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Properties

Label 18.0.46287577738...8023.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 13^{15}\cdot 17^{9}$
Root discriminant $181.63$
Ramified primes $3, 13, 17$
Class number $27813184$ (GRH)
Class group $[2, 14, 14, 70952]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![283790949688, 65824912344, 82416559734, 4904628409, 22810900296, -2896022691, 3633489894, -516066921, 387929952, -33350634, 29343633, -850383, 1346987, -5688, 31761, -4, 318, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 318*x^16 - 4*x^15 + 31761*x^14 - 5688*x^13 + 1346987*x^12 - 850383*x^11 + 29343633*x^10 - 33350634*x^9 + 387929952*x^8 - 516066921*x^7 + 3633489894*x^6 - 2896022691*x^5 + 22810900296*x^4 + 4904628409*x^3 + 82416559734*x^2 + 65824912344*x + 283790949688)
 
gp: K = bnfinit(x^18 + 318*x^16 - 4*x^15 + 31761*x^14 - 5688*x^13 + 1346987*x^12 - 850383*x^11 + 29343633*x^10 - 33350634*x^9 + 387929952*x^8 - 516066921*x^7 + 3633489894*x^6 - 2896022691*x^5 + 22810900296*x^4 + 4904628409*x^3 + 82416559734*x^2 + 65824912344*x + 283790949688, 1)
 

Normalized defining polynomial

\( x^{18} + 318 x^{16} - 4 x^{15} + 31761 x^{14} - 5688 x^{13} + 1346987 x^{12} - 850383 x^{11} + 29343633 x^{10} - 33350634 x^{9} + 387929952 x^{8} - 516066921 x^{7} + 3633489894 x^{6} - 2896022691 x^{5} + 22810900296 x^{4} + 4904628409 x^{3} + 82416559734 x^{2} + 65824912344 x + 283790949688 \)

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:  \(-46287577738090612111387516950954005968023=-\,3^{27}\cdot 13^{15}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $181.63$
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:  $3, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1989=3^{2}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1989}(256,·)$, $\chi_{1989}(1,·)$, $\chi_{1989}(1988,·)$, $\chi_{1989}(1733,·)$, $\chi_{1989}(1225,·)$, $\chi_{1989}(407,·)$, $\chi_{1989}(1427,·)$, $\chi_{1989}(1070,·)$, $\chi_{1989}(662,·)$, $\chi_{1989}(919,·)$, $\chi_{1989}(664,·)$, $\chi_{1989}(1888,·)$, $\chi_{1989}(101,·)$, $\chi_{1989}(1325,·)$, $\chi_{1989}(1582,·)$, $\chi_{1989}(1327,·)$, $\chi_{1989}(562,·)$, $\chi_{1989}(764,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{751316} a^{15} + \frac{4}{73} a^{14} + \frac{9050}{187829} a^{13} - \frac{150663}{751316} a^{12} - \frac{36461}{375658} a^{11} - \frac{99}{751316} a^{10} + \frac{86715}{751316} a^{9} + \frac{116159}{751316} a^{8} + \frac{34743}{375658} a^{7} + \frac{29375}{187829} a^{6} - \frac{55157}{751316} a^{5} + \frac{13030}{187829} a^{4} + \frac{93151}{751316} a^{3} + \frac{34419}{751316} a^{2} - \frac{51603}{375658} a + \frac{1006}{2573}$, $\frac{1}{5259212} a^{16} - \frac{1}{2629606} a^{15} - \frac{158195}{1314803} a^{14} + \frac{15183}{751316} a^{13} - \frac{258883}{2629606} a^{12} - \frac{435753}{5259212} a^{11} - \frac{345351}{5259212} a^{10} - \frac{438075}{5259212} a^{9} + \frac{170278}{1314803} a^{8} - \frac{465935}{1314803} a^{7} + \frac{349645}{751316} a^{6} + \frac{3300}{1314803} a^{5} + \frac{1198221}{5259212} a^{4} - \frac{1026703}{5259212} a^{3} + \frac{337793}{1314803} a^{2} - \frac{59327}{187829} a - \frac{7158}{18011}$, $\frac{1}{89531084893328285890185140493436632472513609314720122589096584531294109412203858884} a^{17} + \frac{2472505217054933845203662893406998776104151803406210548651591217101606824351}{44765542446664142945092570246718316236256804657360061294548292265647054706101929442} a^{16} + \frac{32606666067094612941476083280427774779466039396946077128229999914666291276045}{89531084893328285890185140493436632472513609314720122589096584531294109412203858884} a^{15} - \frac{5669457449371573011795130508755190726242922125957483387701021551022336718974156345}{89531084893328285890185140493436632472513609314720122589096584531294109412203858884} a^{14} - \frac{9996606181551703028349042134357041516285548040528233579609076828665617125185969325}{89531084893328285890185140493436632472513609314720122589096584531294109412203858884} a^{13} + \frac{384478510741788280994379370687362899143958150058690146791917183616804366799396425}{6395077492380591849298938606674045176608114951051437327792613180806722100871704206} a^{12} + \frac{3756880075079090897506815676831429651110722954283617506843684261680505957308205473}{44765542446664142945092570246718316236256804657360061294548292265647054706101929442} a^{11} + \frac{575604979560178796447386642682769815916422572319981453522091411163125835050148097}{12790154984761183698597877213348090353216229902102874655585226361613444201743408412} a^{10} + \frac{4504735399511795895520766634798771838296940587537576246887406422409943470085877573}{22382771223332071472546285123359158118128402328680030647274146132823527353050964721} a^{9} + \frac{7542257991459866732729784264517893802517019292066396707236664150744054400346580533}{89531084893328285890185140493436632472513609314720122589096584531294109412203858884} a^{8} - \frac{3347406772345545713791599949140625071284995316925148630094165197265969803672249409}{89531084893328285890185140493436632472513609314720122589096584531294109412203858884} a^{7} - \frac{32942587197341150992151163552186615874082184738362253419837303173823730903236912125}{89531084893328285890185140493436632472513609314720122589096584531294109412203858884} a^{6} + \frac{2116866690066476246012932520715449386150354303992353772320486517887288276005897285}{22382771223332071472546285123359158118128402328680030647274146132823527353050964721} a^{5} - \frac{2118124474978778188949204559620064761441296257794305352457386632854615076201189483}{44765542446664142945092570246718316236256804657360061294548292265647054706101929442} a^{4} - \frac{4150002396457836350581015757075094484464355928595309408823798743680846505901938509}{22382771223332071472546285123359158118128402328680030647274146132823527353050964721} a^{3} - \frac{6666866289545450893215636559844920694713880975625389318001398105324836313275359775}{22382771223332071472546285123359158118128402328680030647274146132823527353050964721} a^{2} - \frac{2757737862404608549974260436895810448125235929701199089310827810677916668566539965}{44765542446664142945092570246718316236256804657360061294548292265647054706101929442} a - \frac{77147450954725100605659981059493270172259739119928355645956233592630462416977366}{306613304429206458528031303059714494768882223680548365031152686751007224014396777}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{14}\times C_{70952}$, which has order $27813184$ (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:  \( 400417.1364448253 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-663}) \), 3.3.13689.2, \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.169.1, 6.0.35904990664647.3, 6.0.212455566063.4, 6.0.35904990664647.2, 6.0.49252387743.2, 9.9.2565164201769.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
13Data not computed
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$