↑

Properties

Label 18.2.86368861444...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{32}\cdot 5^{9}\cdot 17^{12}$
Root discriminant $165.46$
Ramified primes $2, 3, 5, 17$
Class number $27$ (GRH)
Class group $[3, 9]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37749839, -3206385, -17365923, 9731004, -3956958, -3779586, 1198338, -376128, -60795, -37579, 49635, -17712, 30, 2286, -810, 84, 27, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 27*x^16 + 84*x^15 - 810*x^14 + 2286*x^13 + 30*x^12 - 17712*x^11 + 49635*x^10 - 37579*x^9 - 60795*x^8 - 376128*x^7 + 1198338*x^6 - 3779586*x^5 - 3956958*x^4 + 9731004*x^3 - 17365923*x^2 - 3206385*x + 37749839)
 
gp: K = bnfinit(x^18 - 9*x^17 + 27*x^16 + 84*x^15 - 810*x^14 + 2286*x^13 + 30*x^12 - 17712*x^11 + 49635*x^10 - 37579*x^9 - 60795*x^8 - 376128*x^7 + 1198338*x^6 - 3779586*x^5 - 3956958*x^4 + 9731004*x^3 - 17365923*x^2 - 3206385*x + 37749839, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 27 x^{16} + 84 x^{15} - 810 x^{14} + 2286 x^{13} + 30 x^{12} - 17712 x^{11} + 49635 x^{10} - 37579 x^{9} - 60795 x^{8} - 376128 x^{7} + 1198338 x^{6} - 3779586 x^{5} - 3956958 x^{4} + 9731004 x^{3} - 17365923 x^{2} - 3206385 x + 37749839 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8636886144486190692792838720008000000000=2^{12}\cdot 3^{32}\cdot 5^{9}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $165.46$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{18} a^{8} + \frac{1}{18} a^{7} + \frac{1}{18} a^{5} - \frac{4}{9} a^{3} - \frac{1}{9} a^{2} + \frac{2}{9} a - \frac{7}{18}$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{9} - \frac{1}{18} a^{8} - \frac{1}{18} a^{7} + \frac{1}{18} a^{6} + \frac{1}{9} a^{5} + \frac{1}{18} a^{4} - \frac{1}{9} a + \frac{2}{9}$, $\frac{1}{36} a^{12} - \frac{1}{36} a^{10} - \frac{1}{18} a^{8} + \frac{1}{18} a^{7} - \frac{1}{36} a^{6} + \frac{1}{18} a^{5} - \frac{1}{3} a^{4} - \frac{7}{18} a^{3} + \frac{5}{36} a^{2} - \frac{4}{9} a - \frac{13}{36}$, $\frac{1}{36} a^{13} - \frac{1}{36} a^{11} - \frac{1}{18} a^{9} + \frac{1}{18} a^{8} - \frac{1}{36} a^{7} + \frac{1}{18} a^{6} - \frac{1}{3} a^{5} - \frac{7}{18} a^{4} + \frac{5}{36} a^{3} - \frac{4}{9} a^{2} - \frac{13}{36} a$, $\frac{1}{108} a^{14} - \frac{1}{108} a^{13} - \frac{1}{108} a^{12} + \frac{1}{108} a^{11} + \frac{1}{54} a^{10} - \frac{1}{27} a^{9} - \frac{7}{108} a^{8} + \frac{1}{108} a^{7} - \frac{1}{54} a^{6} - \frac{4}{27} a^{5} + \frac{25}{108} a^{4} + \frac{49}{108} a^{3} + \frac{13}{108} a^{2} - \frac{19}{108} a - \frac{5}{54}$, $\frac{1}{108} a^{15} + \frac{1}{108} a^{13} - \frac{1}{54} a^{10} + \frac{1}{108} a^{9} - \frac{1}{27} a^{7} + \frac{1}{18} a^{6} - \frac{1}{4} a^{5} + \frac{8}{27} a^{4} + \frac{23}{108} a^{3} - \frac{1}{2} a^{2} + \frac{10}{27} a + \frac{2}{27}$, $\frac{1}{324} a^{16} + \frac{1}{324} a^{15} - \frac{1}{324} a^{14} + \frac{1}{162} a^{12} - \frac{7}{324} a^{11} + \frac{1}{324} a^{10} - \frac{1}{108} a^{9} + \frac{1}{81} a^{8} - \frac{1}{108} a^{7} + \frac{25}{324} a^{6} - \frac{95}{324} a^{5} - \frac{49}{324} a^{4} + \frac{5}{27} a^{3} + \frac{25}{162} a^{2} + \frac{17}{324} a - \frac{37}{162}$, $\frac{1}{111943593827764300397871154822502874997693764} a^{17} - \frac{12731284458999779317607924619225942984313}{111943593827764300397871154822502874997693764} a^{16} - \frac{1379060432265763396662044007544600852981}{18657265637960716732978525803750479166282294} a^{15} - \frac{385116609296895534092796572059594867344631}{111943593827764300397871154822502874997693764} a^{14} + \frac{625544202770415686789691905645391104167817}{111943593827764300397871154822502874997693764} a^{13} - \frac{11323803644273636921596723066331646601475}{27985898456941075099467788705625718749423441} a^{12} - \frac{224284160221093773339353633673690852682687}{12438177091973811155319017202500319444188196} a^{11} - \frac{1482056865466674601974085345652756962293185}{111943593827764300397871154822502874997693764} a^{10} - \frac{2317071995649623593891500244388988608074625}{111943593827764300397871154822502874997693764} a^{9} + \frac{789288084176556039986904038226253655447014}{27985898456941075099467788705625718749423441} a^{8} + \frac{4555272903152415497827076082516761499550969}{111943593827764300397871154822502874997693764} a^{7} - \frac{7166303084095263077876525196486038492682745}{111943593827764300397871154822502874997693764} a^{6} - \frac{3158868074047487536074842146426814318682595}{18657265637960716732978525803750479166282294} a^{5} + \frac{34044429924938853703788826817674515171799169}{111943593827764300397871154822502874997693764} a^{4} - \frac{28981297045276104875136561142204393120955713}{111943593827764300397871154822502874997693764} a^{3} - \frac{6401161670978894601294239368438622114892337}{55971796913882150198935577411251437498846882} a^{2} + \frac{5018420722490744842259302711847210202783479}{18657265637960716732978525803750479166282294} a - \frac{5810447632214804031725273920281277758005930}{27985898456941075099467788705625718749423441}$

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

Class group and class number

$C_{3}\times C_{9}$, which has order $27$ (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:  $9$
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:  \( 491197739129.93445 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.867.1, 6.2.93961125.1, 9.1.66498764695119936.21

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.10.3$x^{6} + 36$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.12.22.93$x^{12} + 81 x^{11} - 81 x^{10} + 51 x^{9} + 63 x^{8} - 108 x^{7} - 93 x^{6} + 108 x^{5} + 72 x^{3} + 108 x^{2} - 81 x + 90$$6$$2$$22$$C_6\times S_3$$[2, 5/2]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$