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Properties

Label 18.6.24989675855...3289.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 13^{4}\cdot 43^{6}$
Root discriminant $22.67$
Ramified primes $7, 13, 43$
Class number $1$
Class group Trivial
Galois group 18T473

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![281, -597, -1091, 5864, -11093, 12400, -8750, 3080, 1056, -2340, 1685, -603, -30, 167, -91, 16, 7, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 7*x^16 + 16*x^15 - 91*x^14 + 167*x^13 - 30*x^12 - 603*x^11 + 1685*x^10 - 2340*x^9 + 1056*x^8 + 3080*x^7 - 8750*x^6 + 12400*x^5 - 11093*x^4 + 5864*x^3 - 1091*x^2 - 597*x + 281)
 
gp: K = bnfinit(x^18 - 5*x^17 + 7*x^16 + 16*x^15 - 91*x^14 + 167*x^13 - 30*x^12 - 603*x^11 + 1685*x^10 - 2340*x^9 + 1056*x^8 + 3080*x^7 - 8750*x^6 + 12400*x^5 - 11093*x^4 + 5864*x^3 - 1091*x^2 - 597*x + 281, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 7 x^{16} + 16 x^{15} - 91 x^{14} + 167 x^{13} - 30 x^{12} - 603 x^{11} + 1685 x^{10} - 2340 x^{9} + 1056 x^{8} + 3080 x^{7} - 8750 x^{6} + 12400 x^{5} - 11093 x^{4} + 5864 x^{3} - 1091 x^{2} - 597 x + 281 \)

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:  \(2498967585584686901883289=7^{12}\cdot 13^{4}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{29} a^{15} - \frac{4}{29} a^{14} + \frac{6}{29} a^{13} - \frac{4}{29} a^{12} + \frac{8}{29} a^{11} - \frac{8}{29} a^{10} + \frac{13}{29} a^{9} - \frac{1}{29} a^{8} + \frac{8}{29} a^{7} + \frac{6}{29} a^{6} - \frac{2}{29} a^{5} - \frac{3}{29} a^{4} + \frac{2}{29} a^{3} + \frac{9}{29} a^{2} + \frac{13}{29} a - \frac{11}{29}$, $\frac{1}{29} a^{16} - \frac{10}{29} a^{14} - \frac{9}{29} a^{13} - \frac{8}{29} a^{12} - \frac{5}{29} a^{11} + \frac{10}{29} a^{10} - \frac{7}{29} a^{9} + \frac{4}{29} a^{8} + \frac{9}{29} a^{7} - \frac{7}{29} a^{6} - \frac{11}{29} a^{5} - \frac{10}{29} a^{4} - \frac{12}{29} a^{3} - \frac{9}{29} a^{2} + \frac{12}{29} a + \frac{14}{29}$, $\frac{1}{10908470121486425153} a^{17} + \frac{44760244950623273}{10908470121486425153} a^{16} + \frac{74451096320604494}{10908470121486425153} a^{15} + \frac{2790040122748091870}{10908470121486425153} a^{14} + \frac{2186199117526567214}{10908470121486425153} a^{13} + \frac{2002612581930110909}{10908470121486425153} a^{12} - \frac{1439918674323683795}{10908470121486425153} a^{11} + \frac{3150560250954752289}{10908470121486425153} a^{10} + \frac{4219427574052161895}{10908470121486425153} a^{9} - \frac{85019533209546628}{10908470121486425153} a^{8} + \frac{2609454388580565024}{10908470121486425153} a^{7} + \frac{1257123066742223399}{10908470121486425153} a^{6} - \frac{3303931907503169719}{10908470121486425153} a^{5} - \frac{11989421696192156}{10908470121486425153} a^{4} - \frac{2476426935863623614}{10908470121486425153} a^{3} + \frac{750410982841940056}{10908470121486425153} a^{2} + \frac{1270656606332139869}{10908470121486425153} a + \frac{1682689385638027191}{10908470121486425153}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 159759.944289 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T473:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5184
The 32 conjugacy class representatives for t18n473
Character table for t18n473 is not computed

Intermediate fields

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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$7.9.6.1$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$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
43Data not computed