Properties

Label 17.1.93022763197...4721.1
Degree $17$
Signature $[1, 8]$
Discriminant $991^{8}$
Root discriminant $25.70$
Ramified prime $991$
Class number $1$
Class group Trivial
Galois group $D_{17}$ (as 17T2)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 4*x^15 - 18*x^14 + 11*x^13 + 14*x^12 + 21*x^11 - 62*x^10 + 14*x^9 + 50*x^8 + 54*x^7 - 121*x^6 - 7*x^5 + 36*x^4 + 64*x^3 - 23*x^2 + 43*x + 1)
 
gp: K = bnfinit(x^17 - x^16 + 4*x^15 - 18*x^14 + 11*x^13 + 14*x^12 + 21*x^11 - 62*x^10 + 14*x^9 + 50*x^8 + 54*x^7 - 121*x^6 - 7*x^5 + 36*x^4 + 64*x^3 - 23*x^2 + 43*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 43, -23, 64, 36, -7, -121, 54, 50, 14, -62, 21, 14, 11, -18, 4, -1, 1]);
 

Normalized defining polynomial

\( x^{17} - x^{16} + 4 x^{15} - 18 x^{14} + 11 x^{13} + 14 x^{12} + 21 x^{11} - 62 x^{10} + 14 x^{9} + 50 x^{8} + 54 x^{7} - 121 x^{6} - 7 x^{5} + 36 x^{4} + 64 x^{3} - 23 x^{2} + 43 x + 1 \)

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

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(930227631978098127294721=991^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $25.70$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $991$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{63} a^{13} - \frac{1}{63} a^{12} + \frac{1}{7} a^{11} - \frac{2}{21} a^{10} - \frac{5}{63} a^{9} - \frac{1}{63} a^{8} + \frac{1}{7} a^{7} + \frac{17}{63} a^{5} - \frac{26}{63} a^{4} + \frac{1}{7} a^{3} + \frac{8}{21} a^{2} - \frac{4}{63} a - \frac{17}{63}$, $\frac{1}{63} a^{14} + \frac{1}{63} a^{12} - \frac{4}{63} a^{11} - \frac{4}{63} a^{10} + \frac{8}{63} a^{9} + \frac{8}{63} a^{8} - \frac{5}{63} a^{7} + \frac{31}{63} a^{6} - \frac{23}{63} a^{5} + \frac{4}{63} a^{4} + \frac{26}{63} a^{3} + \frac{3}{7} a^{2} + \frac{2}{9} a - \frac{1}{21}$, $\frac{1}{63} a^{15} - \frac{1}{21} a^{12} + \frac{8}{63} a^{11} - \frac{1}{9} a^{10} - \frac{8}{63} a^{9} - \frac{4}{63} a^{8} + \frac{1}{63} a^{7} - \frac{2}{63} a^{6} + \frac{29}{63} a^{5} + \frac{10}{63} a^{4} - \frac{8}{21} a^{3} - \frac{31}{63} a^{2} + \frac{1}{63} a - \frac{25}{63}$, $\frac{1}{929061441} a^{16} + \frac{674213}{309687147} a^{15} + \frac{106045}{929061441} a^{14} - \frac{4358033}{929061441} a^{13} + \frac{13622674}{309687147} a^{12} - \frac{13836964}{929061441} a^{11} - \frac{2470456}{132723063} a^{10} - \frac{10593104}{103229049} a^{9} + \frac{18158150}{929061441} a^{8} + \frac{23823571}{929061441} a^{7} + \frac{28153693}{929061441} a^{6} - \frac{26253538}{103229049} a^{5} - \frac{80177674}{929061441} a^{4} + \frac{523508}{3428271} a^{3} + \frac{42304928}{309687147} a^{2} - \frac{252479222}{929061441} a - \frac{25772827}{929061441}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 323467.453718 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$D_{17}$ (as 17T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 34
The 10 conjugacy class representatives for $D_{17}$
Character table for $D_{17}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Galois closure: 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 $17$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $17$ $17$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $17$ $17$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
991Data not computed