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Properties

Label 18.14.5050489291...5625.1
Degree $18$
Signature $[14, 2]$
Discriminant $5^{4}\cdot 13^{15}\cdot 181\cdot 2953332121^{2}$
Root discriminant $182.51$
Ramified primes $5, 13, 181, 2953332121$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T857

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2273671009, -7912659482, 11572272509, -8745926570, 2997941500, 300860113, -646229978, 173217555, 26331093, -21266729, 1618181, 1065482, -189044, -26371, 7337, 299, -135, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 135*x^16 + 299*x^15 + 7337*x^14 - 26371*x^13 - 189044*x^12 + 1065482*x^11 + 1618181*x^10 - 21266729*x^9 + 26331093*x^8 + 173217555*x^7 - 646229978*x^6 + 300860113*x^5 + 2997941500*x^4 - 8745926570*x^3 + 11572272509*x^2 - 7912659482*x + 2273671009)
 
gp: K = bnfinit(x^18 - x^17 - 135*x^16 + 299*x^15 + 7337*x^14 - 26371*x^13 - 189044*x^12 + 1065482*x^11 + 1618181*x^10 - 21266729*x^9 + 26331093*x^8 + 173217555*x^7 - 646229978*x^6 + 300860113*x^5 + 2997941500*x^4 - 8745926570*x^3 + 11572272509*x^2 - 7912659482*x + 2273671009, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 135 x^{16} + 299 x^{15} + 7337 x^{14} - 26371 x^{13} - 189044 x^{12} + 1065482 x^{11} + 1618181 x^{10} - 21266729 x^{9} + 26331093 x^{8} + 173217555 x^{7} - 646229978 x^{6} + 300860113 x^{5} + 2997941500 x^{4} - 8745926570 x^{3} + 11572272509 x^{2} - 7912659482 x + 2273671009 \)

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:  \(50504892913025399729453656274066127435625=5^{4}\cdot 13^{15}\cdot 181\cdot 2953332121^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.51$
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:  $5, 13, 181, 2953332121$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{2300291976661834502114058255} a^{17} + \frac{334221574178508912435729632}{2300291976661834502114058255} a^{16} - \frac{3586165305251457758816917}{48942382482166691534341665} a^{15} + \frac{104426770344243766283555354}{766763992220611500704686085} a^{14} + \frac{321049108970330547631466181}{766763992220611500704686085} a^{13} - \frac{984910565749190185782680282}{2300291976661834502114058255} a^{12} - \frac{64560774953325673283549465}{153352798444122300140937217} a^{11} - \frac{28788083326047942294363763}{2300291976661834502114058255} a^{10} - \frac{91391641223309667756022696}{766763992220611500704686085} a^{9} + \frac{446995954338326149657045477}{2300291976661834502114058255} a^{8} - \frac{868481699629130578962022246}{2300291976661834502114058255} a^{7} + \frac{40378986579983789513919}{766763992220611500704686085} a^{6} + \frac{311191641247099034397720268}{2300291976661834502114058255} a^{5} - \frac{309528861645440416129976291}{766763992220611500704686085} a^{4} + \frac{561115309863593087013021466}{2300291976661834502114058255} a^{3} - \frac{659849813247035904259862152}{2300291976661834502114058255} a^{2} - \frac{51900430484335653620543237}{2300291976661834502114058255} a - \frac{107746546739912062154771956}{766763992220611500704686085}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 48516743393100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{13})^+\)

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

Sibling fields

Degree 18 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ $18$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$181$$\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$Trivial$[\ ]$
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}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
2953332121Data not computed