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Properties

Label 18.8.31035847405...5247.2
Degree $18$
Signature $[8, 5]$
Discriminant $-\,3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 167$
Root discriminant $38.27$
Ramified primes $3, 7, 29, 167$
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![-448, -672, 4032, 1576, -7596, -780, 5413, 393, -3804, 968, 1734, -1680, 704, 9, -174, 88, -12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 12*x^16 + 88*x^15 - 174*x^14 + 9*x^13 + 704*x^12 - 1680*x^11 + 1734*x^10 + 968*x^9 - 3804*x^8 + 393*x^7 + 5413*x^6 - 780*x^5 - 7596*x^4 + 1576*x^3 + 4032*x^2 - 672*x - 448)
 
gp: K = bnfinit(x^18 - 3*x^17 - 12*x^16 + 88*x^15 - 174*x^14 + 9*x^13 + 704*x^12 - 1680*x^11 + 1734*x^10 + 968*x^9 - 3804*x^8 + 393*x^7 + 5413*x^6 - 780*x^5 - 7596*x^4 + 1576*x^3 + 4032*x^2 - 672*x - 448, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 12 x^{16} + 88 x^{15} - 174 x^{14} + 9 x^{13} + 704 x^{12} - 1680 x^{11} + 1734 x^{10} + 968 x^{9} - 3804 x^{8} + 393 x^{7} + 5413 x^{6} - 780 x^{5} - 7596 x^{4} + 1576 x^{3} + 4032 x^{2} - 672 x - 448 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-31035847405427991068883845247=-\,3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 167\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.27$
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:  $3, 7, 29, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{56} a^{15} + \frac{1}{56} a^{14} - \frac{1}{14} a^{13} + \frac{3}{14} a^{12} + \frac{9}{28} a^{11} - \frac{1}{8} a^{10} - \frac{5}{14} a^{9} - \frac{1}{2} a^{8} - \frac{13}{28} a^{7} + \frac{2}{7} a^{6} + \frac{5}{14} a^{5} + \frac{17}{56} a^{4} + \frac{1}{56} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{112} a^{16} - \frac{1}{112} a^{15} - \frac{3}{56} a^{14} + \frac{5}{28} a^{13} - \frac{3}{56} a^{12} + \frac{13}{112} a^{11} - \frac{3}{56} a^{10} + \frac{3}{28} a^{9} + \frac{15}{56} a^{8} - \frac{11}{28} a^{7} + \frac{11}{28} a^{6} + \frac{33}{112} a^{5} + \frac{23}{112} a^{4} + \frac{13}{56} a^{3} - \frac{1}{2} a$, $\frac{1}{2036327068432143438759968693344} a^{17} - \frac{786902307610194153754682753}{290903866918877634108566956192} a^{16} - \frac{31025029868381931653580703}{9090745841214926065892717381} a^{15} + \frac{3861603435225331247096963715}{36362983364859704263570869524} a^{14} - \frac{30089505299657536281639712177}{145451933459438817054283478096} a^{13} - \frac{63374813741343245580941649993}{290903866918877634108566956192} a^{12} + \frac{3858968427822934214263765969}{72725966729719408527141739048} a^{11} - \frac{28887526226550711800124287051}{127270441777008964922498043334} a^{10} - \frac{33181183705074726769054787659}{145451933459438817054283478096} a^{9} + \frac{8722609827704517255814700357}{18181491682429852131785434762} a^{8} + \frac{9046857986208181634766272127}{72725966729719408527141739048} a^{7} - \frac{62712007918233347444067408001}{290903866918877634108566956192} a^{6} + \frac{21769049812198867794700803735}{290903866918877634108566956192} a^{5} + \frac{1582907077520775732593635637}{9090745841214926065892717381} a^{4} - \frac{11167800999381780678484045959}{509081767108035859689992173336} a^{3} + \frac{2993102816145697312656449801}{36362983364859704263570869524} a^{2} - \frac{442504573432059818329210242}{9090745841214926065892717381} a - \frac{3996755356480894120243386329}{9090745841214926065892717381}$

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:  $12$
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:  \( 52992980.8614 \) (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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R $18$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ $18$

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
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$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$$0$Trivial$[\ ]$
$\Q_{29}$$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$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$167$$\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$$[\ ]^{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
167.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$