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Properties

Label 18.0.95719345891...2211.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{15}\cdot 31^{9}$
Root discriminant $146.42$
Ramified primes $3, 7, 31$
Class number $10027136$ (GRH)
Class group $[2, 4, 76, 16492]$ (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![2780371629001, -609765780432, 930282815520, -182182097162, 185880800082, -18496427643, 23827510937, -809469819, 1882862586, -15798444, 89391399, -133146, 2500302, -321, 39447, -1, 318, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 318*x^16 - x^15 + 39447*x^14 - 321*x^13 + 2500302*x^12 - 133146*x^11 + 89391399*x^10 - 15798444*x^9 + 1882862586*x^8 - 809469819*x^7 + 23827510937*x^6 - 18496427643*x^5 + 185880800082*x^4 - 182182097162*x^3 + 930282815520*x^2 - 609765780432*x + 2780371629001)
 
gp: K = bnfinit(x^18 + 318*x^16 - x^15 + 39447*x^14 - 321*x^13 + 2500302*x^12 - 133146*x^11 + 89391399*x^10 - 15798444*x^9 + 1882862586*x^8 - 809469819*x^7 + 23827510937*x^6 - 18496427643*x^5 + 185880800082*x^4 - 182182097162*x^3 + 930282815520*x^2 - 609765780432*x + 2780371629001, 1)
 

Normalized defining polynomial

\( x^{18} + 318 x^{16} - x^{15} + 39447 x^{14} - 321 x^{13} + 2500302 x^{12} - 133146 x^{11} + 89391399 x^{10} - 15798444 x^{9} + 1882862586 x^{8} - 809469819 x^{7} + 23827510937 x^{6} - 18496427643 x^{5} + 185880800082 x^{4} - 182182097162 x^{3} + 930282815520 x^{2} - 609765780432 x + 2780371629001 \)

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:  \(-957193458914897226781087429876810532211=-\,3^{27}\cdot 7^{15}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $146.42$
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:  $3, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1953=3^{2}\cdot 7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1953}(1024,·)$, $\chi_{1953}(1,·)$, $\chi_{1953}(836,·)$, $\chi_{1953}(650,·)$, $\chi_{1953}(1675,·)$, $\chi_{1953}(652,·)$, $\chi_{1953}(1487,·)$, $\chi_{1953}(466,·)$, $\chi_{1953}(1301,·)$, $\chi_{1953}(278,·)$, $\chi_{1953}(1303,·)$, $\chi_{1953}(1117,·)$, $\chi_{1953}(1952,·)$, $\chi_{1953}(929,·)$, $\chi_{1953}(1768,·)$, $\chi_{1953}(1580,·)$, $\chi_{1953}(373,·)$, $\chi_{1953}(185,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4927} a^{15} + \frac{1739}{4927} a^{13} - \frac{1829}{4927} a^{12} - \frac{2204}{4927} a^{11} + \frac{1149}{4927} a^{10} + \frac{1980}{4927} a^{9} + \frac{401}{4927} a^{8} - \frac{369}{4927} a^{7} - \frac{176}{379} a^{6} - \frac{1310}{4927} a^{5} + \frac{1897}{4927} a^{4} + \frac{2231}{4927} a^{3} - \frac{852}{4927} a^{2} + \frac{2266}{4927} a - \frac{2380}{4927}$, $\frac{1}{1156338273169235} a^{16} - \frac{1738744548}{1156338273169235} a^{15} - \frac{255024579118364}{1156338273169235} a^{14} + \frac{70022205343209}{1156338273169235} a^{13} - \frac{39569100359832}{88949097936095} a^{12} - \frac{575216761372062}{1156338273169235} a^{11} - \frac{117763819244471}{1156338273169235} a^{10} + \frac{565746459549614}{1156338273169235} a^{9} + \frac{181293719512683}{1156338273169235} a^{8} - \frac{82332169058957}{1156338273169235} a^{7} - \frac{295244575330481}{1156338273169235} a^{6} - \frac{378404602194904}{1156338273169235} a^{5} + \frac{73923338032711}{231267654633847} a^{4} - \frac{147274860848654}{1156338273169235} a^{3} - \frac{449134314082706}{1156338273169235} a^{2} + \frac{71355509157656}{231267654633847} a + \frac{398540573189091}{1156338273169235}$, $\frac{1}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{17} - \frac{564248812947546375072813433570414116655970433098947117731177844}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{16} - \frac{2220673773755322964940979101356382553487487484839518210858010688965936894396}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{15} - \frac{1622280383704850006589511433790525094554815228242822735256689664299179890825452}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{14} - \frac{392968416929745415864803186155441482764029554950391818928415182383650561984818}{5886919471918362863853895726833967253866100136580729824047139060451331756486905} a^{13} - \frac{9375664979210257178993594825357115305061408181597969365112247085233707724682136}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{12} + \frac{9391903328932641301307255161024357529471226761291966085609776683551185029759686}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{11} - \frac{2828371488937960937455013954787012660350211462603063346073609075725932560773191}{5886919471918362863853895726833967253866100136580729824047139060451331756486905} a^{10} - \frac{11868441770586454916754956806857222698076559993710266376584003887698844954188306}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{9} + \frac{1568272259423547758112164006981600678772232776220574074282404029440532149458594}{5886919471918362863853895726833967253866100136580729824047139060451331756486905} a^{8} - \frac{13893495279862530725792894078434596480207563280460409198010939821948203724057394}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{7} - \frac{739353368001042943755129749715136558724050826172287336130795040198270836621058}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{6} + \frac{822543346549628137966450853656012375877423407176839470107785034960421691621203}{2264199796891678024559190664166910482256192360223357624633515023250512214033425} a^{5} - \frac{5134953738861801444845164629266963189052781704326014767633219330989242120379934}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{4} + \frac{291549778621533092496589521672259093969387366632855255141663108477608631255281}{2264199796891678024559190664166910482256192360223357624633515023250512214033425} a^{3} - \frac{6090375338020121964951700356163629980569054376716235917217413762058533036052069}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a^{2} - \frac{12272802796409795687368567044662749694472795249041046406929497206073627511673819}{29434597359591814319269478634169836269330500682903649120235695302256658782434525} a - \frac{74693187898424206532317521107613134066424978648933810993078767415997416958567}{2264199796891678024559190664166910482256192360223357624633515023250512214033425}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{76}\times C_{16492}$, which has order $10027136$ (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:  \( 54408.48888868202 \) (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

\(\Q(\sqrt{-651}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.0.201127054779.4, 6.0.9855225684171.7, 6.0.13518828099.1, 6.0.9855225684171.8, 9.9.62523502209.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
7Data not computed
$31$31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$