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Properties

Label 17.17.8972555039...3201.1
Degree $17$
Signature $[17, 0]$
Discriminant $17^{12}\cdot 137^{16}$
Root discriminant $757.85$
Ramified primes $17, 137$
Class number $17$ (GRH)
Class group $[17]$ (GRH)
Galois group $C_{17}:C_{4}$ (as 17T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-12878683864992, 27480501953536, -21491689501032, 6800548404356, -7660737412, -524744382701, 95353221324, 9723361580, -4267795762, 108448584, 80531352, -6121180, -736812, 82996, 3026, -476, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^16 - 476*x^15 + 3026*x^14 + 82996*x^13 - 736812*x^12 - 6121180*x^11 + 80531352*x^10 + 108448584*x^9 - 4267795762*x^8 + 9723361580*x^7 + 95353221324*x^6 - 524744382701*x^5 - 7660737412*x^4 + 6800548404356*x^3 - 21491689501032*x^2 + 27480501953536*x - 12878683864992)
 
gp: K = bnfinit(x^17 - 4*x^16 - 476*x^15 + 3026*x^14 + 82996*x^13 - 736812*x^12 - 6121180*x^11 + 80531352*x^10 + 108448584*x^9 - 4267795762*x^8 + 9723361580*x^7 + 95353221324*x^6 - 524744382701*x^5 - 7660737412*x^4 + 6800548404356*x^3 - 21491689501032*x^2 + 27480501953536*x - 12878683864992, 1)
 

Normalized defining polynomial

\( x^{17} - 4 x^{16} - 476 x^{15} + 3026 x^{14} + 82996 x^{13} - 736812 x^{12} - 6121180 x^{11} + 80531352 x^{10} + 108448584 x^{9} - 4267795762 x^{8} + 9723361580 x^{7} + 95353221324 x^{6} - 524744382701 x^{5} - 7660737412 x^{4} + 6800548404356 x^{3} - 21491689501032 x^{2} + 27480501953536 x - 12878683864992 \)

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

Invariants

Degree:  $17$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[17, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8972555039016074378226050552414591426720707963201=17^{12}\cdot 137^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $757.85$
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:  $17, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} + \frac{3}{16} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{5}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{11} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{3}{16} a^{7} + \frac{1}{8} a^{5} + \frac{3}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{13} + \frac{1}{48} a^{12} + \frac{1}{48} a^{11} - \frac{1}{48} a^{10} + \frac{5}{48} a^{9} - \frac{5}{48} a^{8} - \frac{11}{48} a^{7} - \frac{5}{48} a^{6} - \frac{5}{48} a^{5} - \frac{11}{48} a^{4} - \frac{1}{48} a^{3} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{288} a^{15} + \frac{1}{48} a^{13} + \frac{1}{36} a^{12} - \frac{1}{24} a^{11} - \frac{13}{144} a^{10} + \frac{1}{48} a^{9} - \frac{1}{18} a^{8} + \frac{13}{144} a^{7} + \frac{1}{9} a^{6} - \frac{1}{72} a^{5} + \frac{11}{48} a^{4} + \frac{89}{288} a^{3} + \frac{7}{18} a^{2} + \frac{17}{36} a + \frac{1}{4}$, $\frac{1}{702898237230931617154238730367736828358829109072467274370816} a^{16} + \frac{32712467553338663169238444540711338524251803387138527399}{351449118615465808577119365183868414179414554536233637185408} a^{15} - \frac{19576155863968144443343475124759506635160576147866662187}{14643713275644408690713306882661183924142273105676401549392} a^{14} + \frac{5050452187901756458867026907682813766647772317221281325177}{351449118615465808577119365183868414179414554536233637185408} a^{13} - \frac{2655837026747541801719505830599481514135454207166800233773}{87862279653866452144279841295967103544853638634058409296352} a^{12} - \frac{4485894281101177567687423000563714702945775572870844166603}{175724559307732904288559682591934207089707277268116818592704} a^{11} - \frac{11980197617316673477695110286747764789850499030202888206949}{175724559307732904288559682591934207089707277268116818592704} a^{10} + \frac{4758005646516582653220412517731282478920842843859631617199}{43931139826933226072139920647983551772426819317029204648176} a^{9} + \frac{10587122231105403807859820161717022602147768434642236327483}{87862279653866452144279841295967103544853638634058409296352} a^{8} + \frac{2684612057836492379554504608946242647854696308340132467935}{351449118615465808577119365183868414179414554536233637185408} a^{7} - \frac{14269928511606052314695219115055537643267878525844428514923}{87862279653866452144279841295967103544853638634058409296352} a^{6} - \frac{40073300534771204530800398696446644049995647538682459894141}{175724559307732904288559682591934207089707277268116818592704} a^{5} - \frac{175372745835932842751734148300011597733152907668868132127733}{702898237230931617154238730367736828358829109072467274370816} a^{4} - \frac{36646196222464461617893563949485628366570430111922839838197}{117149706205155269525706455061289471393138184845411212395136} a^{3} + \frac{9841243936800494943081857707577077550732650480484458493917}{87862279653866452144279841295967103544853638634058409296352} a^{2} + \frac{20431418805512748947193285495992321465283197110882072004669}{87862279653866452144279841295967103544853638634058409296352} a + \frac{596212624291383251180088179506608605788893823622093632655}{1627079252849378743412589653629020436015808122852933505488}$

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

Class group and class number

$C_{17}$, which has order $17$ (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:  $16$
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:  \( 31945964915800000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{17}.C_2$ (as 17T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 68
The 8 conjugacy class representatives for $C_{17}:C_{4}$
Character table for $C_{17}:C_{4}$

Intermediate fields

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $17$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
137Data not computed