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Properties

Label 18.2.14610422921...5693.1
Degree $18$
Signature $[2, 8]$
Discriminant $19^{16}\cdot 37^{3}$
Root discriminant $25.01$
Ramified primes $19, 37$
Class number $2$
Class group $[2]$
Galois group 18T460

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-647, 898, -1580, 1581, -1148, 1071, -684, 612, -370, 74, -13, -70, 57, -27, 31, -15, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 2*x^16 - 15*x^15 + 31*x^14 - 27*x^13 + 57*x^12 - 70*x^11 - 13*x^10 + 74*x^9 - 370*x^8 + 612*x^7 - 684*x^6 + 1071*x^5 - 1148*x^4 + 1581*x^3 - 1580*x^2 + 898*x - 647)
 
gp: K = bnfinit(x^18 - 2*x^17 + 2*x^16 - 15*x^15 + 31*x^14 - 27*x^13 + 57*x^12 - 70*x^11 - 13*x^10 + 74*x^9 - 370*x^8 + 612*x^7 - 684*x^6 + 1071*x^5 - 1148*x^4 + 1581*x^3 - 1580*x^2 + 898*x - 647, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 2 x^{16} - 15 x^{15} + 31 x^{14} - 27 x^{13} + 57 x^{12} - 70 x^{11} - 13 x^{10} + 74 x^{9} - 370 x^{8} + 612 x^{7} - 684 x^{6} + 1071 x^{5} - 1148 x^{4} + 1581 x^{3} - 1580 x^{2} + 898 x - 647 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14610422921440715006545693=19^{16}\cdot 37^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.01$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{129673999571069021832815923} a^{17} + \frac{2058619786312105405061134}{129673999571069021832815923} a^{16} + \frac{60795350327438167638265307}{129673999571069021832815923} a^{15} - \frac{1799813331071096174987062}{129673999571069021832815923} a^{14} - \frac{58930641611343517239338493}{129673999571069021832815923} a^{13} - \frac{59599327459081010494253391}{129673999571069021832815923} a^{12} - \frac{14189327705095785333527602}{129673999571069021832815923} a^{11} + \frac{29528954427093285367331121}{129673999571069021832815923} a^{10} + \frac{60104799862743886111320344}{129673999571069021832815923} a^{9} + \frac{40444419157739915425698584}{129673999571069021832815923} a^{8} - \frac{52205174439304600298615368}{129673999571069021832815923} a^{7} + \frac{51021664631129412067331828}{129673999571069021832815923} a^{6} + \frac{60813631270716611019705875}{129673999571069021832815923} a^{5} + \frac{43148360972389913926387934}{129673999571069021832815923} a^{4} - \frac{62780309025624681928731619}{129673999571069021832815923} a^{3} - \frac{42892087149049303902008616}{129673999571069021832815923} a^{2} - \frac{12902565624800466215135595}{129673999571069021832815923} a - \frac{60766845913199508017018178}{129673999571069021832815923}$

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

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  \( 68387.8334575 \)
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$$0$Trivial$[\ ]$
$\Q_{37}$$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$$0$Trivial$[\ ]$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{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}$
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}$