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Properties

Label 18.2.93446253200...0000.2
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 7^{15}$
Root discriminant $40.69$
Ramified primes $2, 3, 5, 7$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_2\times C_3:S_3$ (as 18T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-20, 180, -460, -388, 372, 3232, 1187, -75, 407, 416, 203, 117, 170, 91, -3, 14, -1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - x^16 + 14*x^15 - 3*x^14 + 91*x^13 + 170*x^12 + 117*x^11 + 203*x^10 + 416*x^9 + 407*x^8 - 75*x^7 + 1187*x^6 + 3232*x^5 + 372*x^4 - 388*x^3 - 460*x^2 + 180*x - 20)
 
gp: K = bnfinit(x^18 - 3*x^17 - x^16 + 14*x^15 - 3*x^14 + 91*x^13 + 170*x^12 + 117*x^11 + 203*x^10 + 416*x^9 + 407*x^8 - 75*x^7 + 1187*x^6 + 3232*x^5 + 372*x^4 - 388*x^3 - 460*x^2 + 180*x - 20, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - x^{16} + 14 x^{15} - 3 x^{14} + 91 x^{13} + 170 x^{12} + 117 x^{11} + 203 x^{10} + 416 x^{9} + 407 x^{8} - 75 x^{7} + 1187 x^{6} + 3232 x^{5} + 372 x^{4} - 388 x^{3} - 460 x^{2} + 180 x - 20 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(93446253200208069000000000000=2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.69$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{1}{4} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{5}{12} a^{5} - \frac{1}{12} a^{4} + \frac{1}{12} a^{3} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{24} a^{12} - \frac{1}{12} a^{10} + \frac{11}{24} a^{9} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} + \frac{1}{3} a^{4} + \frac{3}{8} a^{3} + \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{48} a^{13} - \frac{1}{48} a^{12} - \frac{1}{24} a^{11} - \frac{1}{16} a^{10} + \frac{7}{16} a^{9} - \frac{5}{12} a^{8} - \frac{19}{48} a^{7} + \frac{23}{48} a^{6} - \frac{1}{6} a^{5} + \frac{3}{16} a^{4} - \frac{23}{48} a^{3} - \frac{11}{24} a^{2} + \frac{7}{24} a - \frac{1}{24}$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{12} - \frac{1}{48} a^{11} + \frac{1}{24} a^{10} + \frac{1}{16} a^{9} + \frac{17}{48} a^{8} + \frac{1}{12} a^{7} + \frac{3}{16} a^{6} + \frac{13}{48} a^{5} + \frac{7}{24} a^{4} + \frac{17}{48} a^{3} - \frac{1}{12} a^{2} - \frac{5}{12} a - \frac{11}{24}$, $\frac{1}{1632} a^{15} - \frac{1}{136} a^{14} + \frac{11}{1632} a^{13} - \frac{23}{1632} a^{12} - \frac{23}{816} a^{11} - \frac{97}{1632} a^{10} - \frac{103}{544} a^{9} + \frac{20}{51} a^{8} - \frac{105}{544} a^{7} + \frac{535}{1632} a^{6} - \frac{389}{816} a^{5} - \frac{253}{544} a^{4} - \frac{203}{816} a^{3} - \frac{41}{204} a^{2} - \frac{55}{816} a - \frac{15}{136}$, $\frac{1}{4896} a^{16} + \frac{1}{4896} a^{15} - \frac{43}{4896} a^{14} + \frac{1}{272} a^{13} - \frac{5}{4896} a^{12} - \frac{49}{4896} a^{11} - \frac{3}{136} a^{10} + \frac{2335}{4896} a^{9} - \frac{607}{1632} a^{8} - \frac{199}{2448} a^{7} + \frac{427}{1632} a^{6} + \frac{1061}{4896} a^{5} - \frac{2147}{4896} a^{4} - \frac{107}{816} a^{3} - \frac{929}{2448} a^{2} + \frac{1133}{2448} a + \frac{163}{1224}$, $\frac{1}{510771273315421536} a^{17} + \frac{4681280591461}{46433752119583776} a^{16} + \frac{10689950176583}{170257091105140512} a^{15} + \frac{98560188531475}{11608438029895944} a^{14} - \frac{1866999552680903}{510771273315421536} a^{13} + \frac{594916301726607}{56752363701713504} a^{12} + \frac{6070767612430795}{255385636657710768} a^{11} + \frac{15724883868287881}{510771273315421536} a^{10} - \frac{197816218819590209}{510771273315421536} a^{9} + \frac{329279049769019}{2902109507473986} a^{8} + \frac{20570716588337321}{46433752119583776} a^{7} + \frac{129427222917402005}{510771273315421536} a^{6} - \frac{57008847379879111}{170257091105140512} a^{5} + \frac{126945423117882029}{255385636657710768} a^{4} + \frac{57140729931293197}{255385636657710768} a^{3} + \frac{33609890615100301}{85128545552570256} a^{2} - \frac{50353929847809391}{127692818328855384} a + \frac{434158221450227}{63846409164427692}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  $9$
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:  \( 10827008.366670528 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3$ (as 18T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2\times C_3:S_3$
Character table for $C_2\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{21}) \), 3.1.300.1, 3.1.3675.1, 3.1.14700.1, 3.1.588.1, 6.2.283618125.1, 6.2.92610000.1, 6.2.4537890000.1, 6.2.7260624.1, 9.1.9529569000000.1

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

Sibling fields

Galois closure: data not computed
Degree 18 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 R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{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}$
7Data not computed