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Properties

Label 18.6.39487629517...5289.1
Degree $18$
Signature $[6, 6]$
Discriminant $19^{16}\cdot 37^{2}$
Root discriminant $20.46$
Ramified primes $19, 37$
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, 14, -8, -3, 0, 20, -89, 111, 54, -270, 241, 7, -180, 149, -38, -21, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 21*x^16 - 21*x^15 - 38*x^14 + 149*x^13 - 180*x^12 + 7*x^11 + 241*x^10 - 270*x^9 + 54*x^8 + 111*x^7 - 89*x^6 + 20*x^5 - 3*x^3 - 8*x^2 + 14*x - 1)
 
gp: K = bnfinit(x^18 - 7*x^17 + 21*x^16 - 21*x^15 - 38*x^14 + 149*x^13 - 180*x^12 + 7*x^11 + 241*x^10 - 270*x^9 + 54*x^8 + 111*x^7 - 89*x^6 + 20*x^5 - 3*x^3 - 8*x^2 + 14*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 21 x^{16} - 21 x^{15} - 38 x^{14} + 149 x^{13} - 180 x^{12} + 7 x^{11} + 241 x^{10} - 270 x^{9} + 54 x^{8} + 111 x^{7} - 89 x^{6} + 20 x^{5} - 3 x^{3} - 8 x^{2} + 14 x - 1 \)

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:  \(394876295174073378555289=19^{16}\cdot 37^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.46$
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:  $19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{14} + \frac{3}{7} a^{12} + \frac{1}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{14} + \frac{3}{7} a^{13} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{3} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{438993133243} a^{17} - \frac{2178683445}{62713304749} a^{16} - \frac{386772296}{62713304749} a^{15} - \frac{139691858962}{438993133243} a^{14} - \frac{5813636473}{62713304749} a^{13} + \frac{131546557109}{438993133243} a^{12} + \frac{42900055598}{438993133243} a^{11} + \frac{191316783380}{438993133243} a^{10} + \frac{21589896157}{62713304749} a^{9} - \frac{21452698327}{438993133243} a^{8} + \frac{17024374277}{62713304749} a^{7} + \frac{67842787749}{438993133243} a^{6} - \frac{141141519567}{438993133243} a^{5} - \frac{109662435936}{438993133243} a^{4} + \frac{24605586491}{62713304749} a^{3} - \frac{109607476093}{438993133243} a^{2} - \frac{89260888138}{438993133243} a + \frac{53401318210}{438993133243}$

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:  \( 60096.7790439 \)
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 ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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}$
37Data not computed