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Properties

Label 18.0.21005238210...344.10
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{15}\cdot 19^{12}$
Root discriminant $374.51$
Ramified primes $2, 3, 7, 19$
Class number $323481600$ (GRH)
Class group $[2, 2, 2, 2, 4, 12, 180, 2340]$ (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![27422121601, -30542807610, 16365722709, -2732121798, 3867087, 63019572, 77021785, -49796760, 25989579, -6643036, 2514777, -475356, 148828, -19992, 5088, -470, 105, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 105*x^16 - 470*x^15 + 5088*x^14 - 19992*x^13 + 148828*x^12 - 475356*x^11 + 2514777*x^10 - 6643036*x^9 + 25989579*x^8 - 49796760*x^7 + 77021785*x^6 + 63019572*x^5 + 3867087*x^4 - 2732121798*x^3 + 16365722709*x^2 - 30542807610*x + 27422121601)
 
gp: K = bnfinit(x^18 - 6*x^17 + 105*x^16 - 470*x^15 + 5088*x^14 - 19992*x^13 + 148828*x^12 - 475356*x^11 + 2514777*x^10 - 6643036*x^9 + 25989579*x^8 - 49796760*x^7 + 77021785*x^6 + 63019572*x^5 + 3867087*x^4 - 2732121798*x^3 + 16365722709*x^2 - 30542807610*x + 27422121601, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 105 x^{16} - 470 x^{15} + 5088 x^{14} - 19992 x^{13} + 148828 x^{12} - 475356 x^{11} + 2514777 x^{10} - 6643036 x^{9} + 25989579 x^{8} - 49796760 x^{7} + 77021785 x^{6} + 63019572 x^{5} + 3867087 x^{4} - 2732121798 x^{3} + 16365722709 x^{2} - 30542807610 x + 27422121601 \)

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:  \(-21005238210476463669563930923405113491534905344=-\,2^{18}\cdot 3^{27}\cdot 7^{15}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $374.51$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(2819,·)$, $\chi_{4788}(961,·)$, $\chi_{4788}(457,·)$, $\chi_{4788}(1559,·)$, $\chi_{4788}(311,·)$, $\chi_{4788}(3469,·)$, $\chi_{4788}(2063,·)$, $\chi_{4788}(2965,·)$, $\chi_{4788}(2519,·)$, $\chi_{4788}(2015,·)$, $\chi_{4788}(4225,·)$, $\chi_{4788}(1261,·)$, $\chi_{4788}(1717,·)$, $\chi_{4788}(3275,·)$, $\chi_{4788}(4343,·)$, $\chi_{4788}(505,·)$, $\chi_{4788}(3839,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{22} a^{8} - \frac{2}{11} a^{7} + \frac{3}{11} a^{6} - \frac{3}{11} a^{5} - \frac{3}{22} a^{4} + \frac{3}{11} a^{3} - \frac{5}{22} a^{2} + \frac{2}{11} a + \frac{1}{22}$, $\frac{1}{110} a^{9} + \frac{1}{55} a^{8} + \frac{24}{55} a^{7} - \frac{18}{55} a^{6} + \frac{49}{110} a^{5} + \frac{27}{55} a^{4} + \frac{31}{110} a^{3} - \frac{24}{55} a^{2} + \frac{5}{22} a + \frac{3}{55}$, $\frac{1}{110} a^{10} - \frac{1}{110} a^{8} + \frac{24}{55} a^{7} - \frac{39}{110} a^{6} + \frac{3}{55} a^{5} - \frac{26}{55} a^{4} - \frac{5}{11} a^{3} + \frac{8}{55} a^{2} - \frac{2}{55} a + \frac{53}{110}$, $\frac{1}{110} a^{11} - \frac{1}{10} a^{7} - \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{3}{10} a^{3} - \frac{1}{5} a^{2} - \frac{6}{55} a - \frac{2}{5}$, $\frac{1}{110} a^{12} - \frac{1}{110} a^{8} - \frac{4}{11} a^{7} + \frac{27}{110} a^{6} - \frac{8}{55} a^{5} + \frac{47}{110} a^{4} + \frac{19}{55} a^{3} + \frac{24}{55} a^{2} - \frac{2}{55} a + \frac{1}{11}$, $\frac{1}{110} a^{13} + \frac{1}{55} a^{8} + \frac{5}{22} a^{7} - \frac{16}{55} a^{6} - \frac{17}{55} a^{5} - \frac{14}{55} a^{4} - \frac{1}{10} a^{3} - \frac{16}{55} a^{2} - \frac{5}{22} a + \frac{23}{55}$, $\frac{1}{1210} a^{14} + \frac{1}{242} a^{13} + \frac{1}{605} a^{12} - \frac{1}{242} a^{11} - \frac{2}{605} a^{10} + \frac{2}{605} a^{9} + \frac{3}{605} a^{8} - \frac{54}{605} a^{7} + \frac{252}{605} a^{6} + \frac{109}{1210} a^{5} - \frac{93}{605} a^{4} - \frac{117}{605} a^{3} - \frac{147}{605} a^{2} + \frac{229}{1210} a + \frac{91}{242}$, $\frac{1}{1210} a^{15} - \frac{1}{1210} a^{13} - \frac{2}{605} a^{12} - \frac{1}{1210} a^{11} + \frac{1}{605} a^{10} - \frac{3}{1210} a^{9} - \frac{3}{605} a^{8} - \frac{281}{605} a^{7} + \frac{197}{605} a^{6} + \frac{179}{605} a^{5} - \frac{4}{605} a^{4} - \frac{81}{1210} a^{3} - \frac{179}{605} a^{2} - \frac{437}{1210} a - \frac{2}{121}$, $\frac{1}{1217255145031292392665860} a^{16} + \frac{100521667535401684234}{304313786257823098166465} a^{15} - \frac{45238106234780260589}{121725514503129239266586} a^{14} - \frac{189424178954810067163}{304313786257823098166465} a^{13} + \frac{71518669285610586983}{55329779319604199666630} a^{12} + \frac{130278391485267825291}{608627572515646196332930} a^{11} + \frac{82069917093475433834}{27664889659802099833315} a^{10} - \frac{57972926428482504011}{121725514503129239266586} a^{9} + \frac{16816093353012547148299}{1217255145031292392665860} a^{8} + \frac{228484740240029121967163}{608627572515646196332930} a^{7} - \frac{117019699279823134793254}{304313786257823098166465} a^{6} - \frac{113104307167415924321266}{304313786257823098166465} a^{5} - \frac{514591144358121699809879}{1217255145031292392665860} a^{4} - \frac{29057797291433900822858}{304313786257823098166465} a^{3} + \frac{102142839454731278549189}{304313786257823098166465} a^{2} - \frac{4119255393658517698499}{11065955863920839933326} a - \frac{404533084389809055225973}{1217255145031292392665860}$, $\frac{1}{2515938439875943377868813366007934079940} a^{17} + \frac{34186561758971}{2515938439875943377868813366007934079940} a^{16} + \frac{11662786626728794909572449989619379}{628984609968985844467203341501983519985} a^{15} + \frac{1662633763318698883347106797565005}{251593843987594337786881336600793407994} a^{14} + \frac{814703867480981219212203237474594831}{628984609968985844467203341501983519985} a^{13} - \frac{2496050757725777614825087554777767509}{628984609968985844467203341501983519985} a^{12} - \frac{2264563200191120479278291433304966378}{628984609968985844467203341501983519985} a^{11} + \frac{2539448035044794948025813366469446001}{628984609968985844467203341501983519985} a^{10} + \frac{9378047921977296774524343285179751949}{2515938439875943377868813366007934079940} a^{9} - \frac{25725894743288864631502062919741894567}{2515938439875943377868813366007934079940} a^{8} - \frac{290763562407887201552037373636057519591}{1257969219937971688934406683003967039970} a^{7} - \frac{510587908551438667015512828180820614133}{1257969219937971688934406683003967039970} a^{6} + \frac{86856653759351823601317185475944506199}{503187687975188675573762673201586815988} a^{5} + \frac{499973285884884888750609085452703174687}{2515938439875943377868813366007934079940} a^{4} + \frac{46988531391341202699868094915992927172}{125796921993797168893440668300396703997} a^{3} - \frac{15172312298838346497516744649537080127}{1257969219937971688934406683003967039970} a^{2} + \frac{223975651185779064500533702903356159049}{2515938439875943377868813366007934079940} a + \frac{400797131695865954939729852025890213727}{2515938439875943377868813366007934079940}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{12}\times C_{180}\times C_{2340}$, which has order $323481600$ (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:  \( 110172188.8644179 \) (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

\(\Q(\sqrt{-21}) \), 3.3.1432809.2, 3.3.1432809.3, 3.3.361.1, 3.3.3969.2, 6.0.2759153551366464.5, 6.0.2759153551366464.2, 6.0.77241777984.6, 6.0.21171979584.1, 9.9.2941473244627851129.9

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$7$7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$