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Properties

Label 18.10.1276091230...2421.4
Degree $18$
Signature $[10, 4]$
Discriminant $7^{12}\cdot 83^{4}\cdot 181^{5}$
Root discriminant $41.40$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T705

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![491, 1740, -2424, -8900, 6765, 12006, -8603, -6035, 5641, 2419, -891, -744, -40, 304, 70, -26, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 9*x^16 - 26*x^15 + 70*x^14 + 304*x^13 - 40*x^12 - 744*x^11 - 891*x^10 + 2419*x^9 + 5641*x^8 - 6035*x^7 - 8603*x^6 + 12006*x^5 + 6765*x^4 - 8900*x^3 - 2424*x^2 + 1740*x + 491)
 
gp: K = bnfinit(x^18 - 2*x^17 - 9*x^16 - 26*x^15 + 70*x^14 + 304*x^13 - 40*x^12 - 744*x^11 - 891*x^10 + 2419*x^9 + 5641*x^8 - 6035*x^7 - 8603*x^6 + 12006*x^5 + 6765*x^4 - 8900*x^3 - 2424*x^2 + 1740*x + 491, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 9 x^{16} - 26 x^{15} + 70 x^{14} + 304 x^{13} - 40 x^{12} - 744 x^{11} - 891 x^{10} + 2419 x^{9} + 5641 x^{8} - 6035 x^{7} - 8603 x^{6} + 12006 x^{5} + 6765 x^{4} - 8900 x^{3} - 2424 x^{2} + 1740 x + 491 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(127609123015304468465889942421=7^{12}\cdot 83^{4}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.40$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{14} + \frac{3}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} - \frac{2}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7}$, $\frac{1}{21417096285887329572047397240437} a^{17} + \frac{518436827478252930643994760284}{21417096285887329572047397240437} a^{16} - \frac{100613233361629173162939185330}{1647468945068256120926722864649} a^{15} - \frac{266580261491632965183194542927}{1647468945068256120926722864649} a^{14} - \frac{1115221377028355700823352503211}{3059585183698189938863913891491} a^{13} + \frac{1905651495572784131880378032298}{21417096285887329572047397240437} a^{12} - \frac{9279922671082200851777387256836}{21417096285887329572047397240437} a^{11} + \frac{663995065740089086985757979944}{1647468945068256120926722864649} a^{10} + \frac{343161910230830844437942661392}{3059585183698189938863913891491} a^{9} + \frac{8615917288663641086498516355770}{21417096285887329572047397240437} a^{8} + \frac{7146077726981454098293971732829}{21417096285887329572047397240437} a^{7} + \frac{5341507649340175128202319653998}{21417096285887329572047397240437} a^{6} - \frac{862021367061968563062570208051}{21417096285887329572047397240437} a^{5} - \frac{7150111648319230989883474094778}{21417096285887329572047397240437} a^{4} - \frac{708799542666868992044979170822}{1647468945068256120926722864649} a^{3} - \frac{9666591760395719659258680685645}{21417096285887329572047397240437} a^{2} + \frac{8776030842852658689535599927276}{21417096285887329572047397240437} a - \frac{4472821689518205013242214587151}{21417096285887329572047397240437}$

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:  $13$
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:  \( 118123305.523 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T705:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 64 conjugacy class representatives for t18n705 are not computed
Character table for t18n705 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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

Sibling fields

Degree 18 siblings: 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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $18$ $18$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
83Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$