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Properties

Label 18.8.25832232271...8499.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 139$
Root discriminant $37.88$
Ramified primes $3, 7, 29, 139$
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![113, 165, -1818, 4021, -3648, -2415, 8868, -4875, -4254, 4500, 648, -1464, -111, 285, 18, -17, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 - 17*x^15 + 18*x^14 + 285*x^13 - 111*x^12 - 1464*x^11 + 648*x^10 + 4500*x^9 - 4254*x^8 - 4875*x^7 + 8868*x^6 - 2415*x^5 - 3648*x^4 + 4021*x^3 - 1818*x^2 + 165*x + 113)
 
gp: K = bnfinit(x^18 - 9*x^16 - 17*x^15 + 18*x^14 + 285*x^13 - 111*x^12 - 1464*x^11 + 648*x^10 + 4500*x^9 - 4254*x^8 - 4875*x^7 + 8868*x^6 - 2415*x^5 - 3648*x^4 + 4021*x^3 - 1818*x^2 + 165*x + 113, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{16} - 17 x^{15} + 18 x^{14} + 285 x^{13} - 111 x^{12} - 1464 x^{11} + 648 x^{10} + 4500 x^{9} - 4254 x^{8} - 4875 x^{7} + 8868 x^{6} - 2415 x^{5} - 3648 x^{4} + 4021 x^{3} - 1818 x^{2} + 165 x + 113 \)

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:  \(-25832232271583776997454218499=-\,3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 139\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.88$
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:  $3, 7, 29, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{14} a^{16} + \frac{1}{7} a^{14} + \frac{5}{14} a^{13} + \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{5}{14} a^{10} - \frac{1}{7} a^{9} - \frac{1}{2} a^{8} - \frac{3}{7} a^{7} + \frac{5}{14} a^{6} - \frac{1}{14} a^{5} + \frac{5}{14} a^{4} - \frac{3}{7} a^{3} + \frac{3}{14} a^{2} - \frac{3}{14} a + \frac{3}{14}$, $\frac{1}{276691693468234702432180532} a^{17} + \frac{628939053996072478338879}{276691693468234702432180532} a^{16} - \frac{3934496536672088514448465}{69172923367058675608045133} a^{15} + \frac{8802733887610387898472651}{276691693468234702432180532} a^{14} - \frac{15002461811546205061568397}{276691693468234702432180532} a^{13} + \frac{2738732331978940750525729}{19763692390588193030870038} a^{12} + \frac{101936393095847573577070831}{276691693468234702432180532} a^{11} + \frac{24271645227615129214362957}{276691693468234702432180532} a^{10} - \frac{23725286023413583792842173}{276691693468234702432180532} a^{9} - \frac{124957756538630376109548391}{276691693468234702432180532} a^{8} + \frac{35128065220640562533985533}{276691693468234702432180532} a^{7} + \frac{2532638741770624723032929}{9881846195294096515435019} a^{6} + \frac{11337245778631151768995248}{69172923367058675608045133} a^{5} - \frac{57253336892860283649988259}{276691693468234702432180532} a^{4} - \frac{135994547002645397234129}{9541092878214989739040708} a^{3} - \frac{40774698298426305491557583}{138345846734117351216090266} a^{2} - \frac{23870521444683689517940680}{69172923367058675608045133} a - \frac{100284043851800288300145131}{276691693468234702432180532}$

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:  \( 40203580.4379 \) (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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ $18$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$139$139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$
139.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
139.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$