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Properties

Label 18.6.21000303933...2233.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 113$
Root discriminant $37.45$
Ramified primes $3, 7, 29, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, -768, -1920, 2688, 1056, -768, -1592, 456, 186, 139, 93, 114, -199, -48, 33, 42, -15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 15*x^16 + 42*x^15 + 33*x^14 - 48*x^13 - 199*x^12 + 114*x^11 + 93*x^10 + 139*x^9 + 186*x^8 + 456*x^7 - 1592*x^6 - 768*x^5 + 1056*x^4 + 2688*x^3 - 1920*x^2 - 768*x + 512)
 
gp: K = bnfinit(x^18 - 3*x^17 - 15*x^16 + 42*x^15 + 33*x^14 - 48*x^13 - 199*x^12 + 114*x^11 + 93*x^10 + 139*x^9 + 186*x^8 + 456*x^7 - 1592*x^6 - 768*x^5 + 1056*x^4 + 2688*x^3 - 1920*x^2 - 768*x + 512, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 15 x^{16} + 42 x^{15} + 33 x^{14} - 48 x^{13} - 199 x^{12} + 114 x^{11} + 93 x^{10} + 139 x^{9} + 186 x^{8} + 456 x^{7} - 1592 x^{6} - 768 x^{5} + 1056 x^{4} + 2688 x^{3} - 1920 x^{2} - 768 x + 512 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21000303933014149645412422233=3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 113\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.45$
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:  $3, 7, 29, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{56} a^{12} + \frac{1}{56} a^{11} + \frac{1}{56} a^{10} + \frac{5}{28} a^{9} + \frac{5}{56} a^{8} - \frac{1}{2} a^{7} - \frac{27}{56} a^{6} + \frac{1}{4} a^{5} - \frac{15}{56} a^{4} - \frac{25}{56} a^{3} + \frac{1}{28} a^{2} + \frac{1}{14} a + \frac{1}{7}$, $\frac{1}{112} a^{13} - \frac{1}{112} a^{12} - \frac{1}{112} a^{11} + \frac{1}{14} a^{10} - \frac{15}{112} a^{9} + \frac{9}{56} a^{8} + \frac{29}{112} a^{7} + \frac{3}{28} a^{6} - \frac{43}{112} a^{5} + \frac{5}{112} a^{4} + \frac{13}{28} a^{3} - \frac{1}{2} a - \frac{1}{7}$, $\frac{1}{224} a^{14} - \frac{1}{224} a^{13} - \frac{1}{224} a^{12} + \frac{1}{28} a^{11} - \frac{15}{224} a^{10} - \frac{47}{112} a^{9} - \frac{83}{224} a^{8} - \frac{25}{56} a^{7} + \frac{69}{224} a^{6} + \frac{5}{224} a^{5} + \frac{13}{56} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{14} a$, $\frac{1}{448} a^{15} - \frac{1}{448} a^{14} - \frac{1}{448} a^{13} - \frac{23}{448} a^{11} - \frac{51}{224} a^{10} - \frac{163}{448} a^{9} + \frac{3}{16} a^{8} - \frac{155}{448} a^{7} - \frac{3}{448} a^{6} + \frac{41}{112} a^{5} + \frac{1}{56} a^{4} + \frac{1}{14} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{977536} a^{16} + \frac{995}{977536} a^{15} + \frac{183}{977536} a^{14} + \frac{493}{122192} a^{13} - \frac{2731}{977536} a^{12} + \frac{25379}{488768} a^{11} + \frac{189905}{977536} a^{10} - \frac{33275}{122192} a^{9} - \frac{487007}{977536} a^{8} + \frac{326217}{977536} a^{7} + \frac{63349}{244384} a^{6} - \frac{111229}{244384} a^{5} - \frac{4365}{17456} a^{4} - \frac{21569}{61096} a^{3} - \frac{454}{7637} a^{2} - \frac{96}{7637} a - \frac{2180}{7637}$, $\frac{1}{13685504} a^{17} - \frac{1}{1955072} a^{16} - \frac{10543}{13685504} a^{15} - \frac{6795}{6842752} a^{14} + \frac{16621}{13685504} a^{13} + \frac{7293}{3421376} a^{12} + \frac{240813}{13685504} a^{11} - \frac{1667705}{6842752} a^{10} - \frac{2009687}{13685504} a^{9} - \frac{1961985}{13685504} a^{8} + \frac{189241}{6842752} a^{7} - \frac{71089}{3421376} a^{6} - \frac{280575}{855344} a^{5} + \frac{204805}{427672} a^{4} + \frac{8738}{53459} a^{3} + \frac{11388}{53459} a^{2} - \frac{2948}{7637} a + \frac{16454}{53459}$

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:  $11$
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:  \( 33090400.5204 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ $18$

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
3Data not computed
$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}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
113Data not computed