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Properties

Label 18.2.42774537890...8848.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{44}\cdot 13^{9}$
Root discriminant $83.94$
Ramified primes $2, 3, 13$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-201483, 692145, -1119177, 721170, -245160, 337122, -265014, 26334, 77481, -15875, -17145, 5022, 2712, -1080, -216, 120, 9, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 9*x^16 + 120*x^15 - 216*x^14 - 1080*x^13 + 2712*x^12 + 5022*x^11 - 17145*x^10 - 15875*x^9 + 77481*x^8 + 26334*x^7 - 265014*x^6 + 337122*x^5 - 245160*x^4 + 721170*x^3 - 1119177*x^2 + 692145*x - 201483)
 
gp: K = bnfinit(x^18 - 9*x^17 + 9*x^16 + 120*x^15 - 216*x^14 - 1080*x^13 + 2712*x^12 + 5022*x^11 - 17145*x^10 - 15875*x^9 + 77481*x^8 + 26334*x^7 - 265014*x^6 + 337122*x^5 - 245160*x^4 + 721170*x^3 - 1119177*x^2 + 692145*x - 201483, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 9 x^{16} + 120 x^{15} - 216 x^{14} - 1080 x^{13} + 2712 x^{12} + 5022 x^{11} - 17145 x^{10} - 15875 x^{9} + 77481 x^{8} + 26334 x^{7} - 265014 x^{6} + 337122 x^{5} - 245160 x^{4} + 721170 x^{3} - 1119177 x^{2} + 692145 x - 201483 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(42774537890835454566245668796878848=2^{12}\cdot 3^{44}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.94$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} + \frac{1}{6} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{12} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} + \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{6} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{24} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{24} a^{15} - \frac{1}{24} a^{13} - \frac{1}{24} a^{12} + \frac{1}{24} a^{10} + \frac{1}{24} a^{9} - \frac{1}{4} a^{8} - \frac{1}{6} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{48} a^{16} + \frac{1}{8} a^{9} - \frac{1}{16} a^{8} + \frac{1}{12} a^{7} + \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{1}{16}$, $\frac{1}{931712696276011579745448112150527125971056} a^{17} - \frac{1801395594405523789971138101893560846803}{232928174069002894936362028037631781492764} a^{16} + \frac{4397732404386856809455181419101450093489}{465856348138005789872724056075263562985528} a^{15} - \frac{4670923397974121917634136631685157320021}{465856348138005789872724056075263562985528} a^{14} - \frac{2793585929588122121941241708399340683009}{232928174069002894936362028037631781492764} a^{13} + \frac{4578860777123628386751093908925665761}{9911837194425655103674979916494969425224} a^{12} + \frac{15039806147305386746198772611430879867625}{465856348138005789872724056075263562985528} a^{11} + \frac{11088572397062153138801233607918043560783}{155285449379335263290908018691754520995176} a^{10} - \frac{99874102929407367768394453337867841530273}{931712696276011579745448112150527125971056} a^{9} - \frac{10508304807107935332186127142521865948757}{155285449379335263290908018691754520995176} a^{8} + \frac{13720309496022130275126611965074585958115}{155285449379335263290908018691754520995176} a^{7} + \frac{106919323079383678097072501971058114702995}{465856348138005789872724056075263562985528} a^{6} - \frac{213573994103997196379219840944009063689929}{465856348138005789872724056075263562985528} a^{5} + \frac{17355503353452820785397372956805137647625}{155285449379335263290908018691754520995176} a^{4} - \frac{130702005699246880255202848949307212253}{19410681172416907911363502336469315124397} a^{3} + \frac{27589508292697157903743658747757172819795}{77642724689667631645454009345877260497588} a^{2} - \frac{67452139030794514296343723561918220798103}{310570898758670526581816037383509041990352} a + \frac{3549683111774858470893638765300512233917}{38821362344833815822727004672938630248794}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $9$
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:  \( 7883902648.832988 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.1.243.1, 6.2.129730653.3, 9.1.2008387814976.4

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.9.22.46$x^{9} + 18 x^{5} + 3$$9$$1$$22$$C_3^2 : C_6$$[2, 5/2, 17/6]_{2}$
3.9.22.46$x^{9} + 18 x^{5} + 3$$9$$1$$22$$C_3^2 : C_6$$[2, 5/2, 17/6]_{2}$
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$