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Properties

Label 18.10.8659616134...7136.3
Degree $18$
Signature $[10, 4]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}$
Root discriminant $67.59$
Ramified primes $2, 3, 7, 41$
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![-26711, 16766, -127059, -125407, -35028, 175749, 195573, -162914, -104951, 81126, 17249, -17411, -851, 1373, 134, -1, -26, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 26*x^16 - x^15 + 134*x^14 + 1373*x^13 - 851*x^12 - 17411*x^11 + 17249*x^10 + 81126*x^9 - 104951*x^8 - 162914*x^7 + 195573*x^6 + 175749*x^5 - 35028*x^4 - 125407*x^3 - 127059*x^2 + 16766*x - 26711)
 
gp: K = bnfinit(x^18 - 3*x^17 - 26*x^16 - x^15 + 134*x^14 + 1373*x^13 - 851*x^12 - 17411*x^11 + 17249*x^10 + 81126*x^9 - 104951*x^8 - 162914*x^7 + 195573*x^6 + 175749*x^5 - 35028*x^4 - 125407*x^3 - 127059*x^2 + 16766*x - 26711, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 26 x^{16} - x^{15} + 134 x^{14} + 1373 x^{13} - 851 x^{12} - 17411 x^{11} + 17249 x^{10} + 81126 x^{9} - 104951 x^{8} - 162914 x^{7} + 195573 x^{6} + 175749 x^{5} - 35028 x^{4} - 125407 x^{3} - 127059 x^{2} + 16766 x - 26711 \)

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:  \(865961613414533621361938102747136=2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.59$
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:  $2, 3, 7, 41$
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}{2014014478547898088671622077723826858222066844047} a^{17} - \frac{198022260751529425733151440682476674126208159458}{2014014478547898088671622077723826858222066844047} a^{16} - \frac{450404266003819096227780985570832993024589782426}{2014014478547898088671622077723826858222066844047} a^{15} - \frac{943942467161378398342069701591101207847307461570}{2014014478547898088671622077723826858222066844047} a^{14} - \frac{689684527049655149538336317729460830135957276106}{2014014478547898088671622077723826858222066844047} a^{13} + \frac{758450083760671073146757332006100399750629626837}{2014014478547898088671622077723826858222066844047} a^{12} - \frac{452722897101829560343788772934164654125153083393}{2014014478547898088671622077723826858222066844047} a^{11} - \frac{308100417888445650887720395131354077663166137576}{2014014478547898088671622077723826858222066844047} a^{10} - \frac{248106148775314273360705098782891214034737566744}{2014014478547898088671622077723826858222066844047} a^{9} + \frac{505474333159259321489083797377511842456918927837}{2014014478547898088671622077723826858222066844047} a^{8} - \frac{965087530043467400548294484716700989092299706269}{2014014478547898088671622077723826858222066844047} a^{7} - \frac{470646987769700753547982032161014870361384747339}{2014014478547898088671622077723826858222066844047} a^{6} - \frac{530032742831653235310937885749368040806453111582}{2014014478547898088671622077723826858222066844047} a^{5} - \frac{1000199663479017494052431844302948424559692527134}{2014014478547898088671622077723826858222066844047} a^{4} - \frac{833383542720158792799896199462565393525307244822}{2014014478547898088671622077723826858222066844047} a^{3} - \frac{29337645486582433679468355836111959641721309450}{69448775122341313402469726818062995111105753243} a^{2} + \frac{531554946682092393389776512143385066427683613618}{2014014478547898088671622077723826858222066844047} a + \frac{896777866711958426983641684343646174100978737220}{2014014478547898088671622077723826858222066844047}$

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:  \( 9322830981.08 \) (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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.64$x^{12} + 14 x^{11} + 12 x^{10} + 4 x^{9} + 10 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 16 x - 8$$4$$3$$18$12T51$[2, 2, 2, 2]^{6}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$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$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
41Data not computed