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Properties

Label 18.10.7103059591...3637.1
Degree $18$
Signature $[10, 4]$
Discriminant $11^{4}\cdot 37^{9}\cdot 139^{4}$
Root discriminant $31.03$
Ramified primes $11, 37, 139$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T888

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37, -185, -645, 3167, -2161, -1829, -50, 2685, 1205, -3498, -20, 1958, -349, -533, 136, 72, -20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 20*x^16 + 72*x^15 + 136*x^14 - 533*x^13 - 349*x^12 + 1958*x^11 - 20*x^10 - 3498*x^9 + 1205*x^8 + 2685*x^7 - 50*x^6 - 1829*x^5 - 2161*x^4 + 3167*x^3 - 645*x^2 - 185*x + 37)
 
gp: K = bnfinit(x^18 - 4*x^17 - 20*x^16 + 72*x^15 + 136*x^14 - 533*x^13 - 349*x^12 + 1958*x^11 - 20*x^10 - 3498*x^9 + 1205*x^8 + 2685*x^7 - 50*x^6 - 1829*x^5 - 2161*x^4 + 3167*x^3 - 645*x^2 - 185*x + 37, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 20 x^{16} + 72 x^{15} + 136 x^{14} - 533 x^{13} - 349 x^{12} + 1958 x^{11} - 20 x^{10} - 3498 x^{9} + 1205 x^{8} + 2685 x^{7} - 50 x^{6} - 1829 x^{5} - 2161 x^{4} + 3167 x^{3} - 645 x^{2} - 185 x + 37 \)

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:  \(710305959195813821519073637=11^{4}\cdot 37^{9}\cdot 139^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.03$
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:  $11, 37, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{67385254896809543501553} a^{17} + \frac{8451310507515296850341}{67385254896809543501553} a^{16} + \frac{2005459388249532250220}{67385254896809543501553} a^{15} - \frac{456695233734353566221}{22461751632269847833851} a^{14} - \frac{8301049185677451840416}{22461751632269847833851} a^{13} + \frac{924616401253671734440}{67385254896809543501553} a^{12} + \frac{3862921274255112301522}{67385254896809543501553} a^{11} + \frac{19861890466514227079476}{67385254896809543501553} a^{10} - \frac{8776803993780956701623}{22461751632269847833851} a^{9} - \frac{17574652402029918906716}{67385254896809543501553} a^{8} + \frac{31140800044539602064677}{67385254896809543501553} a^{7} - \frac{9731717118450403139902}{22461751632269847833851} a^{6} + \frac{4570135326437641673316}{22461751632269847833851} a^{5} + \frac{16400694118575853206415}{67385254896809543501553} a^{4} - \frac{32095429011837011956361}{67385254896809543501553} a^{3} + \frac{27777198176151736464844}{67385254896809543501553} a^{2} + \frac{8118954262763195130233}{67385254896809543501553} a - \frac{21106788769985147667367}{67385254896809543501553}$

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:  $13$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13968206.257 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T888:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 362880
The 36 conjugacy class representatives for t18n888
Character table for t18n888 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 9.5.3200504329.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ R $18$ $18$ $18$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$11$11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
37Data not computed
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
139.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$