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Properties

Label 18.12.1105415479...5939.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{12}\cdot 92779^{3}$
Root discriminant $24.62$
Ramified primes $7, 92779$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T362

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -10, 43, -15, -178, 198, 317, -352, -275, 174, 200, -61, -82, 40, 15, -12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 12*x^16 + 15*x^15 + 40*x^14 - 82*x^13 - 61*x^12 + 200*x^11 + 174*x^10 - 275*x^9 - 352*x^8 + 317*x^7 + 198*x^6 - 178*x^5 - 15*x^4 + 43*x^3 - 10*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^18 - x^17 - 12*x^16 + 15*x^15 + 40*x^14 - 82*x^13 - 61*x^12 + 200*x^11 + 174*x^10 - 275*x^9 - 352*x^8 + 317*x^7 + 198*x^6 - 178*x^5 - 15*x^4 + 43*x^3 - 10*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 12 x^{16} + 15 x^{15} + 40 x^{14} - 82 x^{13} - 61 x^{12} + 200 x^{11} + 174 x^{10} - 275 x^{9} - 352 x^{8} + 317 x^{7} + 198 x^{6} - 178 x^{5} - 15 x^{4} + 43 x^{3} - 10 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-11054154796697849551765939=-\,7^{12}\cdot 92779^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.62$
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:  $7, 92779$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5511988137590299} a^{17} - \frac{1826080897015035}{5511988137590299} a^{16} - \frac{37110344153405}{190068556468631} a^{15} + \frac{1283596305777849}{5511988137590299} a^{14} + \frac{2327085021361570}{5511988137590299} a^{13} + \frac{63489088618084}{5511988137590299} a^{12} + \frac{997145141720627}{5511988137590299} a^{11} - \frac{2066011993356703}{5511988137590299} a^{10} + \frac{2053202940676436}{5511988137590299} a^{9} + \frac{207301445969709}{5511988137590299} a^{8} - \frac{233209828746226}{5511988137590299} a^{7} + \frac{48550777627593}{190068556468631} a^{6} - \frac{1756640369420760}{5511988137590299} a^{5} - \frac{2392223489164279}{5511988137590299} a^{4} + \frac{739534248595252}{5511988137590299} a^{3} - \frac{298186179379583}{5511988137590299} a^{2} + \frac{498544863036933}{5511988137590299} a - \frac{2648272165882341}{5511988137590299}$

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:  $14$
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:  \( 1178343.03068 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T362:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2160
The 33 conjugacy class representatives for t18n362
Character table for t18n362 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.92779.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ $15{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ $15{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
92779Data not computed