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Properties

Label 18.14.5838293633...4521.1
Degree $18$
Signature $[14, 2]$
Discriminant $7^{14}\cdot 13^{2}\cdot 83^{4}\cdot 181^{4}$
Root discriminant $51.20$
Ramified primes $7, 13, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T646

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121807, -126112, -395780, 765309, -257587, -405001, 409413, -52634, -116233, 66167, -1385, -10341, 3770, -281, -271, 129, -13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 13*x^16 + 129*x^15 - 271*x^14 - 281*x^13 + 3770*x^12 - 10341*x^11 - 1385*x^10 + 66167*x^9 - 116233*x^8 - 52634*x^7 + 409413*x^6 - 405001*x^5 - 257587*x^4 + 765309*x^3 - 395780*x^2 - 126112*x + 121807)
 
gp: K = bnfinit(x^18 - 5*x^17 - 13*x^16 + 129*x^15 - 271*x^14 - 281*x^13 + 3770*x^12 - 10341*x^11 - 1385*x^10 + 66167*x^9 - 116233*x^8 - 52634*x^7 + 409413*x^6 - 405001*x^5 - 257587*x^4 + 765309*x^3 - 395780*x^2 - 126112*x + 121807, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 13 x^{16} + 129 x^{15} - 271 x^{14} - 281 x^{13} + 3770 x^{12} - 10341 x^{11} - 1385 x^{10} + 66167 x^{9} - 116233 x^{8} - 52634 x^{7} + 409413 x^{6} - 405001 x^{5} - 257587 x^{4} + 765309 x^{3} - 395780 x^{2} - 126112 x + 121807 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5838293633644951952298533774521=7^{14}\cdot 13^{2}\cdot 83^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.20$
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:  $7, 13, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{13} - \frac{3}{7} a^{11} + \frac{1}{7} a^{9} + \frac{2}{7} a^{7} - \frac{3}{7} a^{5} + \frac{1}{7} a^{3}$, $\frac{1}{1597687} a^{16} - \frac{25342}{1597687} a^{15} - \frac{748809}{1597687} a^{14} + \frac{338187}{1597687} a^{13} + \frac{716461}{1597687} a^{12} + \frac{100554}{1597687} a^{11} - \frac{545551}{1597687} a^{10} - \frac{657498}{1597687} a^{9} + \frac{114018}{1597687} a^{8} - \frac{89366}{1597687} a^{7} - \frac{576999}{1597687} a^{6} - \frac{11479}{122899} a^{5} + \frac{530818}{1597687} a^{4} + \frac{431016}{1597687} a^{3} + \frac{70219}{228241} a^{2} - \frac{41984}{228241} a + \frac{12020}{228241}$, $\frac{1}{508641919686732195001764319921} a^{17} - \frac{152966185803646694873129}{508641919686732195001764319921} a^{16} - \frac{6177547584295738907782368174}{508641919686732195001764319921} a^{15} + \frac{67856182493561213553014214748}{508641919686732195001764319921} a^{14} - \frac{70103571680551983807776528775}{508641919686732195001764319921} a^{13} + \frac{88270388908328033787044752188}{508641919686732195001764319921} a^{12} + \frac{46181993178768444012580970974}{508641919686732195001764319921} a^{11} - \frac{97587837662818195245076580113}{508641919686732195001764319921} a^{10} - \frac{192496191441560257935431203795}{508641919686732195001764319921} a^{9} - \frac{3088458439533185540768227867}{12405900480164199878091812681} a^{8} - \frac{20414970507148647677784032894}{508641919686732195001764319921} a^{7} + \frac{31998679198448226341722715780}{508641919686732195001764319921} a^{6} - \frac{72306867505328396934266250703}{508641919686732195001764319921} a^{5} + \frac{163748253517899881525455110111}{508641919686732195001764319921} a^{4} - \frac{13584299901130218160423366230}{39126301514364015000135716917} a^{3} + \frac{30919858213912018768749567951}{72663131383818885000252045703} a^{2} - \frac{3321616524497942503624805397}{10380447340545555000036006529} a - \frac{25951269002815061680130366281}{72663131383818885000252045703}$

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:  $15$
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:  \( 2924095221.58 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T646:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20736
The 32 conjugacy class representatives for t18n646
Character table for t18n646 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$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}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$