Properties

Label 17.17.220...001.1
Degree $17$
Signature $[17, 0]$
Discriminant $2.200\times 10^{42}$
Root discriminant $309.55$
Ramified prime $443$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{17}$ (as 17T1)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 208*x^15 - 17*x^14 + 15287*x^13 + 13881*x^12 - 487578*x^11 - 703261*x^10 + 6754359*x^9 + 10540902*x^8 - 41136753*x^7 - 57683825*x^6 + 92010954*x^5 + 95287840*x^4 - 17501435*x^3 - 25026156*x^2 - 563260*x + 1246103)
 
gp: K = bnfinit(x^17 - x^16 - 208*x^15 - 17*x^14 + 15287*x^13 + 13881*x^12 - 487578*x^11 - 703261*x^10 + 6754359*x^9 + 10540902*x^8 - 41136753*x^7 - 57683825*x^6 + 92010954*x^5 + 95287840*x^4 - 17501435*x^3 - 25026156*x^2 - 563260*x + 1246103, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1246103, -563260, -25026156, -17501435, 95287840, 92010954, -57683825, -41136753, 10540902, 6754359, -703261, -487578, 13881, 15287, -17, -208, -1, 1]);
 

\(x^{17} - x^{16} - 208 x^{15} - 17 x^{14} + 15287 x^{13} + 13881 x^{12} - 487578 x^{11} - 703261 x^{10} + 6754359 x^{9} + 10540902 x^{8} - 41136753 x^{7} - 57683825 x^{6} + 92010954 x^{5} + 95287840 x^{4} - 17501435 x^{3} - 25026156 x^{2} - 563260 x + 1246103\)  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:  \(2200187128095499475530336818113367454680001\)\(\medspace = 443^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $309.55$
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:  $443$
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:  \(443\)
Dirichlet character group:    $\lbrace$$\chi_{443}(1,·)$, $\chi_{443}(67,·)$, $\chi_{443}(324,·)$, $\chi_{443}(267,·)$, $\chi_{443}(13,·)$, $\chi_{443}(270,·)$, $\chi_{443}(209,·)$, $\chi_{443}(409,·)$, $\chi_{443}(225,·)$, $\chi_{443}(123,·)$, $\chi_{443}(425,·)$, $\chi_{443}(428,·)$, $\chi_{443}(370,·)$, $\chi_{443}(169,·)$, $\chi_{443}(248,·)$, $\chi_{443}(59,·)$, $\chi_{443}(380,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4013480870186551123112427443978541538197334827965530746341757461} a^{16} + \frac{30664611433497702720987163865545312304482917972317611699004451}{4013480870186551123112427443978541538197334827965530746341757461} a^{15} - \frac{234236602191203470251879063625934586476098163260364116994871423}{4013480870186551123112427443978541538197334827965530746341757461} a^{14} - \frac{1538525439450223170396067485610034784683093539704487494351943862}{4013480870186551123112427443978541538197334827965530746341757461} a^{13} + \frac{55342538172956023481629299767885008096663009551709221443365116}{4013480870186551123112427443978541538197334827965530746341757461} a^{12} + \frac{753763549332543509880444461910703558803979931791746488591544835}{4013480870186551123112427443978541538197334827965530746341757461} a^{11} + \frac{449079695525428969055400104617954013904982282494392624345855044}{4013480870186551123112427443978541538197334827965530746341757461} a^{10} + \frac{1356331504173895330061633890226502143904723258261751560217500635}{4013480870186551123112427443978541538197334827965530746341757461} a^{9} - \frac{258115246454719936497475617167093245823083651779523923072083989}{4013480870186551123112427443978541538197334827965530746341757461} a^{8} - \frac{1439096506119506255900602412328539151888891788935030943061379179}{4013480870186551123112427443978541538197334827965530746341757461} a^{7} - \frac{264370601676178270686168837198796059220002990634243712724781297}{4013480870186551123112427443978541538197334827965530746341757461} a^{6} - \frac{180010373031027196207070542203811906918368022587662836705394813}{4013480870186551123112427443978541538197334827965530746341757461} a^{5} + \frac{782240170953201337633372345915642504015417842262032178961326220}{4013480870186551123112427443978541538197334827965530746341757461} a^{4} - \frac{425224591713287603242465941484685733198764997139724614627529825}{4013480870186551123112427443978541538197334827965530746341757461} a^{3} + \frac{1591411055048919701518357482737603500517050089451817421319409739}{4013480870186551123112427443978541538197334827965530746341757461} a^{2} + \frac{1107332544781032139606978167678448335717904050624033843189535906}{4013480870186551123112427443978541538197334827965530746341757461} a - \frac{1274193145411167372061357994658018891769389950257661519381602369}{4013480870186551123112427443978541538197334827965530746341757461}$  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:  \( 2360552434044002.5 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 2360552434044002.5 \cdot 1}{2\sqrt{2200187128095499475530336818113367454680001}}\approx 0.104295068223382$ (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
443Data not computed