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Properties

Label 18.0.40019255390...8352.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{12}\cdot 107^{9}$
Root discriminant $34.16$
Ramified primes $2, 3, 107$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_3:S_3:S_4$ (as 18T155)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![877, -5565, 14889, -18930, 27005, -26741, 22549, -17477, 12946, -9906, 7049, -4227, 2199, -917, 323, -101, 25, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 25*x^16 - 101*x^15 + 323*x^14 - 917*x^13 + 2199*x^12 - 4227*x^11 + 7049*x^10 - 9906*x^9 + 12946*x^8 - 17477*x^7 + 22549*x^6 - 26741*x^5 + 27005*x^4 - 18930*x^3 + 14889*x^2 - 5565*x + 877)
 
gp: K = bnfinit(x^18 - 6*x^17 + 25*x^16 - 101*x^15 + 323*x^14 - 917*x^13 + 2199*x^12 - 4227*x^11 + 7049*x^10 - 9906*x^9 + 12946*x^8 - 17477*x^7 + 22549*x^6 - 26741*x^5 + 27005*x^4 - 18930*x^3 + 14889*x^2 - 5565*x + 877, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 25 x^{16} - 101 x^{15} + 323 x^{14} - 917 x^{13} + 2199 x^{12} - 4227 x^{11} + 7049 x^{10} - 9906 x^{9} + 12946 x^{8} - 17477 x^{7} + 22549 x^{6} - 26741 x^{5} + 27005 x^{4} - 18930 x^{3} + 14889 x^{2} - 5565 x + 877 \)

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:  \(-4001925539052664141228388352=-\,2^{12}\cdot 3^{12}\cdot 107^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.16$
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:  $2, 3, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} - \frac{4}{11} a^{15} + \frac{5}{11} a^{14} + \frac{1}{11} a^{13} + \frac{1}{11} a^{12} - \frac{3}{11} a^{11} + \frac{3}{11} a^{10} - \frac{5}{11} a^{9} - \frac{4}{11} a^{8} + \frac{2}{11} a^{7} - \frac{4}{11} a^{6} + \frac{3}{11} a^{5} - \frac{2}{11} a^{4} + \frac{4}{11} a^{3} - \frac{1}{11} a^{2} - \frac{5}{11} a - \frac{3}{11}$, $\frac{1}{902642945799972723745898076368197} a^{17} - \frac{29452627690310896139802798781964}{902642945799972723745898076368197} a^{16} + \frac{85009079218917464932995445962144}{902642945799972723745898076368197} a^{15} + \frac{127924330939358556407956462781367}{902642945799972723745898076368197} a^{14} + \frac{993528093624665096604848610537}{82058449618179338522354370578927} a^{13} + \frac{88158191944822469169810380031395}{902642945799972723745898076368197} a^{12} + \frac{354416860640851020781411483855867}{902642945799972723745898076368197} a^{11} + \frac{407428148466206473979538630862675}{902642945799972723745898076368197} a^{10} + \frac{414052753868236939473621295930635}{902642945799972723745898076368197} a^{9} - \frac{212082882540360916909844436171100}{902642945799972723745898076368197} a^{8} - \frac{309657173008507765236477916507708}{902642945799972723745898076368197} a^{7} + \frac{27906042888968278701771656925512}{902642945799972723745898076368197} a^{6} - \frac{134176204726140652642896772356237}{902642945799972723745898076368197} a^{5} + \frac{47652868587755795549440791263250}{902642945799972723745898076368197} a^{4} - \frac{19984994033497066267536310232088}{902642945799972723745898076368197} a^{3} + \frac{69893567619156068977335332712489}{902642945799972723745898076368197} a^{2} + \frac{398867937858297549110622745696788}{902642945799972723745898076368197} a - \frac{268237608772222586510415753162919}{902642945799972723745898076368197}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 217021.279396 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T155):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.321.1, 6.0.99228483.4, 9.3.6350622912.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.9.2$x^{12} - 9 x^{4} + 27$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$
107Data not computed