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Properties

Label 18.6.10522631208...0561.1
Degree $18$
Signature $[6, 6]$
Discriminant $19^{16}\cdot 191^{2}$
Root discriminant $24.55$
Ramified primes $19, 191$
Class number $1$
Class group Trivial
Galois group 18T177

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 15, -56, -12, 78, 445, -373, 499, -17, -357, 499, -230, -64, 151, -79, 7, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 12*x^16 + 7*x^15 - 79*x^14 + 151*x^13 - 64*x^12 - 230*x^11 + 499*x^10 - 357*x^9 - 17*x^8 + 499*x^7 - 373*x^6 + 445*x^5 + 78*x^4 - 12*x^3 - 56*x^2 + 15*x - 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 12*x^16 + 7*x^15 - 79*x^14 + 151*x^13 - 64*x^12 - 230*x^11 + 499*x^10 - 357*x^9 - 17*x^8 + 499*x^7 - 373*x^6 + 445*x^5 + 78*x^4 - 12*x^3 - 56*x^2 + 15*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 12 x^{16} + 7 x^{15} - 79 x^{14} + 151 x^{13} - 64 x^{12} - 230 x^{11} + 499 x^{10} - 357 x^{9} - 17 x^{8} + 499 x^{7} - 373 x^{6} + 445 x^{5} + 78 x^{4} - 12 x^{3} - 56 x^{2} + 15 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10522631208360387818170561=19^{16}\cdot 191^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.55$
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:  $19, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{56} a^{12} - \frac{1}{14} a^{11} + \frac{13}{56} a^{10} + \frac{25}{56} a^{9} + \frac{9}{56} a^{8} - \frac{1}{2} a^{7} + \frac{25}{56} a^{6} - \frac{3}{8} a^{5} - \frac{9}{56} a^{3} + \frac{1}{56} a^{2} - \frac{19}{56} a + \frac{15}{56}$, $\frac{1}{112} a^{13} - \frac{1}{112} a^{12} + \frac{1}{112} a^{11} + \frac{1}{14} a^{10} - \frac{1}{4} a^{9} - \frac{1}{112} a^{8} + \frac{53}{112} a^{7} - \frac{1}{56} a^{6} - \frac{1}{16} a^{5} + \frac{47}{112} a^{4} - \frac{13}{56} a^{3} + \frac{5}{14} a^{2} + \frac{1}{8} a + \frac{45}{112}$, $\frac{1}{224} a^{14} + \frac{9}{224} a^{11} - \frac{5}{56} a^{10} + \frac{83}{224} a^{9} + \frac{13}{56} a^{8} + \frac{51}{224} a^{7} - \frac{9}{224} a^{6} + \frac{5}{28} a^{5} + \frac{3}{32} a^{4} - \frac{7}{16} a^{3} - \frac{29}{112} a^{2} - \frac{53}{224} a + \frac{45}{224}$, $\frac{1}{448} a^{15} - \frac{1}{448} a^{14} + \frac{1}{448} a^{12} + \frac{3}{448} a^{11} - \frac{1}{448} a^{10} - \frac{1}{64} a^{9} - \frac{73}{448} a^{8} - \frac{15}{112} a^{7} + \frac{73}{448} a^{6} + \frac{149}{448} a^{5} - \frac{17}{64} a^{4} + \frac{1}{4} a^{3} - \frac{3}{448} a^{2} + \frac{13}{224} a - \frac{165}{448}$, $\frac{1}{135296} a^{16} - \frac{17}{16912} a^{15} + \frac{3}{135296} a^{14} + \frac{193}{135296} a^{13} + \frac{71}{33824} a^{12} - \frac{349}{67648} a^{11} - \frac{287}{1208} a^{10} - \frac{15409}{33824} a^{9} + \frac{661}{19328} a^{8} - \frac{46175}{135296} a^{7} - \frac{6035}{67648} a^{6} - \frac{17125}{67648} a^{5} - \frac{62691}{135296} a^{4} - \frac{55659}{135296} a^{3} - \frac{40521}{135296} a^{2} - \frac{39591}{135296} a + \frac{13487}{135296}$, $\frac{1}{270592} a^{17} - \frac{1}{270592} a^{16} - \frac{237}{270592} a^{15} + \frac{299}{135296} a^{14} - \frac{237}{270592} a^{13} + \frac{97}{135296} a^{12} - \frac{6411}{135296} a^{11} + \frac{6561}{67648} a^{10} + \frac{4471}{270592} a^{9} - \frac{54441}{135296} a^{8} + \frac{29865}{270592} a^{7} + \frac{16539}{67648} a^{6} - \frac{27185}{270592} a^{5} + \frac{2865}{8456} a^{4} - \frac{26403}{135296} a^{3} + \frac{43369}{135296} a^{2} + \frac{51747}{135296} a + \frac{59481}{270592}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 274434.575676 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T177:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 16 conjugacy class representatives for t18n177
Character table for t18n177

Intermediate fields

3.3.361.1, \(\Q(\zeta_{19})^+\)

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
$191$$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
191.2.0.1$x^{2} - x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.0.1$x^{2} - x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.0.1$x^{2} - x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.2.0.1$x^{2} - x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$