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Properties

Label 18.0.95090394182...1296.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 7^{12}\cdot 13^{15}$
Root discriminant $87.75$
Ramified primes $2, 7, 13$
Class number $54432$ (GRH)
Class group $[2, 2, 6, 6, 378]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31314401, -26053208, 36433958, -17759238, 15664617, -8245054, 6181459, -2975076, 1620055, -615760, 304596, -107954, 39726, -9226, 2577, -370, 81, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 81*x^16 - 370*x^15 + 2577*x^14 - 9226*x^13 + 39726*x^12 - 107954*x^11 + 304596*x^10 - 615760*x^9 + 1620055*x^8 - 2975076*x^7 + 6181459*x^6 - 8245054*x^5 + 15664617*x^4 - 17759238*x^3 + 36433958*x^2 - 26053208*x + 31314401)
 
gp: K = bnfinit(x^18 - 6*x^17 + 81*x^16 - 370*x^15 + 2577*x^14 - 9226*x^13 + 39726*x^12 - 107954*x^11 + 304596*x^10 - 615760*x^9 + 1620055*x^8 - 2975076*x^7 + 6181459*x^6 - 8245054*x^5 + 15664617*x^4 - 17759238*x^3 + 36433958*x^2 - 26053208*x + 31314401, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 81 x^{16} - 370 x^{15} + 2577 x^{14} - 9226 x^{13} + 39726 x^{12} - 107954 x^{11} + 304596 x^{10} - 615760 x^{9} + 1620055 x^{8} - 2975076 x^{7} + 6181459 x^{6} - 8245054 x^{5} + 15664617 x^{4} - 17759238 x^{3} + 36433958 x^{2} - 26053208 x + 31314401 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-95090394182193546591849307801911296=-\,2^{27}\cdot 7^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.75$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(728=2^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{728}(1,·)$, $\chi_{728}(387,·)$, $\chi_{728}(179,·)$, $\chi_{728}(81,·)$, $\chi_{728}(417,·)$, $\chi_{728}(9,·)$, $\chi_{728}(459,·)$, $\chi_{728}(529,·)$, $\chi_{728}(667,·)$, $\chi_{728}(491,·)$, $\chi_{728}(289,·)$, $\chi_{728}(155,·)$, $\chi_{728}(113,·)$, $\chi_{728}(43,·)$, $\chi_{728}(625,·)$, $\chi_{728}(51,·)$, $\chi_{728}(393,·)$, $\chi_{728}(571,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{12} - \frac{1}{5} a^{11} - \frac{1}{10} a^{10} + \frac{1}{5} a^{9} + \frac{1}{20} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{7}{20}$, $\frac{1}{60} a^{13} + \frac{1}{5} a^{11} - \frac{1}{15} a^{10} - \frac{13}{60} a^{9} - \frac{1}{5} a^{7} - \frac{1}{15} a^{6} - \frac{1}{2} a^{5} + \frac{4}{15} a^{4} + \frac{7}{15} a^{3} - \frac{2}{15} a^{2} - \frac{3}{20} a + \frac{7}{15}$, $\frac{1}{60} a^{14} + \frac{7}{30} a^{11} + \frac{11}{60} a^{10} + \frac{1}{5} a^{9} + \frac{1}{10} a^{8} + \frac{7}{30} a^{7} + \frac{1}{5} a^{6} - \frac{7}{30} a^{5} + \frac{7}{15} a^{4} + \frac{1}{6} a^{3} - \frac{9}{20} a^{2} + \frac{4}{15} a - \frac{2}{5}$, $\frac{1}{60} a^{15} - \frac{1}{60} a^{12} + \frac{11}{60} a^{11} + \frac{1}{5} a^{10} + \frac{1}{10} a^{9} - \frac{1}{60} a^{8} + \frac{1}{5} a^{7} - \frac{7}{30} a^{6} + \frac{7}{15} a^{5} + \frac{1}{6} a^{4} - \frac{9}{20} a^{3} - \frac{7}{30} a^{2} - \frac{2}{5} a - \frac{1}{4}$, $\frac{1}{13412170194518632687860} a^{16} + \frac{1622522407082285901}{894144679634575512524} a^{15} - \frac{1481222776129341159}{4470723398172877562620} a^{14} - \frac{16902844557740936039}{4470723398172877562620} a^{13} - \frac{243482409923327111809}{13412170194518632687860} a^{12} + \frac{842853083777601030079}{4470723398172877562620} a^{11} - \frac{2084593197549765584137}{13412170194518632687860} a^{10} + \frac{3293605658631588970033}{13412170194518632687860} a^{9} + \frac{16280653869503631059}{223536169908643878131} a^{8} - \frac{62449671870975360577}{670608509725931634393} a^{7} + \frac{85802057258947110184}{1117680849543219390655} a^{6} + \frac{540344966915077193632}{3353042548629658171965} a^{5} + \frac{5924774226553872184831}{13412170194518632687860} a^{4} + \frac{5527754967976096363373}{13412170194518632687860} a^{3} + \frac{140188917571211584513}{13412170194518632687860} a^{2} + \frac{381885013509356842329}{4470723398172877562620} a + \frac{630795528410924925211}{1341217019451863268786}$, $\frac{1}{159134341970510288869196250067915260} a^{17} - \frac{2752476294547}{159134341970510288869196250067915260} a^{16} + \frac{134167337336997052458639745120861}{159134341970510288869196250067915260} a^{15} + \frac{194265871761964694187711558356009}{31826868394102057773839250013583052} a^{14} - \frac{74717869214186618181881561678429}{31826868394102057773839250013583052} a^{13} - \frac{221320389706745801004139949950262}{13261195164209190739099687505659605} a^{12} + \frac{9345426746242354113541035404632561}{53044780656836762956398750022638420} a^{11} + \frac{20929044351130996746405726437607193}{159134341970510288869196250067915260} a^{10} - \frac{17271095899609683468835232627690723}{79567170985255144434598125033957630} a^{9} - \frac{7551259816270419042749590700389097}{53044780656836762956398750022638420} a^{8} - \frac{1135444163070514424078894767226962}{7956717098525514443459812503395763} a^{7} + \frac{18828007329556555138633267203676741}{79567170985255144434598125033957630} a^{6} + \frac{73220380252436827057315870662415637}{159134341970510288869196250067915260} a^{5} - \frac{1651409381258034267624479024625219}{3536318710455784197093250001509228} a^{4} - \frac{539805455634737652834814707458453}{3536318710455784197093250001509228} a^{3} - \frac{5481339919757467033212780790893179}{17681593552278920985466250007546140} a^{2} + \frac{13149543370195872021309922871761993}{79567170985255144434598125033957630} a - \frac{2834700531415879685882072448576259}{159134341970510288869196250067915260}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{378}$, which has order $54432$ (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:  $8$
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:  \( 205236.825908 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-26}) \), 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 3.3.8281.2, 6.0.190102016.1, 6.0.2700798464.1, 6.0.456434940416.3, 6.0.456434940416.5, 9.9.567869252041.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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$$[\ ]_{3}^{3}$
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$