $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.243.1, 6.0.22143375.1, 9.1.2008387814976.4

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{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}$
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$