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Properties

Label 18.10.2799928238...0217.3
Degree $18$
Signature $[10, 4]$
Discriminant $7^{12}\cdot 53^{6}\cdot 97^{3}$
Root discriminant $29.46$
Ramified primes $7, 53, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T176

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83, 618, -2012, -2873, 2707, 4947, -1266, -4056, -995, 1596, 1363, -63, -726, -96, 190, 19, -23, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 23*x^16 + 19*x^15 + 190*x^14 - 96*x^13 - 726*x^12 - 63*x^11 + 1363*x^10 + 1596*x^9 - 995*x^8 - 4056*x^7 - 1266*x^6 + 4947*x^5 + 2707*x^4 - 2873*x^3 - 2012*x^2 + 618*x + 83)
 
gp: K = bnfinit(x^18 - x^17 - 23*x^16 + 19*x^15 + 190*x^14 - 96*x^13 - 726*x^12 - 63*x^11 + 1363*x^10 + 1596*x^9 - 995*x^8 - 4056*x^7 - 1266*x^6 + 4947*x^5 + 2707*x^4 - 2873*x^3 - 2012*x^2 + 618*x + 83, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 23 x^{16} + 19 x^{15} + 190 x^{14} - 96 x^{13} - 726 x^{12} - 63 x^{11} + 1363 x^{10} + 1596 x^{9} - 995 x^{8} - 4056 x^{7} - 1266 x^{6} + 4947 x^{5} + 2707 x^{4} - 2873 x^{3} - 2012 x^{2} + 618 x + 83 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(279992823820843547402730217=7^{12}\cdot 53^{6}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.46$
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:  $7, 53, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{18700363955370780191742113} a^{17} + \frac{809578688373180069238204}{18700363955370780191742113} a^{16} - \frac{2528463122821288828674846}{18700363955370780191742113} a^{15} - \frac{826850932624127372319963}{18700363955370780191742113} a^{14} + \frac{3139808616323682271817464}{18700363955370780191742113} a^{13} - \frac{2310969724474177456974024}{18700363955370780191742113} a^{12} - \frac{3373257620487047884106695}{18700363955370780191742113} a^{11} - \frac{2207447064262140246279669}{18700363955370780191742113} a^{10} + \frac{665460917590269501098864}{18700363955370780191742113} a^{9} + \frac{4139041111184659668514307}{18700363955370780191742113} a^{8} - \frac{7303335281679759356682000}{18700363955370780191742113} a^{7} - \frac{3271358812119454314664564}{18700363955370780191742113} a^{6} - \frac{346078751267537229007178}{18700363955370780191742113} a^{5} + \frac{346349978873198260280291}{18700363955370780191742113} a^{4} + \frac{2555274066299322191513946}{18700363955370780191742113} a^{3} + \frac{7436733183283827380632042}{18700363955370780191742113} a^{2} + \frac{4310802376727267543271176}{18700363955370780191742113} a + \frac{7791154896868281546251991}{18700363955370780191742113}$

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:  $13$
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:  \( 3141426.17133 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T176:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 40 conjugacy class representatives for t18n176
Character table for t18n176 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.12.6.1$x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
97Data not computed