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Properties

Label 18.14.6252847027...8629.1
Degree $18$
Signature $[14, 2]$
Discriminant $7^{14}\cdot 83^{4}\cdot 181^{5}$
Root discriminant $51.39$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T705

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16393, 43838, 12449, -111650, 46791, 100613, -83181, -33512, 55119, -4004, -17526, 6027, 2472, -1645, -42, 178, -20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 20*x^16 + 178*x^15 - 42*x^14 - 1645*x^13 + 2472*x^12 + 6027*x^11 - 17526*x^10 - 4004*x^9 + 55119*x^8 - 33512*x^7 - 83181*x^6 + 100613*x^5 + 46791*x^4 - 111650*x^3 + 12449*x^2 + 43838*x - 16393)
 
gp: K = bnfinit(x^18 - 6*x^17 - 20*x^16 + 178*x^15 - 42*x^14 - 1645*x^13 + 2472*x^12 + 6027*x^11 - 17526*x^10 - 4004*x^9 + 55119*x^8 - 33512*x^7 - 83181*x^6 + 100613*x^5 + 46791*x^4 - 111650*x^3 + 12449*x^2 + 43838*x - 16393, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 20 x^{16} + 178 x^{15} - 42 x^{14} - 1645 x^{13} + 2472 x^{12} + 6027 x^{11} - 17526 x^{10} - 4004 x^{9} + 55119 x^{8} - 33512 x^{7} - 83181 x^{6} + 100613 x^{5} + 46791 x^{4} - 111650 x^{3} + 12449 x^{2} + 43838 x - 16393 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6252847027749918954828607178629=7^{14}\cdot 83^{4}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.39$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{6} + \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{17233697150504228635475400439} a^{17} + \frac{224663584898504738382683657}{17233697150504228635475400439} a^{16} - \frac{19330329259216415444301405}{1325669011577248356575030803} a^{15} - \frac{506427192928812664413297228}{17233697150504228635475400439} a^{14} + \frac{312142509539023948228572246}{17233697150504228635475400439} a^{13} + \frac{18464435405090003707430369}{17233697150504228635475400439} a^{12} + \frac{446518111797482981243546465}{17233697150504228635475400439} a^{11} + \frac{1131658749024427704987017278}{17233697150504228635475400439} a^{10} - \frac{328172041310891437341251770}{1325669011577248356575030803} a^{9} - \frac{98388490098860301393131742}{1325669011577248356575030803} a^{8} - \frac{2057840777214202787629043547}{17233697150504228635475400439} a^{7} - \frac{6267041427232989587708819500}{17233697150504228635475400439} a^{6} + \frac{8041861065978789993805472692}{17233697150504228635475400439} a^{5} + \frac{5093803993734269576986689495}{17233697150504228635475400439} a^{4} + \frac{3278694564706854980981685938}{17233697150504228635475400439} a^{3} - \frac{8556950396341591576446534310}{17233697150504228635475400439} a^{2} - \frac{88620803441764047855883907}{2461956735786318376496485777} a + \frac{560735140355204163901558478}{1325669011577248356575030803}$

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:  $15$
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:  \( 2398849537.39 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T705:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 64 conjugacy class representatives for t18n705 are not computed
Character table for t18n705 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $18$ $18$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
83Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$