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Properties

Label 18.0.44245913779...5091.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 14731^{3}$
Root discriminant $18.12$
Ramified primes $7, 14731$
Class number $1$
Class group Trivial
Galois group 18T362

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -6, 11, 5, -66, 124, 55, -100, -53, 6, 30, 31, -8, -4, 5, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 2*x^16 + 5*x^15 - 4*x^14 - 8*x^13 + 31*x^12 + 30*x^11 + 6*x^10 - 53*x^9 - 100*x^8 + 55*x^7 + 124*x^6 - 66*x^5 + 5*x^4 + 11*x^3 - 6*x^2 + 1)
 
gp: K = bnfinit(x^18 - x^17 - 2*x^16 + 5*x^15 - 4*x^14 - 8*x^13 + 31*x^12 + 30*x^11 + 6*x^10 - 53*x^9 - 100*x^8 + 55*x^7 + 124*x^6 - 66*x^5 + 5*x^4 + 11*x^3 - 6*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 2 x^{16} + 5 x^{15} - 4 x^{14} - 8 x^{13} + 31 x^{12} + 30 x^{11} + 6 x^{10} - 53 x^{9} - 100 x^{8} + 55 x^{7} + 124 x^{6} - 66 x^{5} + 5 x^{4} + 11 x^{3} - 6 x^{2} + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-44245913779931177475091=-\,7^{12}\cdot 14731^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.12$
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:  $7, 14731$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{47067244695002069} a^{17} + \frac{16265863192082529}{47067244695002069} a^{16} + \frac{7666249607392987}{47067244695002069} a^{15} - \frac{6085685935672992}{47067244695002069} a^{14} - \frac{19238080067914686}{47067244695002069} a^{13} - \frac{13068121190697700}{47067244695002069} a^{12} - \frac{178484170804804}{47067244695002069} a^{11} + \frac{10817341590572674}{47067244695002069} a^{10} - \frac{5734307503871055}{47067244695002069} a^{9} - \frac{11478671331695479}{47067244695002069} a^{8} - \frac{20430570688225965}{47067244695002069} a^{7} + \frac{22119010140921326}{47067244695002069} a^{6} - \frac{22177412261921295}{47067244695002069} a^{5} - \frac{10614580058444677}{47067244695002069} a^{4} - \frac{9573042596625882}{47067244695002069} a^{3} + \frac{6406871727131895}{47067244695002069} a^{2} + \frac{21783828513289023}{47067244695002069} a + \frac{16262410704347486}{47067244695002069}$

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:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5532.42380459 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T362:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2160
The 33 conjugacy class representatives for t18n362
Character table for t18n362 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.14731.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ $15{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $15{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ $15{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ $15{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ $15{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
$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}$
14731Data not computed