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Properties

Label 18.12.3431637129...1696.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{12}\cdot 7^{12}\cdot 37^{9}\cdot 167^{3}$
Root discriminant $82.92$
Ramified primes $2, 7, 37, 167$
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![-1012607, 444781, 3889164, -2434847, -3497873, 2196473, 1205481, -600376, -184219, 9355, 17814, 9132, -300, 255, -266, -40, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 5*x^16 - 40*x^15 - 266*x^14 + 255*x^13 - 300*x^12 + 9132*x^11 + 17814*x^10 + 9355*x^9 - 184219*x^8 - 600376*x^7 + 1205481*x^6 + 2196473*x^5 - 3497873*x^4 - 2434847*x^3 + 3889164*x^2 + 444781*x - 1012607)
 
gp: K = bnfinit(x^18 - 2*x^17 + 5*x^16 - 40*x^15 - 266*x^14 + 255*x^13 - 300*x^12 + 9132*x^11 + 17814*x^10 + 9355*x^9 - 184219*x^8 - 600376*x^7 + 1205481*x^6 + 2196473*x^5 - 3497873*x^4 - 2434847*x^3 + 3889164*x^2 + 444781*x - 1012607, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 5 x^{16} - 40 x^{15} - 266 x^{14} + 255 x^{13} - 300 x^{12} + 9132 x^{11} + 17814 x^{10} + 9355 x^{9} - 184219 x^{8} - 600376 x^{7} + 1205481 x^{6} + 2196473 x^{5} - 3497873 x^{4} - 2434847 x^{3} + 3889164 x^{2} + 444781 x - 1012607 \)

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:  \(-34316371298286047031542175649181696=-\,2^{12}\cdot 7^{12}\cdot 37^{9}\cdot 167^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.92$
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:  $2, 7, 37, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{13611} a^{16} + \frac{98}{1047} a^{15} + \frac{394}{4537} a^{14} - \frac{94}{13611} a^{13} - \frac{1302}{4537} a^{12} + \frac{1036}{4537} a^{11} + \frac{125}{4537} a^{10} - \frac{5503}{13611} a^{9} - \frac{6206}{13611} a^{8} - \frac{2435}{13611} a^{7} - \frac{57}{349} a^{6} + \frac{6098}{13611} a^{5} + \frac{1822}{4537} a^{4} - \frac{3433}{13611} a^{3} - \frac{2861}{13611} a^{2} - \frac{319}{4537} a - \frac{5399}{13611}$, $\frac{1}{60640896607743580056983441141895638367274305574557} a^{17} - \frac{887076080310552225031945174637300446880668192}{60640896607743580056983441141895638367274305574557} a^{16} + \frac{6297351971377554305571823069555436334713808146704}{60640896607743580056983441141895638367274305574557} a^{15} - \frac{2191991160176157512352080820613558121235578274404}{20213632202581193352327813713965212789091435191519} a^{14} - \frac{3875483605495042925506360888305113245487195047541}{60640896607743580056983441141895638367274305574557} a^{13} + \frac{14329117241363023234152743265841229894960056198483}{60640896607743580056983441141895638367274305574557} a^{12} - \frac{5976463764268847276292136919763156791724906715445}{60640896607743580056983441141895638367274305574557} a^{11} - \frac{8441683519424846477619065317875487705358425080470}{60640896607743580056983441141895638367274305574557} a^{10} + \frac{3432128387179099350305393653144625422120672458199}{60640896607743580056983441141895638367274305574557} a^{9} + \frac{7547590598551281807276598559363571883046141688686}{60640896607743580056983441141895638367274305574557} a^{8} - \frac{10115595883127265954259303109613936805443348484076}{60640896607743580056983441141895638367274305574557} a^{7} + \frac{8403376714720014518951907109491093014834027885561}{20213632202581193352327813713965212789091435191519} a^{6} + \frac{4916217550891381505356523590683111617232849770153}{20213632202581193352327813713965212789091435191519} a^{5} + \frac{20637979067406341079411635372416339603338894833870}{60640896607743580056983441141895638367274305574557} a^{4} + \frac{8953188967856019209815892516462486246188290085933}{20213632202581193352327813713965212789091435191519} a^{3} + \frac{135428674431599851340724938769176757300032871973}{625163882554057526360654032390676684198704181181} a^{2} - \frac{8758255667696753133984459226047326399458681827074}{60640896607743580056983441141895638367274305574557} a + \frac{25668619810104116574618566640172963845240985845179}{60640896607743580056983441141895638367274305574557}$

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:  \( 87278650108.0 \) (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

\(\Q(\zeta_{7})^+\), 3.3.148.1, 9.9.381393587008.1

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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$37$37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$
37.12.9.2$x^{12} - 74 x^{8} + 1369 x^{4} - 202612$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
167Data not computed