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Properties

Label 18.0.36284103920...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 5^{9}\cdot 7^{12}$
Root discriminant $23.14$
Ramified primes $2, 5, 7$
Class number $6$
Class group $[6]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1561, -294, 1463, 1940, 1721, 1802, 1081, -824, -735, -514, 5, 416, 172, 28, -10, -32, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 5*x^16 - 32*x^15 - 10*x^14 + 28*x^13 + 172*x^12 + 416*x^11 + 5*x^10 - 514*x^9 - 735*x^8 - 824*x^7 + 1081*x^6 + 1802*x^5 + 1721*x^4 + 1940*x^3 + 1463*x^2 - 294*x + 1561)
 
gp: K = bnfinit(x^18 - 2*x^17 + 5*x^16 - 32*x^15 - 10*x^14 + 28*x^13 + 172*x^12 + 416*x^11 + 5*x^10 - 514*x^9 - 735*x^8 - 824*x^7 + 1081*x^6 + 1802*x^5 + 1721*x^4 + 1940*x^3 + 1463*x^2 - 294*x + 1561, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 5 x^{16} - 32 x^{15} - 10 x^{14} + 28 x^{13} + 172 x^{12} + 416 x^{11} + 5 x^{10} - 514 x^{9} - 735 x^{8} - 824 x^{7} + 1081 x^{6} + 1802 x^{5} + 1721 x^{4} + 1940 x^{3} + 1463 x^{2} - 294 x + 1561 \)

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:  \(-3628410392018944000000000=-\,2^{27}\cdot 5^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.14$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{11} - \frac{1}{20} a^{10} + \frac{1}{5} a^{9} - \frac{1}{20} a^{8} - \frac{1}{2} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{3}{10} a^{3} + \frac{1}{20} a^{2} + \frac{3}{10} a + \frac{1}{5}$, $\frac{1}{20} a^{13} + \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{1}{4} a^{7} - \frac{2}{5} a^{6} - \frac{1}{10} a^{4} - \frac{7}{20} a^{3} - \frac{1}{10} a^{2} + \frac{1}{20} a + \frac{2}{5}$, $\frac{1}{20} a^{14} + \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{4} a^{8} - \frac{2}{5} a^{7} - \frac{1}{10} a^{5} - \frac{7}{20} a^{4} - \frac{1}{10} a^{3} + \frac{1}{20} a^{2} + \frac{2}{5} a$, $\frac{1}{20} a^{15} + \frac{1}{20} a^{11} + \frac{1}{10} a^{9} + \frac{1}{5} a^{8} + \frac{1}{4} a^{7} - \frac{1}{10} a^{6} - \frac{1}{20} a^{5} + \frac{3}{10} a^{4} - \frac{1}{20} a^{3} - \frac{1}{5} a^{2} - \frac{7}{20} a - \frac{2}{5}$, $\frac{1}{140} a^{16} + \frac{3}{140} a^{15} - \frac{1}{70} a^{14} - \frac{3}{140} a^{13} - \frac{1}{70} a^{12} - \frac{3}{28} a^{10} - \frac{9}{140} a^{9} - \frac{1}{35} a^{8} - \frac{3}{20} a^{7} + \frac{67}{140} a^{6} + \frac{1}{140} a^{5} - \frac{31}{140} a^{4} + \frac{5}{14} a^{3} - \frac{1}{5} a^{2} - \frac{3}{20} a - \frac{1}{2}$, $\frac{1}{445533439136420569190459140} a^{17} + \frac{258334722178596433773096}{111383359784105142297614785} a^{16} - \frac{4462665977484466481672897}{222766719568210284595229570} a^{15} - \frac{77997674975358538673411}{445533439136420569190459140} a^{14} + \frac{2122412808935295015419265}{89106687827284113838091828} a^{13} - \frac{7368116613220810179865239}{445533439136420569190459140} a^{12} + \frac{3861870520943457472686143}{111383359784105142297614785} a^{11} - \frac{4847926321230412054816621}{111383359784105142297614785} a^{10} - \frac{102730008198313786017814309}{445533439136420569190459140} a^{9} + \frac{102278818918707062193567337}{445533439136420569190459140} a^{8} + \frac{168796002330425726174206929}{445533439136420569190459140} a^{7} + \frac{66206839854781884340138943}{445533439136420569190459140} a^{6} + \frac{157607533624583752884504923}{445533439136420569190459140} a^{5} - \frac{116393941668837649550304111}{445533439136420569190459140} a^{4} - \frac{11001562993138899494463233}{445533439136420569190459140} a^{3} - \frac{13221714429235544953838013}{31823817081172897799318510} a^{2} + \frac{205755248188432550477741}{31823817081172897799318510} a - \frac{26224416175315632709133}{285415399831147065464740}$

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

Class group and class number

$C_{6}$, which has order $6$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 19821.7051984 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-10}) \), 3.1.1960.1 x3, \(\Q(\zeta_{7})^+\), 6.0.153664000.2, 6.0.3136000.1 x2, 6.0.153664000.1, 9.3.7529536000.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.3136000.1
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$