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Properties

Label 18.8.27391316277...1712.2
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 83$
Root discriminant $55.79$
Ramified primes $2, 3, 7, 41, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4544, -8448, 1968, -58160, -12272, 61776, 35307, -38698, -34315, 26362, 8287, -8638, 807, 434, -115, 64, -13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 13*x^16 + 64*x^15 - 115*x^14 + 434*x^13 + 807*x^12 - 8638*x^11 + 8287*x^10 + 26362*x^9 - 34315*x^8 - 38698*x^7 + 35307*x^6 + 61776*x^5 - 12272*x^4 - 58160*x^3 + 1968*x^2 - 8448*x + 4544)
 
gp: K = bnfinit(x^18 - 4*x^17 - 13*x^16 + 64*x^15 - 115*x^14 + 434*x^13 + 807*x^12 - 8638*x^11 + 8287*x^10 + 26362*x^9 - 34315*x^8 - 38698*x^7 + 35307*x^6 + 61776*x^5 - 12272*x^4 - 58160*x^3 + 1968*x^2 - 8448*x + 4544, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 13 x^{16} + 64 x^{15} - 115 x^{14} + 434 x^{13} + 807 x^{12} - 8638 x^{11} + 8287 x^{10} + 26362 x^{9} - 34315 x^{8} - 38698 x^{7} + 35307 x^{6} + 61776 x^{5} - 12272 x^{4} - 58160 x^{3} + 1968 x^{2} - 8448 x + 4544 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-27391316277974958297652767731712=-\,2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 83\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.79$
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:  $2, 3, 7, 41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{56} a^{15} + \frac{1}{28} a^{14} - \frac{13}{56} a^{13} - \frac{1}{28} a^{12} - \frac{11}{56} a^{11} + \frac{1}{7} a^{10} + \frac{19}{56} a^{9} - \frac{1}{2} a^{8} - \frac{25}{56} a^{7} + \frac{2}{7} a^{6} + \frac{17}{56} a^{5} + \frac{2}{7} a^{4} + \frac{19}{56} a^{3} - \frac{9}{28} a^{2} - \frac{5}{14} a - \frac{1}{7}$, $\frac{1}{336} a^{16} - \frac{17}{336} a^{14} - \frac{1}{84} a^{13} - \frac{3}{16} a^{12} + \frac{5}{56} a^{11} + \frac{1}{112} a^{10} - \frac{5}{168} a^{9} + \frac{1}{112} a^{8} + \frac{19}{168} a^{7} - \frac{71}{336} a^{6} + \frac{5}{168} a^{5} + \frac{5}{112} a^{4} - \frac{1}{12} a^{3} + \frac{13}{28} a^{2} + \frac{2}{21} a + \frac{8}{21}$, $\frac{1}{30515433651957172684325494659120472608} a^{17} - \frac{17290515060895219038381139955457913}{15257716825978586342162747329560236304} a^{16} + \frac{197986202920891514871202068692192539}{30515433651957172684325494659120472608} a^{15} - \frac{7168302636433279667046144734341663}{5085905608659528780720915776520078768} a^{14} + \frac{1800988399199066395341766371880892957}{30515433651957172684325494659120472608} a^{13} + \frac{51944610157034793648782802190078619}{847650934776588130120152629420013128} a^{12} + \frac{47978098551618927212749145075369773}{350752110942036467635925225966901984} a^{11} - \frac{22833186029466717699498246541522955}{953607301623661646385171708097514769} a^{10} + \frac{10354183511252752868222198734195090931}{30515433651957172684325494659120472608} a^{9} + \frac{922994088465156189518034659724309397}{3814429206494646585540686832390059076} a^{8} + \frac{44562056010647071136843612224848951}{484371962729478931497230073954293216} a^{7} - \frac{714328954694441213659185916673802217}{1907214603247323292770343416195029538} a^{6} - \frac{13495914047707128687505238611124565065}{30515433651957172684325494659120472608} a^{5} + \frac{1821178890496813743075889169224645027}{15257716825978586342162747329560236304} a^{4} + \frac{141127006503992972695445413538086396}{953607301623661646385171708097514769} a^{3} + \frac{45209091064356529082962324430612963}{272459229035331898967191916599289934} a^{2} - \frac{676299913269217955731548867137743913}{1907214603247323292770343416195029538} a - \frac{21736911208311764512632232799826347}{953607301623661646385171708097514769}$

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:  $12$
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:  \( 1602330059.69 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$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}$
41Data not computed
83Data not computed