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Properties

Label 18.0.95431754801...0007.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 23^{6}\cdot 167^{3}$
Root discriminant $24.42$
Ramified primes $7, 23, 167$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_3\times A_4$ (as 18T60)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, 0, 768, -64, 544, -304, 344, -400, 270, -211, 106, -77, 44, -30, 14, -8, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 5*x^16 - 8*x^15 + 14*x^14 - 30*x^13 + 44*x^12 - 77*x^11 + 106*x^10 - 211*x^9 + 270*x^8 - 400*x^7 + 344*x^6 - 304*x^5 + 544*x^4 - 64*x^3 + 768*x^2 + 512)
 
gp: K = bnfinit(x^18 - 2*x^17 + 5*x^16 - 8*x^15 + 14*x^14 - 30*x^13 + 44*x^12 - 77*x^11 + 106*x^10 - 211*x^9 + 270*x^8 - 400*x^7 + 344*x^6 - 304*x^5 + 544*x^4 - 64*x^3 + 768*x^2 + 512, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 5 x^{16} - 8 x^{15} + 14 x^{14} - 30 x^{13} + 44 x^{12} - 77 x^{11} + 106 x^{10} - 211 x^{9} + 270 x^{8} - 400 x^{7} + 344 x^{6} - 304 x^{5} + 544 x^{4} - 64 x^{3} + 768 x^{2} + 512 \)

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:  \(-9543175480174580217820007=-\,7^{12}\cdot 23^{6}\cdot 167^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.42$
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:  $7, 23, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{48} a^{13} + \frac{5}{48} a^{11} + \frac{1}{24} a^{10} - \frac{1}{8} a^{9} - \frac{5}{24} a^{8} - \frac{1}{2} a^{7} - \frac{13}{48} a^{6} - \frac{19}{48} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{96} a^{14} + \frac{5}{96} a^{12} + \frac{1}{48} a^{11} - \frac{1}{16} a^{10} + \frac{7}{48} a^{9} - \frac{1}{4} a^{8} + \frac{35}{96} a^{7} - \frac{19}{96} a^{5} + \frac{1}{12} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{384} a^{15} - \frac{1}{192} a^{14} + \frac{1}{384} a^{13} - \frac{1}{48} a^{12} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{7}{96} a^{9} - \frac{23}{128} a^{8} + \frac{37}{192} a^{7} - \frac{53}{128} a^{6} + \frac{71}{192} a^{5} + \frac{5}{32} a^{4} - \frac{13}{48} a^{3} + \frac{5}{24} a^{2} - \frac{5}{12} a + \frac{1}{6}$, $\frac{1}{612096} a^{16} - \frac{47}{38256} a^{15} + \frac{181}{612096} a^{14} + \frac{423}{102016} a^{13} + \frac{6155}{102016} a^{12} + \frac{31535}{306048} a^{11} - \frac{1549}{25504} a^{10} - \frac{44559}{204032} a^{9} - \frac{1751}{12752} a^{8} + \frac{60447}{204032} a^{7} - \frac{799}{2391} a^{6} - \frac{4373}{12752} a^{5} + \frac{587}{38256} a^{4} - \frac{1799}{6376} a^{3} - \frac{715}{9564} a^{2} + \frac{575}{4782} a - \frac{2257}{4782}$, $\frac{1}{921816576} a^{17} + \frac{35}{230454144} a^{16} - \frac{275291}{921816576} a^{15} - \frac{2181485}{460908288} a^{14} + \frac{4395797}{460908288} a^{13} + \frac{2366723}{460908288} a^{12} - \frac{1048103}{19204512} a^{11} + \frac{21436915}{921816576} a^{10} - \frac{55660159}{230454144} a^{9} - \frac{126600851}{921816576} a^{8} + \frac{59920783}{230454144} a^{7} + \frac{4701865}{9602256} a^{6} - \frac{4903345}{14403384} a^{5} - \frac{392779}{7201692} a^{4} + \frac{2235565}{7201692} a^{3} + \frac{2448869}{7201692} a^{2} - \frac{2242195}{7201692} a + \frac{699362}{1800423}$

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:  \( -1 \) (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:  \( 204453.252453 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\times A_4$ (as 18T60):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 24 conjugacy class representatives for $C_2\times S_3\times A_4$
Character table for $C_2\times S_3\times A_4$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.1.23.1, 6.0.400967.1, 9.3.1431435383.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$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}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$23$23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$167$$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.1.1$x^{2} - 167$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.1$x^{2} - 167$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.1.1$x^{2} - 167$$2$$1$$1$$C_2$$[\ ]_{2}$