Properties

Label 17.17.397...041.1
Degree $17$
Signature $[17, 0]$
Discriminant $3.975\times 10^{44}$
Root discriminant $420.23$
Ramified prime $613$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{17}$ (as 17T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 288*x^15 + 265*x^14 + 26034*x^13 - 40228*x^12 - 875968*x^11 + 2022008*x^10 + 8464009*x^9 - 27681440*x^8 - 8855367*x^7 + 101412811*x^6 - 87313302*x^5 - 38624139*x^4 + 67164168*x^3 - 7149746*x^2 - 7878215*x - 664471)
 
gp: K = bnfinit(x^17 - x^16 - 288*x^15 + 265*x^14 + 26034*x^13 - 40228*x^12 - 875968*x^11 + 2022008*x^10 + 8464009*x^9 - 27681440*x^8 - 8855367*x^7 + 101412811*x^6 - 87313302*x^5 - 38624139*x^4 + 67164168*x^3 - 7149746*x^2 - 7878215*x - 664471, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-664471, -7878215, -7149746, 67164168, -38624139, -87313302, 101412811, -8855367, -27681440, 8464009, 2022008, -875968, -40228, 26034, 265, -288, -1, 1]);
 

\(x^{17} - x^{16} - 288 x^{15} + 265 x^{14} + 26034 x^{13} - 40228 x^{12} - 875968 x^{11} + 2022008 x^{10} + 8464009 x^{9} - 27681440 x^{8} - 8855367 x^{7} + 101412811 x^{6} - 87313302 x^{5} - 38624139 x^{4} + 67164168 x^{3} - 7149746 x^{2} - 7878215 x - 664471\)  Toggle raw display

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

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[17, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(397527879693854754943959770157821268619247041\)\(\medspace = 613^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $420.23$
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:  $613$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $17$
This field is Galois and abelian over $\Q$.
Conductor:  \(613\)
Dirichlet character group:    $\lbrace$$\chi_{613}(1,·)$, $\chi_{613}(387,·)$, $\chi_{613}(197,·)$, $\chi_{613}(198,·)$, $\chi_{613}(583,·)$, $\chi_{613}(585,·)$, $\chi_{613}(586,·)$, $\chi_{613}(143,·)$, $\chi_{613}(220,·)$, $\chi_{613}(287,·)$, $\chi_{613}(546,·)$, $\chi_{613}(227,·)$, $\chi_{613}(37,·)$, $\chi_{613}(171,·)$, $\chi_{613}(430,·)$, $\chi_{613}(116,·)$, $\chi_{613}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{83} a^{15} - \frac{17}{83} a^{14} + \frac{3}{83} a^{13} - \frac{23}{83} a^{12} - \frac{18}{83} a^{11} - \frac{39}{83} a^{10} - \frac{36}{83} a^{9} - \frac{37}{83} a^{8} - \frac{8}{83} a^{7} - \frac{21}{83} a^{6} + \frac{4}{83} a^{5} - \frac{40}{83} a^{4} + \frac{11}{83} a^{3} - \frac{29}{83} a^{2} - \frac{4}{83} a - \frac{32}{83}$, $\frac{1}{8005016753506549989469084281029751031104417898106301188902501} a^{16} - \frac{21742052392920060370221755237911823333762807269570076890675}{8005016753506549989469084281029751031104417898106301188902501} a^{15} + \frac{1318262744276119314600470957188333247003568665005465830337918}{8005016753506549989469084281029751031104417898106301188902501} a^{14} + \frac{978985556839464408917368141669338015554258870105656961439672}{8005016753506549989469084281029751031104417898106301188902501} a^{13} + \frac{2769652849413101156955094782253666473242821723068975047662913}{8005016753506549989469084281029751031104417898106301188902501} a^{12} - \frac{3865963828740415503858859292080994092134756470880241715944650}{8005016753506549989469084281029751031104417898106301188902501} a^{11} + \frac{1566778693849231577261720198965935208866780018514555776469291}{8005016753506549989469084281029751031104417898106301188902501} a^{10} + \frac{3292053559812149492078265514315409205047286548695243019801671}{8005016753506549989469084281029751031104417898106301188902501} a^{9} + \frac{2015958968976941668569215230872253848562337373638239291297888}{8005016753506549989469084281029751031104417898106301188902501} a^{8} - \frac{3768311406065772447653212035143009470927710873927962180210625}{8005016753506549989469084281029751031104417898106301188902501} a^{7} - \frac{651578051308172587753476565966120883630751279431155212135290}{8005016753506549989469084281029751031104417898106301188902501} a^{6} + \frac{1211160314524927167956762805442153429806021925493763606465554}{8005016753506549989469084281029751031104417898106301188902501} a^{5} + \frac{1576750091352007627088216157619973684725771671970502479230746}{8005016753506549989469084281029751031104417898106301188902501} a^{4} - \frac{916811404478504719979165509046006355992462069581910473626098}{8005016753506549989469084281029751031104417898106301188902501} a^{3} - \frac{3795236126167884475140566458620786336071637918943472071349866}{8005016753506549989469084281029751031104417898106301188902501} a^{2} - \frac{1000132303391176795188573093161245333680963265630325848143544}{8005016753506549989469084281029751031104417898106301188902501} a + \frac{2324389487913912762466131974077911977648826063869399720738164}{8005016753506549989469084281029751031104417898106301188902501}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $16$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
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 (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 40860825172254010 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 40860825172254010 \cdot 1}{2\sqrt{397527879693854754943959770157821268619247041}}\approx 0.134308428412734$ (assuming GRH)

Galois group

$C_{17}$ (as 17T1):

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

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 $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$

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
613Data not computed