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Properties

Label 18.0.83742161529...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 5^{9}\cdot 19^{15}$
Root discriminant $112.53$
Ramified primes $3, 5, 19$
Class number $134784$ (GRH)
Class group $[2, 2, 2, 18, 936]$ (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![22373704, 4539660, 18187044, 12429935, 11848893, 5730762, 4385648, 1689372, 260676, -150228, 50940, 25530, 4202, 2712, 684, -4, 48, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 48*x^16 - 4*x^15 + 684*x^14 + 2712*x^13 + 4202*x^12 + 25530*x^11 + 50940*x^10 - 150228*x^9 + 260676*x^8 + 1689372*x^7 + 4385648*x^6 + 5730762*x^5 + 11848893*x^4 + 12429935*x^3 + 18187044*x^2 + 4539660*x + 22373704)
 
gp: K = bnfinit(x^18 - 3*x^17 + 48*x^16 - 4*x^15 + 684*x^14 + 2712*x^13 + 4202*x^12 + 25530*x^11 + 50940*x^10 - 150228*x^9 + 260676*x^8 + 1689372*x^7 + 4385648*x^6 + 5730762*x^5 + 11848893*x^4 + 12429935*x^3 + 18187044*x^2 + 4539660*x + 22373704, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 48 x^{16} - 4 x^{15} + 684 x^{14} + 2712 x^{13} + 4202 x^{12} + 25530 x^{11} + 50940 x^{10} - 150228 x^{9} + 260676 x^{8} + 1689372 x^{7} + 4385648 x^{6} + 5730762 x^{5} + 11848893 x^{4} + 12429935 x^{3} + 18187044 x^{2} + 4539660 x + 22373704 \)

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:  \(-8374216152942811568456908683240234375=-\,3^{24}\cdot 5^{9}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.53$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(855=3^{2}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{855}(1,·)$, $\chi_{855}(259,·)$, $\chi_{855}(391,·)$, $\chi_{855}(844,·)$, $\chi_{855}(274,·)$, $\chi_{855}(406,·)$, $\chi_{855}(664,·)$, $\chi_{855}(94,·)$, $\chi_{855}(544,·)$, $\chi_{855}(571,·)$, $\chi_{855}(676,·)$, $\chi_{855}(106,·)$, $\chi_{855}(559,·)$, $\chi_{855}(691,·)$, $\chi_{855}(286,·)$, $\chi_{855}(121,·)$, $\chi_{855}(379,·)$, $\chi_{855}(829,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{22} a^{12} + \frac{5}{22} a^{11} + \frac{3}{22} a^{10} + \frac{2}{11} a^{9} + \frac{1}{22} a^{8} - \frac{3}{11} a^{7} - \frac{5}{11} a^{6} + \frac{5}{22} a^{5} - \frac{1}{22} a^{4} - \frac{9}{22} a^{3} - \frac{1}{11} a^{2} - \frac{7}{22} a + \frac{2}{11}$, $\frac{1}{22} a^{13} + \frac{3}{22} a^{9} - \frac{1}{11} a^{7} - \frac{1}{2} a^{6} - \frac{2}{11} a^{5} - \frac{2}{11} a^{4} + \frac{5}{11} a^{3} + \frac{3}{22} a^{2} + \frac{3}{11} a + \frac{1}{11}$, $\frac{1}{22} a^{14} + \frac{3}{22} a^{10} - \frac{1}{11} a^{8} - \frac{1}{2} a^{7} - \frac{2}{11} a^{6} - \frac{2}{11} a^{5} + \frac{5}{11} a^{4} + \frac{3}{22} a^{3} + \frac{3}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{4202} a^{15} - \frac{7}{2101} a^{14} - \frac{45}{2101} a^{13} - \frac{46}{2101} a^{12} - \frac{465}{2101} a^{11} - \frac{511}{2101} a^{10} - \frac{387}{4202} a^{9} - \frac{76}{2101} a^{8} - \frac{835}{2101} a^{7} - \frac{262}{2101} a^{6} - \frac{1029}{2101} a^{5} - \frac{651}{2101} a^{4} - \frac{626}{2101} a^{3} + \frac{107}{4202} a^{2} + \frac{1209}{4202} a - \frac{384}{2101}$, $\frac{1}{310948} a^{16} - \frac{7}{155474} a^{15} - \frac{29}{2101} a^{14} - \frac{141}{7067} a^{13} - \frac{1611}{155474} a^{12} - \frac{10474}{77737} a^{11} - \frac{8644}{77737} a^{10} - \frac{17419}{77737} a^{9} - \frac{14587}{155474} a^{8} + \frac{14767}{77737} a^{7} + \frac{12569}{77737} a^{6} - \frac{6763}{155474} a^{5} + \frac{31584}{77737} a^{4} - \frac{17975}{77737} a^{3} - \frac{58765}{310948} a^{2} - \frac{31}{2101} a + \frac{137}{407}$, $\frac{1}{8890304428408026700925592385731978984044339917315172252} a^{17} - \frac{9963880545437507928374178772349664090574567570065}{8890304428408026700925592385731978984044339917315172252} a^{16} + \frac{150369696624612146499599748993314179313628781930749}{4445152214204013350462796192865989492022169958657586126} a^{15} - \frac{1150704791103495597557296047259158242306132064867514}{202052373372909697748308917857544976910098634484435733} a^{14} - \frac{4814558765455102333136147434851390272246916322226468}{2222576107102006675231398096432994746011084979328793063} a^{13} + \frac{31755765768786232280150598049129409919069737251101773}{2222576107102006675231398096432994746011084979328793063} a^{12} - \frac{228612095242185398503573616803490373153347000659875791}{4445152214204013350462796192865989492022169958657586126} a^{11} - \frac{799994591754624788703721396216891569048078818354158163}{4445152214204013350462796192865989492022169958657586126} a^{10} + \frac{539099792653937505982044240658849934283033328276081693}{4445152214204013350462796192865989492022169958657586126} a^{9} + \frac{380764657464368112046582010220198140490781700144660187}{4445152214204013350462796192865989492022169958657586126} a^{8} - \frac{322198362639455904921142245888645676882589430657981449}{2222576107102006675231398096432994746011084979328793063} a^{7} + \frac{50843908460562364172127057527917970840214269007046678}{202052373372909697748308917857544976910098634484435733} a^{6} + \frac{542914732614175747129985856524302510356360227934000500}{2222576107102006675231398096432994746011084979328793063} a^{5} - \frac{390691029341795138378422970175064155268903260516984623}{4445152214204013350462796192865989492022169958657586126} a^{4} - \frac{4165850995054543043390576761043623554574007364020063767}{8890304428408026700925592385731978984044339917315172252} a^{3} + \frac{3087448176188399495975474603025647367190901272054517703}{8890304428408026700925592385731978984044339917315172252} a^{2} - \frac{37081252361719798721490892687949536260204729931075090}{202052373372909697748308917857544976910098634484435733} a + \frac{41515376949681140875360500294340787527087491270645308}{2222576107102006675231398096432994746011084979328793063}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{18}\times C_{936}$, which has order $134784$ (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:  \( 1472619.082400847 \) (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

\(\Q(\sqrt{-95}) \), \(\Q(\zeta_{9})^+\), 3.3.29241.1, 3.3.29241.2, 3.3.361.1, 6.0.5625237375.5, 6.0.2030710692375.4, 6.0.2030710692375.5, 6.0.309512375.1, 9.9.25002110044521.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$