↑

Properties

Label 18.0.10591107618...0000.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 5^{15}$
Root discriminant $27.91$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![349, -1127, 1316, -1688, 3655, -5117, 5976, -5713, 4584, -3285, 1976, -1003, 534, -287, 135, -58, 24, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 24*x^16 - 58*x^15 + 135*x^14 - 287*x^13 + 534*x^12 - 1003*x^11 + 1976*x^10 - 3285*x^9 + 4584*x^8 - 5713*x^7 + 5976*x^6 - 5117*x^5 + 3655*x^4 - 1688*x^3 + 1316*x^2 - 1127*x + 349)
 
gp: K = bnfinit(x^18 - 7*x^17 + 24*x^16 - 58*x^15 + 135*x^14 - 287*x^13 + 534*x^12 - 1003*x^11 + 1976*x^10 - 3285*x^9 + 4584*x^8 - 5713*x^7 + 5976*x^6 - 5117*x^5 + 3655*x^4 - 1688*x^3 + 1316*x^2 - 1127*x + 349, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 24 x^{16} - 58 x^{15} + 135 x^{14} - 287 x^{13} + 534 x^{12} - 1003 x^{11} + 1976 x^{10} - 3285 x^{9} + 4584 x^{8} - 5713 x^{7} + 5976 x^{6} - 5117 x^{5} + 3655 x^{4} - 1688 x^{3} + 1316 x^{2} - 1127 x + 349 \)

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:  \(-105911076180375000000000000=-\,2^{12}\cdot 3^{25}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.91$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{204369685269274591397760679459} a^{17} + \frac{20453233311595747676301346329}{204369685269274591397760679459} a^{16} - \frac{39931685141228990725791800624}{204369685269274591397760679459} a^{15} + \frac{512983796534560527555523933}{18579062297206781036160061769} a^{14} - \frac{57926271518140079390172927120}{204369685269274591397760679459} a^{13} + \frac{93225952530603534997911280612}{204369685269274591397760679459} a^{12} + \frac{33354255650310195070157228604}{204369685269274591397760679459} a^{11} - \frac{5495458052286661216112427120}{18579062297206781036160061769} a^{10} + \frac{58407038512759647787418846269}{204369685269274591397760679459} a^{9} - \frac{42150513683583893767772139170}{204369685269274591397760679459} a^{8} + \frac{31361839153757866550306435250}{204369685269274591397760679459} a^{7} + \frac{42160218364852491679906620097}{204369685269274591397760679459} a^{6} - \frac{54718398649797436292694000869}{204369685269274591397760679459} a^{5} + \frac{8453152655033010373104235801}{204369685269274591397760679459} a^{4} - \frac{74227674157149188731577352055}{204369685269274591397760679459} a^{3} - \frac{46722899664502770941203624409}{204369685269274591397760679459} a^{2} - \frac{864336377421895819124545365}{204369685269274591397760679459} a - \frac{75147071817643212237162130862}{204369685269274591397760679459}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 253740.49514956117 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

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 $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.300.1, 6.0.1350000.1, 9.3.177147000000.3

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.22.65$x^{12} + 27 x^{11} + 18 x^{10} + 33 x^{9} - 9 x^{8} + 36 x^{7} + 18 x^{6} + 18 x^{3} - 27 x - 18$$6$$2$$22$$C_6\times S_3$$[5/2]_{2}^{6}$
$5$5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$