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Properties

Label 18.16.2414477422...1072.1
Degree $18$
Signature $[16, 1]$
Discriminant $-\,2^{6}\cdot 3^{18}\cdot 7^{15}\cdot 29^{5}$
Root discriminant $48.75$
Ramified primes $2, 3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78253, 7056, -426153, 173607, 683256, -523467, -264292, 296877, 48447, -79555, -13251, 17955, 1709, -2883, 99, 225, -27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 27*x^16 + 225*x^15 + 99*x^14 - 2883*x^13 + 1709*x^12 + 17955*x^11 - 13251*x^10 - 79555*x^9 + 48447*x^8 + 296877*x^7 - 264292*x^6 - 523467*x^5 + 683256*x^4 + 173607*x^3 - 426153*x^2 + 7056*x + 78253)
 
gp: K = bnfinit(x^18 - 6*x^17 - 27*x^16 + 225*x^15 + 99*x^14 - 2883*x^13 + 1709*x^12 + 17955*x^11 - 13251*x^10 - 79555*x^9 + 48447*x^8 + 296877*x^7 - 264292*x^6 - 523467*x^5 + 683256*x^4 + 173607*x^3 - 426153*x^2 + 7056*x + 78253, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 27 x^{16} + 225 x^{15} + 99 x^{14} - 2883 x^{13} + 1709 x^{12} + 17955 x^{11} - 13251 x^{10} - 79555 x^{9} + 48447 x^{8} + 296877 x^{7} - 264292 x^{6} - 523467 x^{5} + 683256 x^{4} + 173607 x^{3} - 426153 x^{2} + 7056 x + 78253 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2414477422103715329143346811072=-\,2^{6}\cdot 3^{18}\cdot 7^{15}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.75$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{6}$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{7}$, $\frac{1}{7} a^{15} - \frac{1}{7} a^{8}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{9}$, $\frac{1}{670271612131780741531081527349} a^{17} + \frac{10462477722938532105157859790}{670271612131780741531081527349} a^{16} - \frac{6171124393281017139890139785}{670271612131780741531081527349} a^{15} + \frac{523112061227795350834074455}{670271612131780741531081527349} a^{14} + \frac{30916099673570149635952915981}{670271612131780741531081527349} a^{13} - \frac{44136563221078783389892271250}{670271612131780741531081527349} a^{12} - \frac{238796023325212497124215957385}{670271612131780741531081527349} a^{11} - \frac{42622488127109708713286853355}{95753087447397248790154503907} a^{10} - \frac{1913928457646709063908932613}{7365622111338249906934961839} a^{9} - \frac{36976025107360249557026048932}{95753087447397248790154503907} a^{8} - \frac{995961647604066474564332219}{95753087447397248790154503907} a^{7} - \frac{30833636266313630611441749887}{95753087447397248790154503907} a^{6} - \frac{1664812358747822727893162530}{95753087447397248790154503907} a^{5} + \frac{5020221744169185973274731464}{13679012492485321255736357701} a^{4} - \frac{5679797719879544772584358990}{13679012492485321255736357701} a^{3} - \frac{562538188486435260482092698}{13679012492485321255736357701} a^{2} - \frac{3782428084180698617045577281}{13679012492485321255736357701} a - \frac{3264287788489103352282131614}{13679012492485321255736357701}$

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:  $16$
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:  \( 2876257708.48 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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 R R $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ $18$

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$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$