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Properties

Label 18.4.12565843477...9184.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,2^{12}\cdot 7^{12}\cdot 53^{6}$
Root discriminant $21.82$
Ramified primes $2, 7, 53$
Class number $1$
Class group Trivial
Galois group 18T367

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 23, 0, 105, 0, 115, 0, -54, 0, -135, 0, -41, 0, 17, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 + 17*x^14 - 41*x^12 - 135*x^10 - 54*x^8 + 115*x^6 + 105*x^4 + 23*x^2 + 1)
 
gp: K = bnfinit(x^18 + 9*x^16 + 17*x^14 - 41*x^12 - 135*x^10 - 54*x^8 + 115*x^6 + 105*x^4 + 23*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} + 17 x^{14} - 41 x^{12} - 135 x^{10} - 54 x^{8} + 115 x^{6} + 105 x^{4} + 23 x^{2} + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1256584347701942722269184=-\,2^{12}\cdot 7^{12}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.82$
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:  $2, 7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} + \frac{4}{9} a^{8} + \frac{2}{9} a^{6} - \frac{1}{9} a^{4} + \frac{2}{9} a^{2} + \frac{4}{9}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{14} - \frac{1}{6} a^{12} + \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{5}{18} a^{9} - \frac{2}{9} a^{8} + \frac{1}{9} a^{7} - \frac{5}{18} a^{6} - \frac{1}{18} a^{5} - \frac{5}{18} a^{4} - \frac{7}{18} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{5}{18}$, $\frac{1}{1746} a^{16} - \frac{11}{1746} a^{14} - \frac{1}{6} a^{13} - \frac{124}{873} a^{12} - \frac{14}{873} a^{10} - \frac{1}{2} a^{9} - \frac{10}{291} a^{8} - \frac{1}{6} a^{7} - \frac{397}{873} a^{6} - \frac{1}{3} a^{5} - \frac{199}{873} a^{4} - \frac{1}{6} a^{3} + \frac{67}{291} a^{2} - \frac{1}{2} a - \frac{839}{1746}$, $\frac{1}{1746} a^{17} - \frac{11}{1746} a^{15} - \frac{1}{18} a^{14} - \frac{124}{873} a^{13} - \frac{14}{873} a^{11} - \frac{1}{18} a^{10} - \frac{10}{291} a^{9} + \frac{5}{18} a^{8} - \frac{397}{873} a^{7} - \frac{1}{9} a^{6} - \frac{199}{873} a^{5} + \frac{1}{18} a^{4} + \frac{67}{291} a^{3} + \frac{7}{18} a^{2} - \frac{839}{1746} a - \frac{2}{9}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 71037.0414965 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T367:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n367
Character table for t18n367 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.5$x^{12} + 52 x^{10} - 11 x^{8} - 8 x^{6} - 45 x^{4} - 44 x^{2} - 9$$2$$6$$12$12T51$[2, 2, 2, 2]^{6}$
7Data not computed
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.12.6.1$x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$