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Properties

Label 18.6.49205531492...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{26}\cdot 3^{18}\cdot 5^{4}\cdot 37^{9}\cdot 233$
Root discriminant $96.14$
Ramified primes $2, 3, 5, 37, 233$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T874

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25600, 384000, -403200, -769280, -149760, -295920, -59324, 55968, -19008, 28352, -1512, 1512, -412, -240, -180, -8, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^16 - 8*x^15 - 180*x^14 - 240*x^13 - 412*x^12 + 1512*x^11 - 1512*x^10 + 28352*x^9 - 19008*x^8 + 55968*x^7 - 59324*x^6 - 295920*x^5 - 149760*x^4 - 769280*x^3 - 403200*x^2 + 384000*x - 25600)
 
gp: K = bnfinit(x^18 - 12*x^16 - 8*x^15 - 180*x^14 - 240*x^13 - 412*x^12 + 1512*x^11 - 1512*x^10 + 28352*x^9 - 19008*x^8 + 55968*x^7 - 59324*x^6 - 295920*x^5 - 149760*x^4 - 769280*x^3 - 403200*x^2 + 384000*x - 25600, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{16} - 8 x^{15} - 180 x^{14} - 240 x^{13} - 412 x^{12} + 1512 x^{11} - 1512 x^{10} + 28352 x^{9} - 19008 x^{8} + 55968 x^{7} - 59324 x^{6} - 295920 x^{5} - 149760 x^{4} - 769280 x^{3} - 403200 x^{2} + 384000 x - 25600 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(492055314924578283041666849832960000=2^{26}\cdot 3^{18}\cdot 5^{4}\cdot 37^{9}\cdot 233\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.14$
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:  $2, 3, 5, 37, 233$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{320} a^{15} + \frac{1}{40} a^{13} - \frac{1}{40} a^{12} - \frac{1}{16} a^{11} - \frac{3}{80} a^{9} - \frac{1}{40} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} + \frac{7}{20} a^{5} + \frac{2}{5} a^{4} + \frac{9}{80} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{1280} a^{16} - \frac{3}{320} a^{14} - \frac{1}{160} a^{13} - \frac{1}{64} a^{12} - \frac{1}{16} a^{11} + \frac{17}{320} a^{10} + \frac{9}{160} a^{9} - \frac{9}{160} a^{8} + \frac{1}{40} a^{7} + \frac{3}{20} a^{6} + \frac{9}{40} a^{5} - \frac{31}{320} a^{4} + \frac{1}{16} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{5907508936338688108366668398018398720} a^{17} + \frac{62255261913257822959367319401231}{184609654260584003386458387438074960} a^{16} + \frac{1871751814530270593433871527470073}{1476877234084672027091667099504599680} a^{15} - \frac{16173721107150264847080563061734679}{738438617042336013545833549752299840} a^{14} - \frac{43538582251509244676712190830205501}{1476877234084672027091667099504599680} a^{13} + \frac{20009168042035414020658040379464333}{369219308521168006772916774876149920} a^{12} - \frac{50473702945668351318055972618815783}{1476877234084672027091667099504599680} a^{11} + \frac{6875769147685404553390846726415781}{738438617042336013545833549752299840} a^{10} - \frac{32554670480599566380495927356760117}{738438617042336013545833549752299840} a^{9} - \frac{621979955882238110220631826350501}{92304827130292001693229193719037480} a^{8} - \frac{7797397112970709003843435135051993}{92304827130292001693229193719037480} a^{7} - \frac{1377970553417513646923280414206567}{36921930852116800677291677487614992} a^{6} + \frac{93229470344086631749722555170844477}{295375446816934405418333419900919936} a^{5} + \frac{59056448588423386893142010159449609}{369219308521168006772916774876149920} a^{4} - \frac{20488814793746400510661143407996649}{92304827130292001693229193719037480} a^{3} + \frac{3258293487392782867895534744193499}{18460965426058400338645838743807496} a^{2} - \frac{888614896849492931804511397132397}{2307620678257300042330729842975937} a + \frac{533485975299674870997370917077640}{2307620678257300042330729842975937}$

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:  $11$
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:  \( 110315804862 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T874:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 174 conjugacy class representatives for t18n874 are not computed
Character table for t18n874 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.22.12$x^{12} + 8 x^{10} + 2 x^{8} - 4 x^{6} + 4 x^{4} + 4 x^{2} + 4$$6$$2$$22$12T106$[4/3, 4/3, 8/3, 8/3, 3]_{3}^{2}$
$3$3.9.9.9$x^{9} + 18 x^{5} + 27 x^{2} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.5$x^{9} + 3 x^{7} + 3 x^{6} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
37Data not computed
233Data not computed