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Properties

Label 18.0.75380206651...1968.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 7^{14}\cdot 13^{10}$
Root discriminant $51.93$
Ramified primes $2, 3, 7, 13$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -16, 122, -534, 1387, -2114, 1995, -2080, 3635, -4392, 2172, 370, -644, -42, 197, -58, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 3*x^16 - 58*x^15 + 197*x^14 - 42*x^13 - 644*x^12 + 370*x^11 + 2172*x^10 - 4392*x^9 + 3635*x^8 - 2080*x^7 + 1995*x^6 - 2114*x^5 + 1387*x^4 - 534*x^3 + 122*x^2 - 16*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 + 3*x^16 - 58*x^15 + 197*x^14 - 42*x^13 - 644*x^12 + 370*x^11 + 2172*x^10 - 4392*x^9 + 3635*x^8 - 2080*x^7 + 1995*x^6 - 2114*x^5 + 1387*x^4 - 534*x^3 + 122*x^2 - 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 3 x^{16} - 58 x^{15} + 197 x^{14} - 42 x^{13} - 644 x^{12} + 370 x^{11} + 2172 x^{10} - 4392 x^{9} + 3635 x^{8} - 2080 x^{7} + 1995 x^{6} - 2114 x^{5} + 1387 x^{4} - 534 x^{3} + 122 x^{2} - 16 x + 1 \)

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:  \(-7538020665121923215559258451968=-\,2^{12}\cdot 3^{9}\cdot 7^{14}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.93$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{28} a^{12} + \frac{1}{28} a^{11} - \frac{1}{28} a^{9} - \frac{1}{28} a^{8} + \frac{1}{4} a^{7} + \frac{1}{28} a^{6} - \frac{1}{28} a^{4} - \frac{2}{7} a^{3} - \frac{1}{2} a^{2} + \frac{1}{28} a + \frac{2}{7}$, $\frac{1}{28} a^{13} - \frac{1}{28} a^{11} - \frac{1}{28} a^{10} + \frac{2}{7} a^{8} - \frac{3}{14} a^{7} - \frac{1}{28} a^{6} - \frac{1}{28} a^{5} - \frac{1}{4} a^{4} - \frac{3}{14} a^{3} - \frac{13}{28} a^{2} + \frac{1}{4} a - \frac{2}{7}$, $\frac{1}{28} a^{14} - \frac{1}{2} a^{8} + \frac{13}{28} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a + \frac{1}{28}$, $\frac{1}{2184} a^{15} - \frac{5}{728} a^{14} - \frac{5}{546} a^{13} - \frac{1}{56} a^{12} + \frac{3}{728} a^{11} + \frac{251}{2184} a^{10} - \frac{17}{273} a^{9} + \frac{1075}{2184} a^{8} - \frac{131}{273} a^{7} + \frac{69}{182} a^{6} + \frac{141}{364} a^{5} + \frac{41}{91} a^{4} + \frac{3}{56} a^{3} - \frac{125}{2184} a^{2} - \frac{5}{14} a - \frac{1091}{2184}$, $\frac{1}{2184} a^{16} - \frac{11}{2184} a^{14} - \frac{9}{728} a^{13} - \frac{5}{364} a^{12} + \frac{37}{1092} a^{11} + \frac{41}{2184} a^{10} + \frac{127}{2184} a^{9} - \frac{445}{2184} a^{8} + \frac{79}{364} a^{7} - \frac{25}{364} a^{6} - \frac{12}{91} a^{5} - \frac{137}{728} a^{4} + \frac{425}{1092} a^{3} - \frac{53}{728} a^{2} + \frac{859}{2184} a + \frac{23}{104}$, $\frac{1}{2184} a^{17} - \frac{3}{182} a^{14} - \frac{2}{273} a^{13} + \frac{5}{312} a^{12} - \frac{125}{1092} a^{11} - \frac{19}{546} a^{10} - \frac{7}{104} a^{9} - \frac{961}{2184} a^{8} + \frac{479}{1092} a^{7} + \frac{131}{364} a^{6} + \frac{157}{728} a^{5} + \frac{5}{12} a^{4} - \frac{111}{364} a^{3} - \frac{47}{364} a^{2} + \frac{23}{104} a - \frac{223}{2184}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{93}{28} a^{17} + \frac{967}{168} a^{16} - \frac{176}{21} a^{15} + \frac{31975}{168} a^{14} - \frac{101377}{168} a^{13} - \frac{170}{7} a^{12} + \frac{89909}{42} a^{11} - \frac{36619}{56} a^{10} - \frac{1246975}{168} a^{9} + \frac{2117371}{168} a^{8} - \frac{360665}{42} a^{7} + \frac{31324}{7} a^{6} - \frac{150937}{28} a^{5} + \frac{311243}{56} a^{4} - \frac{256537}{84} a^{3} + \frac{152675}{168} a^{2} - \frac{3605}{24} a + \frac{2117}{168} \) (order $6$)
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:  \( 35319007.470736034 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.364.1, 6.0.3577392.1, 6.0.10955763.1

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.5.4$x^{6} + 26$$6$$1$$5$$C_6$$[\ ]_{6}$