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Properties

Label 17.1.20072339997...9521.1
Degree $17$
Signature $[1, 8]$
Discriminant $1091^{8}$
Root discriminant $26.89$
Ramified prime $1091$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{17}$ (as 17T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 80, 240, 220, 200, 275, 88, 134, 18, 72, 10, 44, -16, 10, -18, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64)
 
gp: K = bnfinit(x^17 - 2*x^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64, 1)
 

Normalized defining polynomial

\( x^{17} - 2 x^{16} + 4 x^{15} - 18 x^{14} + 10 x^{13} - 16 x^{12} + 44 x^{11} + 10 x^{10} + 72 x^{9} + 18 x^{8} + 134 x^{7} + 88 x^{6} + 275 x^{5} + 200 x^{4} + 220 x^{3} + 240 x^{2} + 80 x + 64 \)

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

Invariants

Degree:  $17$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2007233999721653909999521=1091^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.89$
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:  $1091$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{3}{16} a^{5} + \frac{1}{16} a^{4} + \frac{7}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} - \frac{1}{32} a^{10} + \frac{1}{16} a^{9} + \frac{1}{32} a^{8} + \frac{1}{16} a^{7} + \frac{3}{32} a^{6} + \frac{1}{16} a^{5} - \frac{5}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{16} a^{10} + \frac{1}{32} a^{9} + \frac{1}{16} a^{8} + \frac{3}{32} a^{7} + \frac{1}{16} a^{6} - \frac{5}{32} a^{5} + \frac{1}{4} a^{2}$, $\frac{1}{8286671488} a^{16} - \frac{68849395}{8286671488} a^{15} + \frac{120169979}{8286671488} a^{14} - \frac{31185489}{8286671488} a^{13} - \frac{231372529}{8286671488} a^{12} - \frac{155375939}{8286671488} a^{11} + \frac{626061315}{8286671488} a^{10} + \frac{5191329}{123681664} a^{9} - \frac{370481855}{8286671488} a^{8} + \frac{21699013}{8286671488} a^{7} + \frac{508174493}{8286671488} a^{6} + \frac{100883231}{487451264} a^{5} - \frac{61534825}{258958484} a^{4} + \frac{612237787}{2071667872} a^{3} - \frac{15242284}{64739621} a^{2} + \frac{225047811}{517916968} a - \frac{42857061}{129479242}$

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:  $8$
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:  \( 1757145.67579 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $17$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $17$ $17$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1091Data not computed