↑

Properties

Label 18.0.46572979962...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{27}\cdot 5^{9}\cdot 13^{12}$
Root discriminant $181.69$
Ramified primes $2, 3, 5, 13$
Class number $11793600$ (GRH)
Class group $[2, 2, 2, 2, 90, 8190]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![191108188441, 893627904, 82917294006, -10762099726, 15903607875, -3413556846, 2271460011, -463924236, 219817671, -37365600, 13567536, -1842426, 534596, -53706, 12825, -866, 171, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 171*x^16 - 866*x^15 + 12825*x^14 - 53706*x^13 + 534596*x^12 - 1842426*x^11 + 13567536*x^10 - 37365600*x^9 + 219817671*x^8 - 463924236*x^7 + 2271460011*x^6 - 3413556846*x^5 + 15903607875*x^4 - 10762099726*x^3 + 82917294006*x^2 + 893627904*x + 191108188441)
 
gp: K = bnfinit(x^18 - 6*x^17 + 171*x^16 - 866*x^15 + 12825*x^14 - 53706*x^13 + 534596*x^12 - 1842426*x^11 + 13567536*x^10 - 37365600*x^9 + 219817671*x^8 - 463924236*x^7 + 2271460011*x^6 - 3413556846*x^5 + 15903607875*x^4 - 10762099726*x^3 + 82917294006*x^2 + 893627904*x + 191108188441, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 171 x^{16} - 866 x^{15} + 12825 x^{14} - 53706 x^{13} + 534596 x^{12} - 1842426 x^{11} + 13567536 x^{10} - 37365600 x^{9} + 219817671 x^{8} - 463924236 x^{7} + 2271460011 x^{6} - 3413556846 x^{5} + 15903607875 x^{4} - 10762099726 x^{3} + 82917294006 x^{2} + 893627904 x + 191108188441 \)

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:  \(-46572979962512449327252831469568000000000=-\,2^{27}\cdot 3^{27}\cdot 5^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $181.69$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4680=2^{3}\cdot 3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4680}(1,·)$, $\chi_{4680}(3721,·)$, $\chi_{4680}(4109,·)$, $\chi_{4680}(269,·)$, $\chi_{4680}(3149,·)$, $\chi_{4680}(1561,·)$, $\chi_{4680}(601,·)$, $\chi_{4680}(989,·)$, $\chi_{4680}(2401,·)$, $\chi_{4680}(1829,·)$, $\chi_{4680}(2161,·)$, $\chi_{4680}(29,·)$, $\chi_{4680}(3121,·)$, $\chi_{4680}(2549,·)$, $\chi_{4680}(841,·)$, $\chi_{4680}(3961,·)$, $\chi_{4680}(3389,·)$, $\chi_{4680}(1589,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{10} a^{6} - \frac{1}{5} a^{5} + \frac{3}{10} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{7} - \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} - \frac{3}{10} a + \frac{1}{5}$, $\frac{1}{10} a^{8} + \frac{3}{10} a^{4} - \frac{1}{2} a^{2} + \frac{1}{10}$, $\frac{1}{10} a^{9} + \frac{3}{10} a^{5} - \frac{1}{2} a^{3} + \frac{1}{10} a$, $\frac{1}{10} a^{10} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{3}{10} a^{2} - \frac{2}{5} a - \frac{3}{10}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{100} a^{12} - \frac{1}{25} a^{11} + \frac{1}{100} a^{8} + \frac{1}{25} a^{6} + \frac{4}{25} a^{4} + \frac{1}{5} a^{3} + \frac{3}{10} a^{2} + \frac{4}{25} a + \frac{41}{100}$, $\frac{1}{100} a^{13} + \frac{1}{25} a^{11} + \frac{1}{100} a^{9} + \frac{1}{25} a^{8} + \frac{1}{25} a^{7} - \frac{1}{25} a^{6} + \frac{4}{25} a^{5} + \frac{1}{25} a^{4} + \frac{1}{10} a^{3} + \frac{9}{25} a^{2} + \frac{1}{4} a + \frac{6}{25}$, $\frac{1}{100} a^{14} - \frac{1}{25} a^{11} + \frac{1}{100} a^{10} + \frac{1}{25} a^{9} - \frac{1}{25} a^{7} + \frac{11}{25} a^{5} - \frac{17}{50} a^{4} + \frac{9}{25} a^{3} + \frac{9}{20} a^{2} - \frac{1}{5} a - \frac{11}{25}$, $\frac{1}{100} a^{15} - \frac{1}{20} a^{11} + \frac{1}{25} a^{10} - \frac{17}{50} a^{5} - \frac{2}{5} a^{4} - \frac{1}{4} a^{3} + \frac{3}{10} a + \frac{11}{25}$, $\frac{1}{212936601815125123779197172100} a^{16} + \frac{32172680781986759061837319}{53234150453781280944799293025} a^{15} - \frac{66994841979935273816962526}{53234150453781280944799293025} a^{14} + \frac{203150398116925898480255979}{42587320363025024755839434420} a^{13} - \frac{203604154701450479515025031}{212936601815125123779197172100} a^{12} + \frac{829322774068196786212317456}{53234150453781280944799293025} a^{11} - \frac{659895007024284922037892793}{21293660181512512377919717210} a^{10} - \frac{3545825129305360806493607611}{212936601815125123779197172100} a^{9} - \frac{1219430668495371012050674069}{53234150453781280944799293025} a^{8} + \frac{3424633783529262960498610843}{106468300907562561889598586050} a^{7} + \frac{2539584450781823423855193393}{53234150453781280944799293025} a^{6} - \frac{833371921598131620102843500}{2129366018151251237791971721} a^{5} + \frac{6281470316146720816278919143}{42587320363025024755839434420} a^{4} + \frac{13016933896479791570555323454}{53234150453781280944799293025} a^{3} + \frac{3772457586695317527007522111}{21293660181512512377919717210} a^{2} - \frac{30478601674449670769398355457}{212936601815125123779197172100} a - \frac{14248858287432120603714906184}{53234150453781280944799293025}$, $\frac{1}{7347895530831347410296167439674761467187204900} a^{17} + \frac{9343329346114819}{7347895530831347410296167439674761467187204900} a^{16} + \frac{6375645133538689412883881919588567505682943}{1836973882707836852574041859918690366796801225} a^{15} - \frac{32306141436641114989670756659527345339937897}{7347895530831347410296167439674761467187204900} a^{14} - \frac{9589822282705205587070449157383757108606611}{3673947765415673705148083719837380733593602450} a^{13} + \frac{20853976189193879658423523257823362310080559}{7347895530831347410296167439674761467187204900} a^{12} - \frac{132077951030568534286980373424999760802704127}{3673947765415673705148083719837380733593602450} a^{11} - \frac{216840725273047844323151066349827645657700889}{7347895530831347410296167439674761467187204900} a^{10} - \frac{18232916911660479890097928443482123121983271}{1469579106166269482059233487934952293437440980} a^{9} - \frac{17161659770581840350680118178023381628317887}{1836973882707836852574041859918690366796801225} a^{8} + \frac{130718115711837990325130504881299112497052523}{3673947765415673705148083719837380733593602450} a^{7} + \frac{6613479210824886554216003768654802635849441}{3673947765415673705148083719837380733593602450} a^{6} + \frac{552076988452565197223392627425299857704181517}{7347895530831347410296167439674761467187204900} a^{5} - \frac{635853018812334356180700334476016989121200827}{1469579106166269482059233487934952293437440980} a^{4} + \frac{905063579589941270149643590605021652790636092}{1836973882707836852574041859918690366796801225} a^{3} + \frac{2583457025975200881255946352620884024034047047}{7347895530831347410296167439674761467187204900} a^{2} + \frac{1370473661380821880681213598742683158821191391}{7347895530831347410296167439674761467187204900} a + \frac{64713826190793263221118074288895444760695087}{1836973882707836852574041859918690366796801225}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{90}\times C_{8190}$, which has order $11793600$ (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:  \( 400417.1364448253 \) (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

\(\Q(\sqrt{-30}) \), 3.3.13689.2, 3.3.13689.1, 3.3.169.1, \(\Q(\zeta_{9})^+\), 6.0.35978634432000.4, 6.0.35978634432000.3, 6.0.49353408000.11, 6.0.1259712000.1, 9.9.2565164201769.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$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}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$