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Properties

Label 18.8.14190681927...7152.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 43$
Root discriminant $53.79$
Ramified primes $2, 3, 7, 41, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 28, -218, 534, -83, -683, -12, 1251, -1342, 277, 998, -1177, 250, 315, -169, 59, -12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 12*x^16 + 59*x^15 - 169*x^14 + 315*x^13 + 250*x^12 - 1177*x^11 + 998*x^10 + 277*x^9 - 1342*x^8 + 1251*x^7 - 12*x^6 - 683*x^5 - 83*x^4 + 534*x^3 - 218*x^2 + 28*x - 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 12*x^16 + 59*x^15 - 169*x^14 + 315*x^13 + 250*x^12 - 1177*x^11 + 998*x^10 + 277*x^9 - 1342*x^8 + 1251*x^7 - 12*x^6 - 683*x^5 - 83*x^4 + 534*x^3 - 218*x^2 + 28*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 12 x^{16} + 59 x^{15} - 169 x^{14} + 315 x^{13} + 250 x^{12} - 1177 x^{11} + 998 x^{10} + 277 x^{9} - 1342 x^{8} + 1251 x^{7} - 12 x^{6} - 683 x^{5} - 83 x^{4} + 534 x^{3} - 218 x^{2} + 28 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-14190681927143653093964686897152=-\,2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 43\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.79$
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:  $2, 3, 7, 41, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7063} a^{16} - \frac{788}{7063} a^{15} - \frac{68}{7063} a^{14} + \frac{906}{7063} a^{13} + \frac{2004}{7063} a^{12} + \frac{339}{1009} a^{11} + \frac{2734}{7063} a^{10} - \frac{82}{7063} a^{9} - \frac{95}{1009} a^{8} + \frac{1326}{7063} a^{7} - \frac{2549}{7063} a^{6} + \frac{2997}{7063} a^{5} + \frac{2980}{7063} a^{4} - \frac{2596}{7063} a^{3} + \frac{3172}{7063} a^{2} - \frac{2131}{7063} a - \frac{1416}{7063}$, $\frac{1}{115077877742250333967} a^{17} - \frac{191841835756055}{16439696820321476281} a^{16} - \frac{2924677341429370950}{16439696820321476281} a^{15} - \frac{30411090590751714819}{115077877742250333967} a^{14} - \frac{7740348050431854860}{16439696820321476281} a^{13} - \frac{52932652024701462320}{115077877742250333967} a^{12} - \frac{41649208122734789951}{115077877742250333967} a^{11} - \frac{35196121184749837273}{115077877742250333967} a^{10} - \frac{17657681356389098254}{115077877742250333967} a^{9} + \frac{35631285954058831494}{115077877742250333967} a^{8} - \frac{17689424637880235390}{115077877742250333967} a^{7} + \frac{44490230668682741352}{115077877742250333967} a^{6} + \frac{54653845692900533739}{115077877742250333967} a^{5} - \frac{4951620966645966338}{16439696820321476281} a^{4} + \frac{9596281415973274664}{115077877742250333967} a^{3} - \frac{28545151337110114881}{115077877742250333967} a^{2} - \frac{541901641321554461}{16439696820321476281} a - \frac{13848025527996995616}{115077877742250333967}$

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:  $12$
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:  \( 1028189148.66 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ R R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$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}$
41Data not computed
43Data not computed