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Properties

Label 18.10.1067233230...4197.1
Degree $18$
Signature $[10, 4]$
Discriminant $19^{16}\cdot 37$
Root discriminant $16.74$
Ramified primes $19, 37$
Class number $1$
Class group Trivial
Galois group 18T460

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -6, 12, 59, -84, -176, 293, 93, -329, 116, 63, -97, 87, -5, -49, 21, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 7*x^16 + 21*x^15 - 49*x^14 - 5*x^13 + 87*x^12 - 97*x^11 + 63*x^10 + 116*x^9 - 329*x^8 + 93*x^7 + 293*x^6 - 176*x^5 - 84*x^4 + 59*x^3 + 12*x^2 - 6*x - 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 7*x^16 + 21*x^15 - 49*x^14 - 5*x^13 + 87*x^12 - 97*x^11 + 63*x^10 + 116*x^9 - 329*x^8 + 93*x^7 + 293*x^6 - 176*x^5 - 84*x^4 + 59*x^3 + 12*x^2 - 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 7 x^{16} + 21 x^{15} - 49 x^{14} - 5 x^{13} + 87 x^{12} - 97 x^{11} + 63 x^{10} + 116 x^{9} - 329 x^{8} + 93 x^{7} + 293 x^{6} - 176 x^{5} - 84 x^{4} + 59 x^{3} + 12 x^{2} - 6 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10672332302001983204197=19^{16}\cdot 37\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.74$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{379} a^{16} - \frac{133}{379} a^{15} - \frac{115}{379} a^{14} - \frac{56}{379} a^{13} - \frac{41}{379} a^{12} - \frac{182}{379} a^{11} - \frac{124}{379} a^{10} + \frac{48}{379} a^{9} + \frac{129}{379} a^{8} + \frac{151}{379} a^{7} - \frac{65}{379} a^{6} - \frac{91}{379} a^{5} + \frac{24}{379} a^{4} + \frac{155}{379} a^{3} + \frac{189}{379} a^{2} - \frac{9}{379}$, $\frac{1}{143641} a^{17} + \frac{120}{143641} a^{16} + \frac{15127}{143641} a^{15} + \frac{38690}{143641} a^{14} - \frac{8903}{143641} a^{13} + \frac{27345}{143641} a^{12} - \frac{1827}{143641} a^{11} + \frac{56983}{143641} a^{10} - \frac{2129}{143641} a^{9} + \frac{19144}{143641} a^{8} - \frac{30082}{143641} a^{7} - \frac{55573}{143641} a^{6} + \frac{36504}{143641} a^{5} + \frac{2816}{143641} a^{4} + \frac{67450}{143641} a^{3} + \frac{23940}{143641} a^{2} - \frac{9}{143641} a - \frac{1140}{143641}$

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:  $13$
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:  \( 18497.6611092 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T460:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 80 conjugacy class representatives for t18n460 are not computed
Character table for t18n460 is not computed

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 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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
19Data not computed
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$