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Properties

Label 18.0.23427723931...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{30}\cdot 5^{9}\cdot 7^{14}$
Root discriminant $225.88$
Ramified primes $2, 3, 5, 7$
Class number $31564512$ (GRH)
Class group $[2, 2, 6, 6, 219198]$ (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![17320048361, 4225615884, 9411997182, 1548565326, 2169968997, 206392410, 286583163, 9022440, 24975411, -636788, 1570920, -114570, 73002, -8274, 2397, -342, 57, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 57*x^16 - 342*x^15 + 2397*x^14 - 8274*x^13 + 73002*x^12 - 114570*x^11 + 1570920*x^10 - 636788*x^9 + 24975411*x^8 + 9022440*x^7 + 286583163*x^6 + 206392410*x^5 + 2169968997*x^4 + 1548565326*x^3 + 9411997182*x^2 + 4225615884*x + 17320048361)
 
gp: K = bnfinit(x^18 - 6*x^17 + 57*x^16 - 342*x^15 + 2397*x^14 - 8274*x^13 + 73002*x^12 - 114570*x^11 + 1570920*x^10 - 636788*x^9 + 24975411*x^8 + 9022440*x^7 + 286583163*x^6 + 206392410*x^5 + 2169968997*x^4 + 1548565326*x^3 + 9411997182*x^2 + 4225615884*x + 17320048361, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 57 x^{16} - 342 x^{15} + 2397 x^{14} - 8274 x^{13} + 73002 x^{12} - 114570 x^{11} + 1570920 x^{10} - 636788 x^{9} + 24975411 x^{8} + 9022440 x^{7} + 286583163 x^{6} + 206392410 x^{5} + 2169968997 x^{4} + 1548565326 x^{3} + 9411997182 x^{2} + 4225615884 x + 17320048361 \)

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:  \(-2342772393121389342665065120137216000000000=-\,2^{33}\cdot 3^{30}\cdot 5^{9}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $225.88$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{1605183192805670646384870778765366721055940601114464668240846} a^{17} - \frac{44015597884855040212993529835772631222845059107479357917339}{535061064268556882128290259588455573685313533704821556080282} a^{16} + \frac{49572488551400101408161933725707047494896552185198356162549}{802591596402835323192435389382683360527970300557232334120423} a^{15} + \frac{13228469605834518855868006944301354868928966679286750367554}{802591596402835323192435389382683360527970300557232334120423} a^{14} - \frac{648298198901686265344662499528242689143373848771449069381}{23263524533415516614273489547324155377622327552383545916534} a^{13} - \frac{22059323307103415310812648109581240878243441250964316390813}{1605183192805670646384870778765366721055940601114464668240846} a^{12} - \frac{48647551007473259875093065531079701558112396396931739407725}{1605183192805670646384870778765366721055940601114464668240846} a^{11} + \frac{34519060213759439863191671794577409988827003708976969524679}{1605183192805670646384870778765366721055940601114464668240846} a^{10} - \frac{87011294896229279798873653483010655853789436994061280785879}{535061064268556882128290259588455573685313533704821556080282} a^{9} - \frac{102613671056890825813527994870466981990249598383308986536669}{267530532134278441064145129794227786842656766852410778040141} a^{8} - \frac{27227374034955549642692752659371015412556486862369153069669}{267530532134278441064145129794227786842656766852410778040141} a^{7} - \frac{48724975623178381771858952416135188190169728771733658232087}{1605183192805670646384870778765366721055940601114464668240846} a^{6} + \frac{29641530142344608366235398475311513581104457271793412151671}{267530532134278441064145129794227786842656766852410778040141} a^{5} + \frac{206488230267402486407198616141486888968578686974412433226672}{802591596402835323192435389382683360527970300557232334120423} a^{4} - \frac{132843029526544362731916075540658628957603818438794469062783}{267530532134278441064145129794227786842656766852410778040141} a^{3} - \frac{74770647947551445537839388826859381452781402318709085352229}{535061064268556882128290259588455573685313533704821556080282} a^{2} + \frac{194569657363071333725531644110312484774561743923911185701681}{802591596402835323192435389382683360527970300557232334120423} a - \frac{16767079556031820915558003216582875743530391968485394345869}{267530532134278441064145129794227786842656766852410778040141}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{219198}$, which has order $31564512$ (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

\(\Q(\sqrt{-10}) \), 3.3.3969.2, 3.3.756.1, 6.0.1008189504000.20, 6.0.9144576000.34, 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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.1$x^{3} + 14$$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}$