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Properties

Label 18.14.8790992484...8269.1
Degree $18$
Signature $[14, 2]$
Discriminant $7^{12}\cdot 83^{6}\cdot 181^{5}$
Root discriminant $67.64$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T705

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![344357, -1585148, 2353229, -628952, -1806627, 1846244, -176218, -680103, 346564, 53003, -86024, 13604, 7770, -2967, -80, 208, -25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 25*x^16 + 208*x^15 - 80*x^14 - 2967*x^13 + 7770*x^12 + 13604*x^11 - 86024*x^10 + 53003*x^9 + 346564*x^8 - 680103*x^7 - 176218*x^6 + 1846244*x^5 - 1806627*x^4 - 628952*x^3 + 2353229*x^2 - 1585148*x + 344357)
 
gp: K = bnfinit(x^18 - 5*x^17 - 25*x^16 + 208*x^15 - 80*x^14 - 2967*x^13 + 7770*x^12 + 13604*x^11 - 86024*x^10 + 53003*x^9 + 346564*x^8 - 680103*x^7 - 176218*x^6 + 1846244*x^5 - 1806627*x^4 - 628952*x^3 + 2353229*x^2 - 1585148*x + 344357, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 25 x^{16} + 208 x^{15} - 80 x^{14} - 2967 x^{13} + 7770 x^{12} + 13604 x^{11} - 86024 x^{10} + 53003 x^{9} + 346564 x^{8} - 680103 x^{7} - 176218 x^{6} + 1846244 x^{5} - 1806627 x^{4} - 628952 x^{3} + 2353229 x^{2} - 1585148 x + 344357 \)

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:  \(879099248452432483261515813338269=7^{12}\cdot 83^{6}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.64$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} - \frac{4}{13} a^{15} + \frac{1}{13} a^{13} - \frac{1}{13} a^{12} - \frac{1}{13} a^{11} + \frac{5}{13} a^{10} - \frac{5}{13} a^{9} - \frac{6}{13} a^{8} - \frac{6}{13} a^{7} - \frac{1}{13} a^{6} - \frac{1}{13} a^{5} + \frac{6}{13} a^{4} + \frac{1}{13} a^{2} + \frac{2}{13} a$, $\frac{1}{3953436829689919697640038745666579407} a^{17} - \frac{4853796642555096505999914368800003}{3953436829689919697640038745666579407} a^{16} - \frac{1264541388254865642564267487299067928}{3953436829689919697640038745666579407} a^{15} - \frac{1278953950317145061433269348338208942}{3953436829689919697640038745666579407} a^{14} + \frac{929239468408360399959409037374457879}{3953436829689919697640038745666579407} a^{13} - \frac{1436385648981588830593311445360595054}{3953436829689919697640038745666579407} a^{12} + \frac{1337767920603738900076079708148400355}{3953436829689919697640038745666579407} a^{11} + \frac{619845718185440741057616009625245310}{3953436829689919697640038745666579407} a^{10} - \frac{935953006126481236135160962451598278}{3953436829689919697640038745666579407} a^{9} - \frac{1493739591154006950838395857842532343}{3953436829689919697640038745666579407} a^{8} - \frac{124985871745264936215527821646725934}{3953436829689919697640038745666579407} a^{7} - \frac{539234366582333703002083575203765652}{3953436829689919697640038745666579407} a^{6} - \frac{84772632556926447775074380881234071}{304110525360763053664618365051275339} a^{5} + \frac{12438677894298432158740351837665286}{3953436829689919697640038745666579407} a^{4} - \frac{1822206049935216140828213583648715718}{3953436829689919697640038745666579407} a^{3} - \frac{1129463846639182116191782343982018278}{3953436829689919697640038745666579407} a^{2} - \frac{1793749706699090912651716907150922232}{3953436829689919697640038745666579407} a - \frac{125079234805144164474397828590818396}{304110525360763053664618365051275339}$

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:  $15$
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:  \( 40925074691.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T705:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 64 conjugacy class representatives for t18n705 are not computed
Character table for t18n705 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $18$ $18$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
83Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$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}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$