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Properties

Label 18.0.94412655580...5443.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{33}\cdot 19^{8}$
Root discriminant $27.74$
Ramified primes $3, 19$
Class number $3$
Class group $[3]$
Galois group $C_2\times He_3$ (as 18T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 1056, 19056, -25632, 50556, -41604, 55569, -40263, 38334, -19451, 12813, -4620, 2691, -750, 339, -57, 24, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 24*x^16 - 57*x^15 + 339*x^14 - 750*x^13 + 2691*x^12 - 4620*x^11 + 12813*x^10 - 19451*x^9 + 38334*x^8 - 40263*x^7 + 55569*x^6 - 41604*x^5 + 50556*x^4 - 25632*x^3 + 19056*x^2 + 1056*x + 64)
 
gp: K = bnfinit(x^18 - 3*x^17 + 24*x^16 - 57*x^15 + 339*x^14 - 750*x^13 + 2691*x^12 - 4620*x^11 + 12813*x^10 - 19451*x^9 + 38334*x^8 - 40263*x^7 + 55569*x^6 - 41604*x^5 + 50556*x^4 - 25632*x^3 + 19056*x^2 + 1056*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 24 x^{16} - 57 x^{15} + 339 x^{14} - 750 x^{13} + 2691 x^{12} - 4620 x^{11} + 12813 x^{10} - 19451 x^{9} + 38334 x^{8} - 40263 x^{7} + 55569 x^{6} - 41604 x^{5} + 50556 x^{4} - 25632 x^{3} + 19056 x^{2} + 1056 x + 64 \)

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:  \(-94412655580832901097225443=-\,3^{33}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.74$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{9} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{12} - \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{7}{18} a^{4} - \frac{4}{9} a^{3} + \frac{1}{6} a^{2} + \frac{7}{18} a - \frac{1}{9}$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{13} - \frac{1}{18} a^{12} - \frac{1}{36} a^{11} + \frac{1}{36} a^{10} - \frac{1}{4} a^{8} + \frac{1}{6} a^{7} + \frac{1}{4} a^{6} + \frac{11}{36} a^{5} + \frac{1}{9} a^{4} - \frac{7}{36} a^{3} + \frac{7}{36} a^{2} - \frac{5}{18} a + \frac{1}{3}$, $\frac{1}{6552} a^{15} - \frac{1}{2184} a^{14} + \frac{1}{273} a^{13} + \frac{199}{6552} a^{12} + \frac{23}{504} a^{11} + \frac{149}{3276} a^{10} - \frac{79}{2184} a^{9} - \frac{73}{546} a^{8} + \frac{95}{2184} a^{7} + \frac{2141}{6552} a^{6} + \frac{317}{1092} a^{5} + \frac{101}{312} a^{4} - \frac{1303}{6552} a^{3} - \frac{239}{1638} a^{2} + \frac{599}{1638} a + \frac{109}{273}$, $\frac{1}{89906544} a^{16} + \frac{6197}{89906544} a^{15} - \frac{43}{208117} a^{14} + \frac{345559}{89906544} a^{13} + \frac{582539}{89906544} a^{12} + \frac{124805}{4994808} a^{11} - \frac{344699}{12843792} a^{10} - \frac{281665}{22476636} a^{9} + \frac{255539}{1427088} a^{8} + \frac{39869021}{89906544} a^{7} + \frac{292781}{6421896} a^{6} + \frac{1534241}{9989616} a^{5} - \frac{41903191}{89906544} a^{4} + \frac{9795733}{22476636} a^{3} + \frac{427003}{2497404} a^{2} + \frac{1404407}{5619159} a - \frac{1550498}{5619159}$, $\frac{1}{1917939765875685892704} a^{17} - \frac{498169265107}{1917939765875685892704} a^{16} + \frac{3357284207163763}{159828313822973824392} a^{15} + \frac{23198241464824195843}{1917939765875685892704} a^{14} - \frac{41925841714997971117}{1917939765875685892704} a^{13} - \frac{808241134881765707}{24588971357380588368} a^{12} + \frac{49188377641135423783}{1917939765875685892704} a^{11} - \frac{14523954925847097397}{479484941468921473176} a^{10} + \frac{7691064100170312971}{639313255291895297568} a^{9} - \frac{462335888633762170987}{1917939765875685892704} a^{8} + \frac{334427612078258024045}{958969882937842946352} a^{7} + \frac{3861129316171803809}{91330465041699328224} a^{6} - \frac{422823004726507425895}{1917939765875685892704} a^{5} + \frac{23492509182959305289}{239742470734460736588} a^{4} - \frac{17837130249627506689}{79914156911486912196} a^{3} + \frac{14196548293262461591}{59935617683615184147} a^{2} + \frac{2950764138370009616}{8562231097659312021} a + \frac{642751939174314956}{2854077032553104007}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{5568121222310585}{5269065290867268936} a^{17} + \frac{2648474003123738183}{958969882937842946352} a^{16} - \frac{1803752061852910285}{73766914072141765104} a^{15} + \frac{6230069533212895969}{119871235367230368294} a^{14} - \frac{328723940402393836243}{958969882937842946352} a^{13} + \frac{654702817928701150397}{958969882937842946352} a^{12} - \frac{1268893230194125850537}{479484941468921473176} a^{11} + \frac{569581439928751433057}{136995697562548992336} a^{10} - \frac{750369654097157629363}{59935617683615184147} a^{9} + \frac{2405561756782615213369}{136995697562548992336} a^{8} - \frac{34911620905285412518973}{958969882937842946352} a^{7} + \frac{2477780783887309761179}{68497848781274496168} a^{6} - \frac{51125569653615529665641}{958969882937842946352} a^{5} + \frac{36832463135176233751147}{958969882937842946352} a^{4} - \frac{23060522071351300901605}{479484941468921473176} a^{3} + \frac{1411781640701054913328}{59935617683615184147} a^{2} - \frac{2281522485447591035389}{119871235367230368294} a - \frac{39572534057471508887}{59935617683615184147} \) (order $18$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 958684.150365 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3$ (as 18T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 22 conjugacy class representatives for $C_2\times He_3$
Character table for $C_2\times He_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 9.9.5609891727441.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$19$19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$