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Properties

Label 18.0.12853682015...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{22}\cdot 5^{12}$
Root discriminant $28.22$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (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![648, 3024, 7056, 10368, 10800, 8568, 5328, 2592, 936, 180, -36, -12, 18, -24, -32, -6, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^15 - 32*x^14 - 24*x^13 + 18*x^12 - 12*x^11 - 36*x^10 + 180*x^9 + 936*x^8 + 2592*x^7 + 5328*x^6 + 8568*x^5 + 10800*x^4 + 10368*x^3 + 7056*x^2 + 3024*x + 648)
 
gp: K = bnfinit(x^18 - 6*x^15 - 32*x^14 - 24*x^13 + 18*x^12 - 12*x^11 - 36*x^10 + 180*x^9 + 936*x^8 + 2592*x^7 + 5328*x^6 + 8568*x^5 + 10800*x^4 + 10368*x^3 + 7056*x^2 + 3024*x + 648, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{15} - 32 x^{14} - 24 x^{13} + 18 x^{12} - 12 x^{11} - 36 x^{10} + 180 x^{9} + 936 x^{8} + 2592 x^{7} + 5328 x^{6} + 8568 x^{5} + 10800 x^{4} + 10368 x^{3} + 7056 x^{2} + 3024 x + 648 \)

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:  \(-128536820158464000000000000=-\,2^{24}\cdot 3^{22}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.22$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{6}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{7}$, $\frac{1}{84} a^{12} + \frac{1}{14} a^{11} + \frac{1}{21} a^{10} + \frac{3}{14} a^{9} + \frac{1}{21} a^{8} + \frac{3}{14} a^{7} + \frac{4}{21} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{84} a^{13} - \frac{1}{21} a^{11} - \frac{1}{14} a^{10} - \frac{5}{21} a^{9} - \frac{1}{14} a^{8} + \frac{5}{21} a^{7} - \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{84} a^{14} + \frac{1}{21} a^{11} - \frac{1}{21} a^{10} - \frac{3}{14} a^{9} - \frac{1}{14} a^{8} + \frac{4}{21} a^{7} + \frac{5}{42} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{84} a^{15} - \frac{1}{14} a^{10} + \frac{1}{14} a^{9} + \frac{2}{21} a^{7} - \frac{1}{7} a^{6} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{252} a^{16} + \frac{1}{252} a^{12} - \frac{1}{14} a^{10} - \frac{2}{21} a^{9} - \frac{5}{42} a^{8} + \frac{1}{42} a^{7} - \frac{3}{14} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{30651999446628} a^{17} - \frac{1516114375}{851444429073} a^{16} - \frac{8304627713}{1702888858146} a^{15} - \frac{52472492783}{10217333148876} a^{14} - \frac{12445023629}{4378857063804} a^{13} + \frac{4208581829}{5108666574438} a^{12} + \frac{40855091597}{1702888858146} a^{11} + \frac{156943952996}{2554333287219} a^{10} - \frac{418507770955}{1702888858146} a^{9} - \frac{28450154342}{851444429073} a^{8} + \frac{72369396146}{851444429073} a^{7} - \frac{271118520901}{1702888858146} a^{6} - \frac{127481954801}{851444429073} a^{5} - \frac{330541983425}{851444429073} a^{4} - \frac{113558195025}{283814809691} a^{3} + \frac{61786187931}{283814809691} a^{2} - \frac{417030389948}{851444429073} a + \frac{59931139693}{283814809691}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{48045741259}{4378857063804} a^{17} + \frac{4337729276}{364904755317} a^{16} - \frac{2321899505}{162179891252} a^{15} + \frac{60831751489}{729809510634} a^{14} + \frac{282978273920}{1094714265951} a^{13} - \frac{2319442982}{364904755317} a^{12} - \frac{6366705854}{40544972813} a^{11} + \frac{103331976403}{364904755317} a^{10} + \frac{3455407381}{40544972813} a^{9} - \frac{246650242733}{121634918439} a^{8} - \frac{982922858468}{121634918439} a^{7} - \frac{2427672117752}{121634918439} a^{6} - \frac{4598719841878}{121634918439} a^{5} - \frac{6727773298474}{121634918439} a^{4} - \frac{2542942545079}{40544972813} a^{3} - \frac{2090926650252}{40544972813} a^{2} - \frac{3419287637824}{121634918439} a - \frac{321487286303}{40544972813} \) (order $4$)
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:  \( 836391.5690201633 \) (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

\(\Q(\sqrt{-1}) \), 3.1.300.1, 6.0.1440000.1, 9.3.177147000000.2

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.12.22.67$x^{12} + 99 x^{11} + 117 x^{10} - 114 x^{9} - 81 x^{8} + 9 x^{7} - 15 x^{6} + 54 x^{5} - 108 x^{4} + 45 x^{3} - 81 x^{2} - 108 x - 72$$6$$2$$22$$D_6$$[5/2]_{2}^{2}$
$5$5.9.6.1$x^{9} - 25 x^{3} + 250$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
5.9.6.1$x^{9} - 25 x^{3} + 250$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$