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Properties

Label 18.8.27391316277...1712.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 83$
Root discriminant $55.79$
Ramified primes $2, 3, 7, 41, 83$
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![-6929, 44075, -159449, 353670, -520446, 507631, -269950, 20603, 58263, -39902, 14135, -1586, -683, 567, -249, 33, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 3*x^16 + 33*x^15 - 249*x^14 + 567*x^13 - 683*x^12 - 1586*x^11 + 14135*x^10 - 39902*x^9 + 58263*x^8 + 20603*x^7 - 269950*x^6 + 507631*x^5 - 520446*x^4 + 353670*x^3 - 159449*x^2 + 44075*x - 6929)
 
gp: K = bnfinit(x^18 - 3*x^17 + 3*x^16 + 33*x^15 - 249*x^14 + 567*x^13 - 683*x^12 - 1586*x^11 + 14135*x^10 - 39902*x^9 + 58263*x^8 + 20603*x^7 - 269950*x^6 + 507631*x^5 - 520446*x^4 + 353670*x^3 - 159449*x^2 + 44075*x - 6929, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 3 x^{16} + 33 x^{15} - 249 x^{14} + 567 x^{13} - 683 x^{12} - 1586 x^{11} + 14135 x^{10} - 39902 x^{9} + 58263 x^{8} + 20603 x^{7} - 269950 x^{6} + 507631 x^{5} - 520446 x^{4} + 353670 x^{3} - 159449 x^{2} + 44075 x - 6929 \)

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:  \(-27391316277974958297652767731712=-\,2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 83\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.79$
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:  $2, 3, 7, 41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{492} a^{16} + \frac{83}{492} a^{15} - \frac{4}{41} a^{14} - \frac{25}{246} a^{13} + \frac{59}{492} a^{12} + \frac{9}{164} a^{11} + \frac{29}{123} a^{10} - \frac{229}{492} a^{9} - \frac{131}{492} a^{8} + \frac{49}{492} a^{7} + \frac{16}{123} a^{6} + \frac{7}{82} a^{5} - \frac{121}{246} a^{4} - \frac{37}{164} a^{3} - \frac{1}{6} a^{2} + \frac{5}{12} a - \frac{1}{12}$, $\frac{1}{5361239140844571935368952178368730771036} a^{17} + \frac{858725497583894484862280555300440025}{2680619570422285967684476089184365385518} a^{16} - \frac{1259248644287211145916763465511198937141}{5361239140844571935368952178368730771036} a^{15} - \frac{163088787605633242255224481913640942440}{1340309785211142983842238044592182692759} a^{14} + \frac{1723437091703761127083956639884309984371}{5361239140844571935368952178368730771036} a^{13} + \frac{444062669023717756445528256230883675323}{2680619570422285967684476089184365385518} a^{12} - \frac{657622955950375229630678715145832307931}{5361239140844571935368952178368730771036} a^{11} - \frac{334327360111771767733296022159867391839}{1787079713614857311789650726122910257012} a^{10} + \frac{115931545980129182215497169951538015935}{2680619570422285967684476089184365385518} a^{9} + \frac{333547587360828423166190409268056466799}{893539856807428655894825363061455128506} a^{8} + \frac{4900373965324768764530627703965814625}{595693237871619103929883575374303419004} a^{7} - \frac{1314704830273386366296766725353848044663}{2680619570422285967684476089184365385518} a^{6} - \frac{863339567868672325863673500163038402331}{2680619570422285967684476089184365385518} a^{5} - \frac{714266139972763195480983087637404951337}{5361239140844571935368952178368730771036} a^{4} + \frac{1703362502616320523614340153687599482655}{5361239140844571935368952178368730771036} a^{3} - \frac{37248478510206368283179027845421897543}{130761930264501754521193955569969043196} a^{2} - \frac{1047033776216557347240914715609148433}{3632275840680604292255387654721362311} a + \frac{40375178899896100869944886373357999585}{130761930264501754521193955569969043196}$

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:  \( 1470104521.38 \) (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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$[\ ]^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
41Data not computed
$83$83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.6.0.1$x^{6} - x + 34$$1$$6$$0$$C_6$$[\ ]^{6}$
83.6.0.1$x^{6} - x + 34$$1$$6$$0$$C_6$$[\ ]^{6}$