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Properties

Label 18.0.16572404264...4768.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{31}\cdot 7^{14}\cdot 11^{9}$
Root discriminant $251.81$
Ramified primes $2, 3, 7, 11$
Class number $81741312$ (GRH)
Class group $[2, 2, 6, 6, 6, 12, 7884]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![151614141291361, -1029715188852, 31158412601505, -789904049838, 2888612355147, -118816855266, 160131076710, -8515724286, 5906580054, -356745098, 151699395, -9357396, 2737851, -153312, 33861, -1446, 264, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 264*x^16 - 1446*x^15 + 33861*x^14 - 153312*x^13 + 2737851*x^12 - 9357396*x^11 + 151699395*x^10 - 356745098*x^9 + 5906580054*x^8 - 8515724286*x^7 + 160131076710*x^6 - 118816855266*x^5 + 2888612355147*x^4 - 789904049838*x^3 + 31158412601505*x^2 - 1029715188852*x + 151614141291361)
 
gp: K = bnfinit(x^18 - 6*x^17 + 264*x^16 - 1446*x^15 + 33861*x^14 - 153312*x^13 + 2737851*x^12 - 9357396*x^11 + 151699395*x^10 - 356745098*x^9 + 5906580054*x^8 - 8515724286*x^7 + 160131076710*x^6 - 118816855266*x^5 + 2888612355147*x^4 - 789904049838*x^3 + 31158412601505*x^2 - 1029715188852*x + 151614141291361, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 264 x^{16} - 1446 x^{15} + 33861 x^{14} - 153312 x^{13} + 2737851 x^{12} - 9357396 x^{11} + 151699395 x^{10} - 356745098 x^{9} + 5906580054 x^{8} - 8515724286 x^{7} + 160131076710 x^{6} - 118816855266 x^{5} + 2888612355147 x^{4} - 789904049838 x^{3} + 31158412601505 x^{2} - 1029715188852 x + 151614141291361 \)

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:  \(-16572404264697372849747294259176558211104768=-\,2^{24}\cdot 3^{31}\cdot 7^{14}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $251.81$
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:  $2, 3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{54} a^{9} + \frac{1}{9} a^{8} - \frac{1}{18} a^{7} - \frac{1}{18} a^{6} - \frac{1}{18} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{18} a^{2} - \frac{7}{18} a + \frac{17}{54}$, $\frac{1}{54} a^{10} - \frac{1}{18} a^{8} - \frac{1}{18} a^{7} - \frac{1}{18} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{18} a^{3} - \frac{1}{18} a^{2} - \frac{19}{54} a + \frac{1}{9}$, $\frac{1}{54} a^{11} - \frac{1}{18} a^{8} + \frac{1}{9} a^{7} - \frac{1}{18} a^{6} + \frac{1}{18} a^{5} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} - \frac{5}{27} a^{2} - \frac{1}{18} a - \frac{7}{18}$, $\frac{1}{54} a^{12} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{5}{18} a^{4} + \frac{4}{27} a^{3} - \frac{2}{9} a^{2} + \frac{1}{9} a + \frac{5}{18}$, $\frac{1}{54} a^{13} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{18} a^{5} - \frac{5}{27} a^{4} - \frac{2}{9} a^{3} + \frac{4}{9} a^{2} - \frac{7}{18} a + \frac{1}{9}$, $\frac{1}{162} a^{14} + \frac{1}{162} a^{13} + \frac{1}{162} a^{12} - \frac{1}{162} a^{11} - \frac{1}{162} a^{10} - \frac{1}{162} a^{9} + \frac{4}{27} a^{8} + \frac{4}{27} a^{7} + \frac{4}{27} a^{6} + \frac{23}{162} a^{5} - \frac{2}{81} a^{4} + \frac{25}{81} a^{3} - \frac{47}{162} a^{2} - \frac{10}{81} a + \frac{7}{162}$, $\frac{1}{162} a^{15} + \frac{1}{162} a^{12} + \frac{1}{162} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{162} a^{6} + \frac{1}{18} a^{5} + \frac{7}{18} a^{4} - \frac{37}{162} a^{3} + \frac{1}{18} a^{2} - \frac{5}{18} a + \frac{31}{81}$, $\frac{1}{162} a^{16} + \frac{1}{162} a^{13} + \frac{1}{162} a^{10} - \frac{1}{9} a^{8} - \frac{1}{162} a^{7} + \frac{1}{18} a^{6} + \frac{1}{18} a^{5} + \frac{71}{162} a^{4} + \frac{7}{18} a^{3} + \frac{1}{18} a^{2} - \frac{23}{81} a - \frac{4}{9}$, $\frac{1}{12054310478422737360826099723579380996249485473680295012599415843422} a^{17} - \frac{25749743475475486564988393743872483353806445548835562212307624125}{12054310478422737360826099723579380996249485473680295012599415843422} a^{16} + \frac{2929206481349496978949529710783210191124250724999185870954163461}{4018103492807579120275366574526460332083161824560098337533138614474} a^{15} - \frac{6138038805044234878845539953477118905963737550897329217284000491}{2009051746403789560137683287263230166041580912280049168766569307237} a^{14} - \frac{23824123688219088163786891041796109475806820349277686345106053}{6470375994859225636514277897788180889022804870467147081373814194} a^{13} + \frac{17634007494782169500578192883639027832688909991138881622911125589}{6027155239211368680413049861789690498124742736840147506299707921711} a^{12} + \frac{17045366040853845934471027533949527811662554548260829163864378287}{6027155239211368680413049861789690498124742736840147506299707921711} a^{11} + \frac{46624174576727382086996765663087767084503716977851059960125138042}{6027155239211368680413049861789690498124742736840147506299707921711} a^{10} + \frac{64299133735053237322573276513637987659759669457069837508787331767}{12054310478422737360826099723579380996249485473680295012599415843422} a^{9} - \frac{511663862425306543962140520642807743296749391369560081691467939661}{12054310478422737360826099723579380996249485473680295012599415843422} a^{8} + \frac{1251791362306956678716258629680438882656083577363167028849345165429}{12054310478422737360826099723579380996249485473680295012599415843422} a^{7} + \frac{138878554630422573391998892325979494345699065063773696713254555114}{2009051746403789560137683287263230166041580912280049168766569307237} a^{6} + \frac{116885251980760645016463011918507830220057774446225199117415370165}{2009051746403789560137683287263230166041580912280049168766569307237} a^{5} + \frac{17692749461343820294908392314285994525700590809237220085955218529}{4018103492807579120275366574526460332083161824560098337533138614474} a^{4} + \frac{5618935369303872596200296563401200182816948885384851080553560367453}{12054310478422737360826099723579380996249485473680295012599415843422} a^{3} - \frac{455357627343550566051440806026262045777898732890959746101353017947}{6027155239211368680413049861789690498124742736840147506299707921711} a^{2} - \frac{5469239293081049413178221418165015185964003464013704345691095891751}{12054310478422737360826099723579380996249485473680295012599415843422} a + \frac{242592855439764629731088189986743706722840275760739944601522168477}{6027155239211368680413049861789690498124742736840147506299707921711}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{6}\times C_{12}\times C_{7884}$, which has order $81741312$ (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:  \( 4695974.091249611 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-33}) \), 3.3.756.1, 3.3.3969.2, 6.0.36514291968.4, 6.0.4025700689472.9, 9.9.756284282720064.2

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
3Data not computed
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$