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Properties

Label 18.6.14266096289...2561.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 43^{3}\cdot 181\cdot 894708527^{3}$
Root discriminant $283.80$
Ramified primes $7, 43, 181, 894708527$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T926

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-989652329, -546084423, 291125718, 921309390, 469197468, -62244207, -93602247, -19473048, 3818016, 3595266, 634635, -242028, -87018, 7290, 4482, -81, -108, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 108*x^16 - 81*x^15 + 4482*x^14 + 7290*x^13 - 87018*x^12 - 242028*x^11 + 634635*x^10 + 3595266*x^9 + 3818016*x^8 - 19473048*x^7 - 93602247*x^6 - 62244207*x^5 + 469197468*x^4 + 921309390*x^3 + 291125718*x^2 - 546084423*x - 989652329)
 
gp: K = bnfinit(x^18 - 108*x^16 - 81*x^15 + 4482*x^14 + 7290*x^13 - 87018*x^12 - 242028*x^11 + 634635*x^10 + 3595266*x^9 + 3818016*x^8 - 19473048*x^7 - 93602247*x^6 - 62244207*x^5 + 469197468*x^4 + 921309390*x^3 + 291125718*x^2 - 546084423*x - 989652329, 1)
 

Normalized defining polynomial

\( x^{18} - 108 x^{16} - 81 x^{15} + 4482 x^{14} + 7290 x^{13} - 87018 x^{12} - 242028 x^{11} + 634635 x^{10} + 3595266 x^{9} + 3818016 x^{8} - 19473048 x^{7} - 93602247 x^{6} - 62244207 x^{5} + 469197468 x^{4} + 921309390 x^{3} + 291125718 x^{2} - 546084423 x - 989652329 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(142660962891827847357688087745406439211942561=7^{12}\cdot 43^{3}\cdot 181\cdot 894708527^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $283.80$
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:  $7, 43, 181, 894708527$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{4} - \frac{1}{9} a^{2} - \frac{2}{9}$, $\frac{1}{9} a^{5} - \frac{1}{9} a^{3} - \frac{2}{9} a$, $\frac{1}{27} a^{6} - \frac{1}{9} a^{2} - \frac{2}{27}$, $\frac{1}{27} a^{7} - \frac{1}{9} a^{3} - \frac{2}{27} a$, $\frac{1}{81} a^{8} + \frac{1}{81} a^{6} - \frac{1}{27} a^{4} - \frac{5}{81} a^{2} - \frac{2}{81}$, $\frac{1}{81} a^{9} + \frac{1}{81} a^{7} - \frac{1}{27} a^{5} - \frac{5}{81} a^{3} - \frac{2}{81} a$, $\frac{1}{243} a^{10} - \frac{1}{243} a^{8} + \frac{4}{243} a^{6} + \frac{1}{243} a^{4} - \frac{19}{243} a^{2} - \frac{14}{243}$, $\frac{1}{243} a^{11} - \frac{1}{243} a^{9} + \frac{4}{243} a^{7} + \frac{1}{243} a^{5} - \frac{19}{243} a^{3} - \frac{14}{243} a$, $\frac{1}{729} a^{12} + \frac{1}{243} a^{8} + \frac{5}{729} a^{6} - \frac{2}{81} a^{4} - \frac{11}{243} a^{2} - \frac{14}{729}$, $\frac{1}{729} a^{13} + \frac{1}{243} a^{9} + \frac{5}{729} a^{7} - \frac{2}{81} a^{5} - \frac{11}{243} a^{3} - \frac{14}{729} a$, $\frac{1}{2187} a^{14} + \frac{1}{2187} a^{12} + \frac{1}{729} a^{10} + \frac{8}{2187} a^{8} - \frac{13}{2187} a^{6} - \frac{17}{729} a^{4} - \frac{47}{2187} a^{2} - \frac{14}{2187}$, $\frac{1}{2187} a^{15} + \frac{1}{2187} a^{13} + \frac{1}{729} a^{11} + \frac{8}{2187} a^{9} - \frac{13}{2187} a^{7} - \frac{17}{729} a^{5} - \frac{47}{2187} a^{3} - \frac{14}{2187} a$, $\frac{1}{6561} a^{16} - \frac{1}{6561} a^{14} + \frac{1}{6561} a^{12} + \frac{2}{6561} a^{10} - \frac{29}{6561} a^{8} - \frac{25}{6561} a^{6} + \frac{55}{6561} a^{4} + \frac{80}{6561} a^{2} + \frac{28}{6561}$, $\frac{1}{87321085692795868937157717662346267538953} a^{17} + \frac{797403405425871061298247894772485475}{29107028564265289645719239220782089179651} a^{16} + \frac{11054744751920358951896037136378161332}{87321085692795868937157717662346267538953} a^{15} - \frac{1190098026120117367822121495587790072}{9702342854755096548573079740260696393217} a^{14} + \frac{55762634566013565787806175187836480756}{87321085692795868937157717662346267538953} a^{13} + \frac{9507086590375692447816838194376818452}{29107028564265289645719239220782089179651} a^{12} + \frac{74628515209341801865499658980564689325}{87321085692795868937157717662346267538953} a^{11} + \frac{49026358241729841542433929709762939878}{29107028564265289645719239220782089179651} a^{10} + \frac{475877339832064151143454201155469012788}{87321085692795868937157717662346267538953} a^{9} - \frac{44140464979800494908028756205800982707}{9702342854755096548573079740260696393217} a^{8} - \frac{1495288743145305024959909695515436063600}{87321085692795868937157717662346267538953} a^{7} + \frac{257979997796946320437443881480540744506}{29107028564265289645719239220782089179651} a^{6} - \frac{4829732640341675508439655958484422083420}{87321085692795868937157717662346267538953} a^{5} - \frac{759459069320067316733288093851550598776}{29107028564265289645719239220782089179651} a^{4} - \frac{10292188744082239669023488653359475330105}{87321085692795868937157717662346267538953} a^{3} + \frac{163673312667141104790496256767352765764}{1078038094972788505397008860028966265913} a^{2} - \frac{390880608990263231719981147462931087117}{87321085692795868937157717662346267538953} a - \frac{5893450966029429018791953528701621028858}{29107028564265289645719239220782089179651}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 1032214003040000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T926:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1119744
The 267 conjugacy class representatives for t18n926 are not computed
Character table for t18n926 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.92372392453061.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ R $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.6.3.1$x^{6} - 86 x^{4} + 1849 x^{2} - 7950700$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
181Data not computed
894708527Data not computed