↑

Properties

Label 18.18.1322221330...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 5^{15}\cdot 13^{6}\cdot 23^{12}$
Root discriminant $115.42$
Ramified primes $2, 5, 13, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^3:C_2^2$ (as 18T53)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1686161, 7617599, -4364387, -14116983, 16638030, 3194954, -11348071, 2583238, 2749637, -1205780, -247452, 194678, -479, -14481, 1320, 492, -68, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 68*x^16 + 492*x^15 + 1320*x^14 - 14481*x^13 - 479*x^12 + 194678*x^11 - 247452*x^10 - 1205780*x^9 + 2749637*x^8 + 2583238*x^7 - 11348071*x^6 + 3194954*x^5 + 16638030*x^4 - 14116983*x^3 - 4364387*x^2 + 7617599*x - 1686161)
 
gp: K = bnfinit(x^18 - 6*x^17 - 68*x^16 + 492*x^15 + 1320*x^14 - 14481*x^13 - 479*x^12 + 194678*x^11 - 247452*x^10 - 1205780*x^9 + 2749637*x^8 + 2583238*x^7 - 11348071*x^6 + 3194954*x^5 + 16638030*x^4 - 14116983*x^3 - 4364387*x^2 + 7617599*x - 1686161, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 68 x^{16} + 492 x^{15} + 1320 x^{14} - 14481 x^{13} - 479 x^{12} + 194678 x^{11} - 247452 x^{10} - 1205780 x^{9} + 2749637 x^{8} + 2583238 x^{7} - 11348071 x^{6} + 3194954 x^{5} + 16638030 x^{4} - 14116983 x^{3} - 4364387 x^{2} + 7617599 x - 1686161 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13222213305011946698211125000000000000=2^{12}\cdot 5^{15}\cdot 13^{6}\cdot 23^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.42$
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:  $2, 5, 13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{7} + \frac{1}{5} a^{5} - \frac{2}{5} a^{2} - \frac{2}{5}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{10} + \frac{1}{25} a^{6} - \frac{7}{25} a^{5} - \frac{6}{25} a - \frac{3}{25}$, $\frac{1}{25} a^{12} + \frac{1}{25} a^{10} + \frac{1}{25} a^{7} + \frac{6}{25} a^{5} - \frac{6}{25} a^{2} + \frac{9}{25}$, $\frac{1}{25} a^{13} + \frac{2}{25} a^{10} + \frac{1}{25} a^{8} - \frac{8}{25} a^{5} - \frac{6}{25} a^{3} + \frac{8}{25}$, $\frac{1}{125} a^{14} + \frac{2}{125} a^{13} - \frac{1}{125} a^{12} - \frac{2}{125} a^{11} + \frac{1}{125} a^{10} + \frac{11}{125} a^{9} + \frac{12}{125} a^{8} - \frac{1}{125} a^{7} + \frac{8}{125} a^{6} - \frac{9}{125} a^{5} - \frac{26}{125} a^{4} + \frac{18}{125} a^{3} - \frac{19}{125} a^{2} - \frac{8}{125} a - \frac{36}{125}$, $\frac{1}{625} a^{15} - \frac{1}{125} a^{13} - \frac{1}{125} a^{12} - \frac{36}{625} a^{10} - \frac{2}{125} a^{9} + \frac{2}{25} a^{8} + \frac{11}{125} a^{7} + \frac{4}{125} a^{6} + \frac{47}{625} a^{5} + \frac{39}{125} a^{4} + \frac{9}{125} a^{3} - \frac{8}{125} a^{2} + \frac{7}{125} a + \frac{267}{625}$, $\frac{1}{625} a^{16} + \frac{1}{125} a^{13} - \frac{1}{125} a^{12} + \frac{4}{625} a^{11} + \frac{4}{125} a^{10} - \frac{4}{125} a^{9} - \frac{2}{125} a^{8} + \frac{3}{125} a^{7} + \frac{12}{625} a^{6} + \frac{2}{25} a^{5} + \frac{33}{125} a^{4} + \frac{12}{25} a^{3} - \frac{12}{125} a^{2} + \frac{177}{625} a - \frac{16}{125}$, $\frac{1}{2276915285930901602965033853125} a^{17} - \frac{59077248604893011833175062}{133936193290053035468531403125} a^{16} - \frac{19029854167464202758356768}{133936193290053035468531403125} a^{15} - \frac{205398633135537240812557767}{455383057186180320593006770625} a^{14} - \frac{1022480341422456323401317348}{91076611437236064118601354125} a^{13} - \frac{41778348500535765791248294031}{2276915285930901602965033853125} a^{12} - \frac{1199842557789092224114032666}{2276915285930901602965033853125} a^{11} - \frac{135552719405876526237508213349}{2276915285930901602965033853125} a^{10} - \frac{12516812179632335924992951422}{455383057186180320593006770625} a^{9} + \frac{692197023039734565803943311}{91076611437236064118601354125} a^{8} + \frac{92448387020371237521255796787}{2276915285930901602965033853125} a^{7} + \frac{131520877348236685434225878787}{2276915285930901602965033853125} a^{6} - \frac{964702588538196066852242276157}{2276915285930901602965033853125} a^{5} - \frac{120251202195038847961983622338}{455383057186180320593006770625} a^{4} + \frac{2794354703080532463199320282}{18215322287447212823720270825} a^{3} + \frac{364080259968479275434979020542}{2276915285930901602965033853125} a^{2} + \frac{239612372458825342189928922772}{2276915285930901602965033853125} a + \frac{138150769557840531258673739158}{2276915285930901602965033853125}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 10134410489300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^3:C_2^2$ (as 18T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 15 conjugacy class representatives for $C_3^3:C_2^2$
Character table for $C_3^3:C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.10580.1 x3, 6.6.2364656450000.1, 6.6.559682000.1

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
2Data not computed
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23Data not computed