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Properties

Label 18.0.39938750212...2896.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 31^{9}\cdot 79^{14}$
Root discriminant $264.42$
Ramified primes $2, 31, 79$
Class number $25886952$ (GRH)
Class group $[3, 8628984]$ (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![9397502237, -293112723, 3333390128, -329016700, 603793415, -71250155, 73631164, -9409373, 6431560, -819803, 448088, -56341, 22318, -3133, 1015, -98, 26, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 26*x^16 - 98*x^15 + 1015*x^14 - 3133*x^13 + 22318*x^12 - 56341*x^11 + 448088*x^10 - 819803*x^9 + 6431560*x^8 - 9409373*x^7 + 73631164*x^6 - 71250155*x^5 + 603793415*x^4 - 329016700*x^3 + 3333390128*x^2 - 293112723*x + 9397502237)
 
gp: K = bnfinit(x^18 - 5*x^17 + 26*x^16 - 98*x^15 + 1015*x^14 - 3133*x^13 + 22318*x^12 - 56341*x^11 + 448088*x^10 - 819803*x^9 + 6431560*x^8 - 9409373*x^7 + 73631164*x^6 - 71250155*x^5 + 603793415*x^4 - 329016700*x^3 + 3333390128*x^2 - 293112723*x + 9397502237, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 26 x^{16} - 98 x^{15} + 1015 x^{14} - 3133 x^{13} + 22318 x^{12} - 56341 x^{11} + 448088 x^{10} - 819803 x^{9} + 6431560 x^{8} - 9409373 x^{7} + 73631164 x^{6} - 71250155 x^{5} + 603793415 x^{4} - 329016700 x^{3} + 3333390128 x^{2} - 293112723 x + 9397502237 \)

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:  \(-39938750212536083623342446592174713370832896=-\,2^{12}\cdot 31^{9}\cdot 79^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $264.42$
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:  $2, 31, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{3} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3583422} a^{16} + \frac{3019}{1194474} a^{15} - \frac{40043}{1194474} a^{14} - \frac{167701}{1791711} a^{13} - \frac{6057}{199079} a^{12} + \frac{113113}{597237} a^{11} - \frac{803155}{1791711} a^{10} - \frac{402388}{1791711} a^{9} - \frac{422564}{1791711} a^{8} - \frac{74977}{199079} a^{7} - \frac{31610}{199079} a^{6} - \frac{239782}{1791711} a^{5} + \frac{64850}{597237} a^{4} - \frac{466751}{1791711} a^{3} + \frac{124143}{398158} a^{2} + \frac{1577599}{3583422} a - \frac{838529}{3583422}$, $\frac{1}{3119483810763033967575609799768166194816405634549546972463177498} a^{17} - \frac{12602472671875693974866574890976868303091327565517132121}{1039827936921011322525203266589388731605468544849848990821059166} a^{16} - \frac{25080200465577551477162496013545907970123215795180607802091811}{346609312307003774175067755529796243868489514949949663607019722} a^{15} - \frac{80886504462182647376170934868295147152701673165695439838955062}{1559741905381516983787804899884083097408202817274773486231588749} a^{14} - \frac{51472152235527743827896117377475980389157790672390851315968657}{519913968460505661262601633294694365802734272424924495410529583} a^{13} + \frac{47623362468095420470728266403147495863650271976561995587492184}{519913968460505661262601633294694365802734272424924495410529583} a^{12} + \frac{23764551328932789119161546101696188018333264789706025241911541}{1559741905381516983787804899884083097408202817274773486231588749} a^{11} - \frac{642587699624087999368514250461351763761010269767017857943140383}{1559741905381516983787804899884083097408202817274773486231588749} a^{10} + \frac{45536977252810279389662051015552142312158208216915806978793922}{1559741905381516983787804899884083097408202817274773486231588749} a^{9} - \frac{100649556974504519683585306940812966358296632317852794339139077}{519913968460505661262601633294694365802734272424924495410529583} a^{8} + \frac{259431502286395530157091524882979470435117470192413369870992772}{519913968460505661262601633294694365802734272424924495410529583} a^{7} - \frac{248713380100386520057174812266228099727340969971381124340634382}{1559741905381516983787804899884083097408202817274773486231588749} a^{6} + \frac{71218032362658861883713676382681400952751611304453567973839341}{519913968460505661262601633294694365802734272424924495410529583} a^{5} - \frac{298510551231283810613548692003237204473588553416375541874194245}{1559741905381516983787804899884083097408202817274773486231588749} a^{4} - \frac{121489988536141814703599543328059642433684313032773506625811073}{346609312307003774175067755529796243868489514949949663607019722} a^{3} + \frac{1307412821796309424630803859550402939842046966072950739674642465}{3119483810763033967575609799768166194816405634549546972463177498} a^{2} - \frac{966389530198520512389007758473690032690572231010732151578228997}{3119483810763033967575609799768166194816405634549546972463177498} a - \frac{28296561360544872531502176602147710216762260602706633905050752}{173304656153501887087533877764898121934244757474974831803509861}$

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

Class group and class number

$C_{3}\times C_{8628984}$, which has order $25886952$ (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:  \( 9044146.559729666 \) (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

\(\Q(\sqrt{-31}) \), 3.3.6241.1, 3.3.316.1, 6.0.1160361863071.1, 6.0.2974810096.1, 9.9.1229050175114176.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.12.6.1$x^{12} + 178746 x^{6} - 114516604 x^{2} + 7987533129$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$79$79.6.4.1$x^{6} + 632 x^{3} + 168507$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
79.12.10.1$x^{12} - 790 x^{6} + 4549689$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$