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Properties

Label 18.2.13888948748...3077.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 83^{4}\cdot 181^{4}\cdot 197$
Root discriminant $41.59$
Ramified primes $7, 83, 181, 197$
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![-923, 7066, -24349, 51155, -73160, 73982, -51325, 17172, 9718, -22010, 20759, -14070, 7337, -3094, 1050, -284, 61, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 61*x^16 - 284*x^15 + 1050*x^14 - 3094*x^13 + 7337*x^12 - 14070*x^11 + 20759*x^10 - 22010*x^9 + 9718*x^8 + 17172*x^7 - 51325*x^6 + 73982*x^5 - 73160*x^4 + 51155*x^3 - 24349*x^2 + 7066*x - 923)
 
gp: K = bnfinit(x^18 - 9*x^17 + 61*x^16 - 284*x^15 + 1050*x^14 - 3094*x^13 + 7337*x^12 - 14070*x^11 + 20759*x^10 - 22010*x^9 + 9718*x^8 + 17172*x^7 - 51325*x^6 + 73982*x^5 - 73160*x^4 + 51155*x^3 - 24349*x^2 + 7066*x - 923, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 61 x^{16} - 284 x^{15} + 1050 x^{14} - 3094 x^{13} + 7337 x^{12} - 14070 x^{11} + 20759 x^{10} - 22010 x^{9} + 9718 x^{8} + 17172 x^{7} - 51325 x^{6} + 73982 x^{5} - 73160 x^{4} + 51155 x^{3} - 24349 x^{2} + 7066 x - 923 \)

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:  \(138889487480745747446300103077=7^{12}\cdot 83^{4}\cdot 181^{4}\cdot 197\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.59$
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:  $7, 83, 181, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{11} + \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{2}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2}$, $\frac{1}{223582583} a^{16} - \frac{8}{223582583} a^{15} - \frac{11298232}{223582583} a^{14} + \frac{15207026}{223582583} a^{13} - \frac{2074249}{31940369} a^{12} - \frac{6670358}{31940369} a^{11} - \frac{46736827}{223582583} a^{10} - \frac{19251541}{223582583} a^{9} - \frac{97368230}{223582583} a^{8} + \frac{17980450}{223582583} a^{7} - \frac{55347300}{223582583} a^{6} + \frac{76729813}{223582583} a^{5} + \frac{47744811}{223582583} a^{4} + \frac{90908746}{223582583} a^{3} + \frac{48972913}{223582583} a^{2} + \frac{57551365}{223582583} a + \frac{40479515}{223582583}$, $\frac{1}{9614051069} a^{17} + \frac{13}{9614051069} a^{16} + \frac{52582338}{9614051069} a^{15} - \frac{637280643}{9614051069} a^{14} - \frac{14575887}{9614051069} a^{13} - \frac{50229587}{1373435867} a^{12} + \frac{4147060325}{9614051069} a^{11} + \frac{171880079}{1373435867} a^{10} + \frac{370937886}{1373435867} a^{9} + \frac{3243408505}{9614051069} a^{8} - \frac{891491872}{9614051069} a^{7} - \frac{1053623118}{9614051069} a^{6} - \frac{4409599226}{9614051069} a^{5} - \frac{2483771551}{9614051069} a^{4} - \frac{537038496}{1373435867} a^{3} + \frac{63890730}{9614051069} a^{2} + \frac{4059810652}{9614051069} a + \frac{4107987453}{9614051069}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 24995724.3385 \) (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

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

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 $18$ $18$ $18$ R $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.6.4.1$x^{6} + 415 x^{3} + 55112$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\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$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$197$197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.1.2$x^{2} + 394$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
197.6.0.1$x^{6} - x + 13$$1$$6$$0$$C_6$$[\ ]^{6}$