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Properties

Label 18.6.17814348159...3633.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 13^{7}\cdot 29^{5}$
Root discriminant $25.28$
Ramified primes $7, 13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 18, 76, -36, -200, 658, -259, -957, 1338, -249, -909, 886, -280, -140, 211, -121, 42, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 121*x^15 + 211*x^14 - 140*x^13 - 280*x^12 + 886*x^11 - 909*x^10 - 249*x^9 + 1338*x^8 - 957*x^7 - 259*x^6 + 658*x^5 - 200*x^4 - 36*x^3 + 76*x^2 + 18*x + 1)
 
gp: K = bnfinit(x^18 - 9*x^17 + 42*x^16 - 121*x^15 + 211*x^14 - 140*x^13 - 280*x^12 + 886*x^11 - 909*x^10 - 249*x^9 + 1338*x^8 - 957*x^7 - 259*x^6 + 658*x^5 - 200*x^4 - 36*x^3 + 76*x^2 + 18*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 42 x^{16} - 121 x^{15} + 211 x^{14} - 140 x^{13} - 280 x^{12} + 886 x^{11} - 909 x^{10} - 249 x^{9} + 1338 x^{8} - 957 x^{7} - 259 x^{6} + 658 x^{5} - 200 x^{4} - 36 x^{3} + 76 x^{2} + 18 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17814348159507645779393633=7^{12}\cdot 13^{7}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.28$
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:  $7, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{1}{13} a^{13} - \frac{5}{13} a^{12} - \frac{3}{13} a^{11} + \frac{6}{13} a^{10} - \frac{3}{13} a^{9} + \frac{5}{13} a^{8} + \frac{1}{13} a^{7} - \frac{2}{13} a^{6} - \frac{6}{13} a^{5} - \frac{1}{13} a^{4} - \frac{5}{13} a^{3} + \frac{2}{13} a^{2} - \frac{1}{13} a + \frac{3}{13}$, $\frac{1}{13} a^{15} - \frac{6}{13} a^{13} + \frac{2}{13} a^{12} - \frac{4}{13} a^{11} + \frac{4}{13} a^{10} - \frac{5}{13} a^{9} - \frac{4}{13} a^{8} - \frac{3}{13} a^{7} - \frac{4}{13} a^{6} + \frac{5}{13} a^{5} - \frac{4}{13} a^{4} - \frac{6}{13} a^{3} - \frac{3}{13} a^{2} + \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{377} a^{16} + \frac{12}{377} a^{15} - \frac{12}{377} a^{14} + \frac{15}{377} a^{13} + \frac{167}{377} a^{12} + \frac{13}{29} a^{11} + \frac{46}{377} a^{10} - \frac{98}{377} a^{9} - \frac{107}{377} a^{8} + \frac{2}{13} a^{7} + \frac{34}{377} a^{6} + \frac{105}{377} a^{5} - \frac{48}{377} a^{4} - \frac{2}{13} a^{3} - \frac{70}{377} a^{2} - \frac{27}{377} a - \frac{67}{377}$, $\frac{1}{15879508544147261} a^{17} + \frac{6962999216614}{15879508544147261} a^{16} + \frac{139504111515398}{15879508544147261} a^{15} + \frac{64307442887744}{15879508544147261} a^{14} - \frac{2123199290301847}{15879508544147261} a^{13} - \frac{503013507358256}{1221500657242097} a^{12} + \frac{1504218667625018}{15879508544147261} a^{11} + \frac{5398219333872078}{15879508544147261} a^{10} + \frac{4897837006277428}{15879508544147261} a^{9} + \frac{5957476589700515}{15879508544147261} a^{8} + \frac{7865305551680013}{15879508544147261} a^{7} + \frac{2119700432939911}{15879508544147261} a^{6} + \frac{5367025844290309}{15879508544147261} a^{5} - \frac{7547970453143668}{15879508544147261} a^{4} - \frac{4856097127089955}{15879508544147261} a^{3} - \frac{6595096957507434}{15879508544147261} a^{2} - \frac{2311533260696150}{15879508544147261} a - \frac{6232174220126916}{15879508544147261}$

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:  $11$
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:  \( 591454.215792 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.5.16721334721.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$