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Properties

Label 18.12.7555018282...9599.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{12}\cdot 13^{5}\cdot 43^{5}$
Root discriminant $21.21$
Ramified primes $7, 13, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T188

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, -38, 180, 256, -661, -519, 748, 393, -151, -110, -137, 48, 50, -5, -2, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 2*x^15 - 5*x^14 + 50*x^13 + 48*x^12 - 137*x^11 - 110*x^10 - 151*x^9 + 393*x^8 + 748*x^7 - 519*x^6 - 661*x^5 + 256*x^4 + 180*x^3 - 38*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^16 - 2*x^15 - 5*x^14 + 50*x^13 + 48*x^12 - 137*x^11 - 110*x^10 - 151*x^9 + 393*x^8 + 748*x^7 - 519*x^6 - 661*x^5 + 256*x^4 + 180*x^3 - 38*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} - 2 x^{15} - 5 x^{14} + 50 x^{13} + 48 x^{12} - 137 x^{11} - 110 x^{10} - 151 x^{9} + 393 x^{8} + 748 x^{7} - 519 x^{6} - 661 x^{5} + 256 x^{4} + 180 x^{3} - 38 x^{2} - 7 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-755501828200021621499599=-\,7^{12}\cdot 13^{5}\cdot 43^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.21$
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:  $7, 13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2936398106549037199} a^{17} - \frac{683438059746446624}{2936398106549037199} a^{16} - \frac{736564076965438002}{2936398106549037199} a^{15} + \frac{1082038542672158244}{2936398106549037199} a^{14} - \frac{773305439147198816}{2936398106549037199} a^{13} - \frac{1280969169795354875}{2936398106549037199} a^{12} - \frac{726687934131481968}{2936398106549037199} a^{11} - \frac{371693928005537326}{2936398106549037199} a^{10} - \frac{398825832753871634}{2936398106549037199} a^{9} - \frac{716753418301430402}{2936398106549037199} a^{8} - \frac{1431116827205901133}{2936398106549037199} a^{7} - \frac{1006563547178479733}{2936398106549037199} a^{6} - \frac{1201245063082646169}{2936398106549037199} a^{5} + \frac{352827821286273321}{2936398106549037199} a^{4} - \frac{1093201198241022393}{2936398106549037199} a^{3} - \frac{225794299224557545}{2936398106549037199} a^{2} + \frac{1332608863812400008}{2936398106549037199} a + \frac{1146278285645509125}{2936398106549037199}$

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:  $14$
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:  \( 275520.322473 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T188:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 88 conjugacy class representatives for t18n188 are not computed
Character table for t18n188 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.1342159.2, 9.9.36763077169.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R $18$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.6.0.1$x^{6} - x + 26$$1$$6$$0$$C_6$$[\ ]^{6}$
43.6.5.5$x^{6} + 31347$$6$$1$$5$$C_6$$[\ ]_{6}$