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Properties

Label 18.0.10247702651...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}\cdot 5^{9}\cdot 7^{12}$
Root discriminant $100.14$
Ramified primes $2, 3, 5, 7$
Class number $508032$ (GRH)
Class group $[2, 2, 2, 252, 252]$ (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![9505772641, 576974412, 4850734998, 248935216, 1124100009, 47240316, 157769068, 4958358, 15135612, 290492, 1056552, 7350, 54990, -96, 2133, -8, 60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 60*x^16 - 8*x^15 + 2133*x^14 - 96*x^13 + 54990*x^12 + 7350*x^11 + 1056552*x^10 + 290492*x^9 + 15135612*x^8 + 4958358*x^7 + 157769068*x^6 + 47240316*x^5 + 1124100009*x^4 + 248935216*x^3 + 4850734998*x^2 + 576974412*x + 9505772641)
 
gp: K = bnfinit(x^18 + 60*x^16 - 8*x^15 + 2133*x^14 - 96*x^13 + 54990*x^12 + 7350*x^11 + 1056552*x^10 + 290492*x^9 + 15135612*x^8 + 4958358*x^7 + 157769068*x^6 + 47240316*x^5 + 1124100009*x^4 + 248935216*x^3 + 4850734998*x^2 + 576974412*x + 9505772641, 1)
 

Normalized defining polynomial

\( x^{18} + 60 x^{16} - 8 x^{15} + 2133 x^{14} - 96 x^{13} + 54990 x^{12} + 7350 x^{11} + 1056552 x^{10} + 290492 x^{9} + 15135612 x^{8} + 4958358 x^{7} + 157769068 x^{6} + 47240316 x^{5} + 1124100009 x^{4} + 248935216 x^{3} + 4850734998 x^{2} + 576974412 x + 9505772641 \)

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:  \(-1024770265180753855691096064000000000=-\,2^{27}\cdot 3^{24}\cdot 5^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.14$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2520=2^{3}\cdot 3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{2520}(1,·)$, $\chi_{2520}(2179,·)$, $\chi_{2520}(961,·)$, $\chi_{2520}(1801,·)$, $\chi_{2520}(2059,·)$, $\chi_{2520}(1681,·)$, $\chi_{2520}(1219,·)$, $\chi_{2520}(2041,·)$, $\chi_{2520}(1339,·)$, $\chi_{2520}(739,·)$, $\chi_{2520}(361,·)$, $\chi_{2520}(1579,·)$, $\chi_{2520}(1201,·)$, $\chi_{2520}(499,·)$, $\chi_{2520}(841,·)$, $\chi_{2520}(121,·)$, $\chi_{2520}(2419,·)$, $\chi_{2520}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{7} + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{8} + \frac{1}{4} a$, $\frac{1}{2032} a^{16} - \frac{69}{1016} a^{15} + \frac{109}{2032} a^{14} - \frac{199}{508} a^{13} + \frac{187}{1016} a^{12} - \frac{125}{254} a^{11} - \frac{77}{508} a^{10} + \frac{475}{1016} a^{9} - \frac{101}{254} a^{8} + \frac{301}{1016} a^{7} - \frac{16}{127} a^{6} + \frac{17}{254} a^{5} + \frac{47}{508} a^{4} + \frac{51}{508} a^{3} - \frac{899}{2032} a^{2} - \frac{443}{1016} a - \frac{613}{2032}$, $\frac{1}{43314292935604716832250035326731261715053987066144032449668432} a^{17} - \frac{2718035629001091436717302288321298305285944226829680284801}{21657146467802358416125017663365630857526993533072016224834216} a^{16} - \frac{3703122718004283235715028936048349051592416598552286184858283}{43314292935604716832250035326731261715053987066144032449668432} a^{15} + \frac{529790788150124271064549866712858937855352600513254970380879}{10828573233901179208062508831682815428763496766536008112417108} a^{14} - \frac{1842590491890373392891406448350850678606102933385371295529741}{21657146467802358416125017663365630857526993533072016224834216} a^{13} + \frac{1274232272415838618410017390807356417862232513438764370782551}{5414286616950589604031254415841407714381748383268004056208554} a^{12} - \frac{3222262243617279002103475684585789989909586785855151664316085}{10828573233901179208062508831682815428763496766536008112417108} a^{11} - \frac{7243038148284834591038536788456688981375392234902610312655077}{21657146467802358416125017663365630857526993533072016224834216} a^{10} - \frac{2617615238121540652454404970194515735320836861046585167046805}{5414286616950589604031254415841407714381748383268004056208554} a^{9} - \frac{9763618875315240037055649468337262544558472884973421429019883}{21657146467802358416125017663365630857526993533072016224834216} a^{8} - \frac{324853533205926219995977547838249189001692358034645170771485}{2707143308475294802015627207920703857190874191634002028104277} a^{7} + \frac{1850660269186760039090606525559487578169835330973382223624661}{5414286616950589604031254415841407714381748383268004056208554} a^{6} + \frac{810549433699932192571322952899120653335403036335342534692295}{10828573233901179208062508831682815428763496766536008112417108} a^{5} + \frac{4208929308384361105788028701807062121604607137192329177203499}{10828573233901179208062508831682815428763496766536008112417108} a^{4} - \frac{12684373940515044848928317377135233303034651695816116618360851}{43314292935604716832250035326731261715053987066144032449668432} a^{3} - \frac{60596017578869756754469138798158000966694479796823281252953}{170528712344900459969488328057997093366354279787968631691608} a^{2} - \frac{5124189671187115552029629437966568718525321295844139743051565}{43314292935604716832250035326731261715053987066144032449668432} a - \frac{388123050599164570522807505683154073222643911668937421580131}{5414286616950589604031254415841407714381748383268004056208554}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{252}\times C_{252}$, which has order $508032$ (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

\(\Q(\sqrt{-10}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.2, 3.3.3969.1, 6.0.419904000.3, 6.0.153664000.1, 6.0.1008189504000.20, 6.0.1008189504000.19, 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 R R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$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}$