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Properties

Label 18.0.12152532964...7808.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 7^{12}\cdot 17^{9}$
Root discriminant $130.56$
Ramified primes $2, 3, 7, 17$
Class number $5458752$ (GRH)
Class group $[2, 18, 18, 18, 468]$ (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![512126566568, 16896372024, 181053453228, 4777613720, 29298758262, 575805318, 2881098747, 37069122, 192211413, 1266502, 9143631, 15918, 314571, -264, 7677, -8, 123, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 123*x^16 - 8*x^15 + 7677*x^14 - 264*x^13 + 314571*x^12 + 15918*x^11 + 9143631*x^10 + 1266502*x^9 + 192211413*x^8 + 37069122*x^7 + 2881098747*x^6 + 575805318*x^5 + 29298758262*x^4 + 4777613720*x^3 + 181053453228*x^2 + 16896372024*x + 512126566568)
 
gp: K = bnfinit(x^18 + 123*x^16 - 8*x^15 + 7677*x^14 - 264*x^13 + 314571*x^12 + 15918*x^11 + 9143631*x^10 + 1266502*x^9 + 192211413*x^8 + 37069122*x^7 + 2881098747*x^6 + 575805318*x^5 + 29298758262*x^4 + 4777613720*x^3 + 181053453228*x^2 + 16896372024*x + 512126566568, 1)
 

Normalized defining polynomial

\( x^{18} + 123 x^{16} - 8 x^{15} + 7677 x^{14} - 264 x^{13} + 314571 x^{12} + 15918 x^{11} + 9143631 x^{10} + 1266502 x^{9} + 192211413 x^{8} + 37069122 x^{7} + 2881098747 x^{6} + 575805318 x^{5} + 29298758262 x^{4} + 4777613720 x^{3} + 181053453228 x^{2} + 16896372024 x + 512126566568 \)

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:  \(-121525329645053177620052260620194807808=-\,2^{18}\cdot 3^{24}\cdot 7^{12}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.56$
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:  $2, 3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4284=2^{2}\cdot 3^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4284}(1,·)$, $\chi_{4284}(67,·)$, $\chi_{4284}(2311,·)$, $\chi_{4284}(3469,·)$, $\chi_{4284}(205,·)$, $\chi_{4284}(2515,·)$, $\chi_{4284}(1429,·)$, $\chi_{4284}(1495,·)$, $\chi_{4284}(3739,·)$, $\chi_{4284}(1633,·)$, $\chi_{4284}(613,·)$, $\chi_{4284}(3943,·)$, $\chi_{4284}(2857,·)$, $\chi_{4284}(2923,·)$, $\chi_{4284}(883,·)$, $\chi_{4284}(3061,·)$, $\chi_{4284}(2041,·)$, $\chi_{4284}(1087,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2032} a^{16} - \frac{69}{1016} a^{15} + \frac{165}{2032} a^{14} + \frac{285}{1016} a^{13} + \frac{183}{2032} a^{12} + \frac{159}{1016} a^{11} - \frac{183}{2032} a^{10} - \frac{121}{254} a^{9} - \frac{283}{2032} a^{8} - \frac{187}{508} a^{7} + \frac{955}{2032} a^{6} - \frac{177}{508} a^{5} + \frac{149}{2032} a^{4} + \frac{211}{508} a^{3} - \frac{217}{508} a^{2} + \frac{61}{254} a - \frac{139}{508}$, $\frac{1}{1032059654733219313211429795374860741941256856126863115925986896212528} a^{17} - \frac{26091454527676744785730138892359665688624261403343759589176020161}{129007456841652414151428724421857592742657107015857889490748362026566} a^{16} - \frac{62025463123225562550826001022075060247938812041911158123686528437723}{1032059654733219313211429795374860741941256856126863115925986896212528} a^{15} - \frac{13178720765486898667719576568055923610695801861863178505436878213513}{258014913683304828302857448843715185485314214031715778981496724053132} a^{14} - \frac{381030247134647089111404040358681939220794490856645634104454707998105}{1032059654733219313211429795374860741941256856126863115925986896212528} a^{13} + \frac{112643673358775515235583709177132358281500163976564697264372947160273}{258014913683304828302857448843715185485314214031715778981496724053132} a^{12} - \frac{290492408478433984786285275366535504586946096404007196385519961659671}{1032059654733219313211429795374860741941256856126863115925986896212528} a^{11} + \frac{166895297742063388742822397583401244615837178439309188953188312862241}{516029827366609656605714897687430370970628428063431557962993448106264} a^{10} + \frac{399880850496766125480995712039009193561983284362055319474796600322513}{1032059654733219313211429795374860741941256856126863115925986896212528} a^{9} + \frac{68170510106389879608744539697380399522078386123303565465442334078167}{516029827366609656605714897687430370970628428063431557962993448106264} a^{8} + \frac{419896065887096660655182950725897937725259954554445433264019856711583}{1032059654733219313211429795374860741941256856126863115925986896212528} a^{7} + \frac{59775510522215790699366070645535391803562069482943860065219949735257}{516029827366609656605714897687430370970628428063431557962993448106264} a^{6} - \frac{154380154370286361396772058094635349888264971158116834185161951507743}{1032059654733219313211429795374860741941256856126863115925986896212528} a^{5} - \frac{211651629022488972757875672117342908529937962305925514520814910605893}{516029827366609656605714897687430370970628428063431557962993448106264} a^{4} - \frac{10682644495497029963979377208283135121500763295024663179852066806247}{64503728420826207075714362210928796371328553507928944745374181013283} a^{3} + \frac{25736667668895998597616740298886845329374378240708506800265323443620}{64503728420826207075714362210928796371328553507928944745374181013283} a^{2} + \frac{75185356651457161045577186167742175702904408519996574719129167214955}{258014913683304828302857448843715185485314214031715778981496724053132} a - \frac{11542819733573866139230363399214947848560602087407491734504144552523}{129007456841652414151428724421857592742657107015857889490748362026566}$

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

Class group and class number

$C_{2}\times C_{18}\times C_{18}\times C_{18}\times C_{468}$, which has order $5458752$ (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:  \( 54408.48888868202 \) (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

\(\Q(\sqrt{-17}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), 6.0.2062988352.6, 6.0.4953235033152.4, 6.0.4953235033152.5, 6.0.754951232.2, 9.9.62523502209.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.3$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}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$17$17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$