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Properties

Label 18.18.1276091230...2421.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{12}\cdot 83^{4}\cdot 181^{5}$
Root discriminant $41.40$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T199

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-239, 4698, -34619, 129421, -267927, 292831, -95838, -137718, 160783, -34395, -36845, 21655, 811, -3227, 476, 179, -43, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 43*x^16 + 179*x^15 + 476*x^14 - 3227*x^13 + 811*x^12 + 21655*x^11 - 36845*x^10 - 34395*x^9 + 160783*x^8 - 137718*x^7 - 95838*x^6 + 292831*x^5 - 267927*x^4 + 129421*x^3 - 34619*x^2 + 4698*x - 239)
 
gp: K = bnfinit(x^18 - 3*x^17 - 43*x^16 + 179*x^15 + 476*x^14 - 3227*x^13 + 811*x^12 + 21655*x^11 - 36845*x^10 - 34395*x^9 + 160783*x^8 - 137718*x^7 - 95838*x^6 + 292831*x^5 - 267927*x^4 + 129421*x^3 - 34619*x^2 + 4698*x - 239, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 43 x^{16} + 179 x^{15} + 476 x^{14} - 3227 x^{13} + 811 x^{12} + 21655 x^{11} - 36845 x^{10} - 34395 x^{9} + 160783 x^{8} - 137718 x^{7} - 95838 x^{6} + 292831 x^{5} - 267927 x^{4} + 129421 x^{3} - 34619 x^{2} + 4698 x - 239 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(127609123015304468465889942421=7^{12}\cdot 83^{4}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.40$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{11} + \frac{2}{7} a^{10} - \frac{3}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{16} - \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{8483757088811464747} a^{17} - \frac{409069251677859328}{8483757088811464747} a^{16} + \frac{190531787667193783}{8483757088811464747} a^{15} - \frac{411534557495457279}{8483757088811464747} a^{14} - \frac{59122672916997494}{1211965298401637821} a^{13} - \frac{210930240405879927}{8483757088811464747} a^{12} - \frac{2269976867398655187}{8483757088811464747} a^{11} - \frac{3575217649622750580}{8483757088811464747} a^{10} + \frac{596030564167865254}{8483757088811464747} a^{9} - \frac{2243048203259428632}{8483757088811464747} a^{8} - \frac{205315823372500434}{8483757088811464747} a^{7} - \frac{89326576573837683}{8483757088811464747} a^{6} + \frac{2339723026575444375}{8483757088811464747} a^{5} + \frac{776018061034817900}{8483757088811464747} a^{4} + \frac{2966806574712555123}{8483757088811464747} a^{3} - \frac{298010337384250650}{1211965298401637821} a^{2} + \frac{1628789589591195243}{8483757088811464747} a + \frac{1034333685811552972}{8483757088811464747}$

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:  $17$
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:  \( 525289669.639 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T199:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 26 conjugacy class representatives for t18n199
Character table for t18n199 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.434581.1, 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
Arithmetically equvalently 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 ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ $18$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$7.6.4.3$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}$
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$Trivial$[\ ]$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.3.2.1$x^{3} - 83$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
83.3.2.1$x^{3} - 83$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
181Data not computed