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Properties

Label 18.0.63472850187...9123.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 13^{12}$
Root discriminant $35.04$
Ramified primes $3, 7, 13$
Class number $36$ (GRH)
Class group $[6, 6]$ (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![64, -480, 3856, -592, 10076, -328, 19813, 1270, 15631, -2850, 7777, -984, 1810, -206, 297, -16, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 20*x^16 - 16*x^15 + 297*x^14 - 206*x^13 + 1810*x^12 - 984*x^11 + 7777*x^10 - 2850*x^9 + 15631*x^8 + 1270*x^7 + 19813*x^6 - 328*x^5 + 10076*x^4 - 592*x^3 + 3856*x^2 - 480*x + 64)
 
gp: K = bnfinit(x^18 + 20*x^16 - 16*x^15 + 297*x^14 - 206*x^13 + 1810*x^12 - 984*x^11 + 7777*x^10 - 2850*x^9 + 15631*x^8 + 1270*x^7 + 19813*x^6 - 328*x^5 + 10076*x^4 - 592*x^3 + 3856*x^2 - 480*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} + 20 x^{16} - 16 x^{15} + 297 x^{14} - 206 x^{13} + 1810 x^{12} - 984 x^{11} + 7777 x^{10} - 2850 x^{9} + 15631 x^{8} + 1270 x^{7} + 19813 x^{6} - 328 x^{5} + 10076 x^{4} - 592 x^{3} + 3856 x^{2} - 480 x + 64 \)

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:  \(-6347285018761982937208599123=-\,3^{9}\cdot 7^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.04$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(107,·)$, $\chi_{273}(263,·)$, $\chi_{273}(74,·)$, $\chi_{273}(79,·)$, $\chi_{273}(16,·)$, $\chi_{273}(211,·)$, $\chi_{273}(22,·)$, $\chi_{273}(92,·)$, $\chi_{273}(29,·)$, $\chi_{273}(100,·)$, $\chi_{273}(170,·)$, $\chi_{273}(235,·)$, $\chi_{273}(172,·)$, $\chi_{273}(113,·)$, $\chi_{273}(53,·)$, $\chi_{273}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{14} + \frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{12} a^{10} + \frac{5}{12} a^{6} - \frac{1}{6} a^{5} + \frac{1}{4} a^{4} + \frac{1}{6} a^{3} + \frac{5}{12} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{50616} a^{15} - \frac{1052}{6327} a^{13} + \frac{1327}{12654} a^{12} + \frac{2911}{16872} a^{11} + \frac{2521}{25308} a^{10} - \frac{413}{2812} a^{9} + \frac{929}{4218} a^{8} - \frac{403}{50616} a^{7} + \frac{41}{8436} a^{6} - \frac{21857}{50616} a^{5} - \frac{485}{1332} a^{4} + \frac{5065}{50616} a^{3} - \frac{4421}{12654} a^{2} + \frac{1031}{2109} a - \frac{2662}{6327}$, $\frac{1}{54968976} a^{16} + \frac{83}{27484488} a^{15} + \frac{5}{13742244} a^{14} + \frac{50695}{254486} a^{13} + \frac{3192889}{54968976} a^{12} - \frac{111311}{6871122} a^{11} - \frac{373997}{27484488} a^{10} + \frac{22523}{4580748} a^{9} - \frac{2629351}{54968976} a^{8} - \frac{2875}{723276} a^{7} - \frac{16464965}{54968976} a^{6} + \frac{23326}{1145187} a^{5} + \frac{4490603}{18322992} a^{4} + \frac{9113287}{27484488} a^{3} + \frac{139411}{723276} a^{2} - \frac{1176623}{3435561} a + \frac{1609666}{3435561}$, $\frac{1}{57051954497300033729376} a^{17} + \frac{37319006788349}{7131494312162504216172} a^{16} - \frac{119051958642392377}{14262988624325008432344} a^{15} + \frac{16735908616125873841}{792388256906944912908} a^{14} - \frac{6733583922789123860375}{57051954497300033729376} a^{13} - \frac{4578549352118446360499}{28525977248650016864688} a^{12} - \frac{2498023691511885358859}{28525977248650016864688} a^{11} + \frac{100288653717589894616}{594291192680208684681} a^{10} + \frac{1823866922213458693361}{57051954497300033729376} a^{9} + \frac{334625379979588836913}{1501367223613158782352} a^{8} + \frac{6528795451801680127303}{57051954497300033729376} a^{7} + \frac{1851449452673647957745}{9508659082883338954896} a^{6} + \frac{162594491873200446567}{2113035351751853101088} a^{5} - \frac{1695345213900096278609}{7131494312162504216172} a^{4} + \frac{113765588854308518611}{750683611806579391176} a^{3} + \frac{185056620711502465619}{375341805903289695588} a^{2} - \frac{313781748392017689301}{1782873578040626054043} a - \frac{82958857084175263165}{594291192680208684681}$

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

Class group and class number

$C_{6}\times C_{6}$, which has order $36$ (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:  \( -\frac{24446932299421959}{11674228462717420448} a^{17} + \frac{296939433887827}{4377835673519032668} a^{16} - \frac{365920779614098829}{8755671347038065336} a^{15} + \frac{152042758276988009}{4377835673519032668} a^{14} - \frac{21759521690497023965}{35022685388152261344} a^{13} + \frac{7836709499734490303}{17511342694076130672} a^{12} - \frac{66134388845569716973}{17511342694076130672} a^{11} + \frac{2328789676888680184}{1094458918379758167} a^{10} - \frac{9941474005092465101}{614433076985127392} a^{9} + \frac{108505622889031566329}{17511342694076130672} a^{8} - \frac{1127624512393037562235}{35022685388152261344} a^{7} - \frac{48158422219520397431}{17511342694076130672} a^{6} - \frac{1401818185963852548889}{35022685388152261344} a^{5} + \frac{125711044367359535}{230412403869422772} a^{4} - \frac{58838211147321113329}{2918557115679355112} a^{3} - \frac{2825556066484140763}{4377835673519032668} a^{2} - \frac{16647209995558651861}{2188917836759516334} a + \frac{54495262354813001}{57603100967355693} \) (order $6$)
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

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{7})^+\), 3.3.8281.1, 3.3.8281.2, 3.3.169.1, 6.0.64827.1, 6.0.1851523947.2, 6.0.1851523947.1, 6.0.771147.1, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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$$[\ ]_{3}^{3}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$