Properties

Label 17.17.942...921.1
Degree $17$
Signature $[17, 0]$
Discriminant $9.429\times 10^{44}$
Root discriminant $442.14$
Ramified prime $647$
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 - 304*x^15 + 1117*x^14 + 25631*x^13 - 126439*x^12 - 773932*x^11 + 4360454*x^10 + 10731832*x^9 - 64676368*x^8 - 79260104*x^7 + 441919082*x^6 + 345306489*x^5 - 1259087517*x^4 - 718017711*x^3 + 1025767171*x^2 + 183044979*x - 202031659)
 
gp: K = bnfinit(x^17 - x^16 - 304*x^15 + 1117*x^14 + 25631*x^13 - 126439*x^12 - 773932*x^11 + 4360454*x^10 + 10731832*x^9 - 64676368*x^8 - 79260104*x^7 + 441919082*x^6 + 345306489*x^5 - 1259087517*x^4 - 718017711*x^3 + 1025767171*x^2 + 183044979*x - 202031659, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-202031659, 183044979, 1025767171, -718017711, -1259087517, 345306489, 441919082, -79260104, -64676368, 10731832, 4360454, -773932, -126439, 25631, 1117, -304, -1, 1]);
 

\(x^{17} - x^{16} - 304 x^{15} + 1117 x^{14} + 25631 x^{13} - 126439 x^{12} - 773932 x^{11} + 4360454 x^{10} + 10731832 x^{9} - 64676368 x^{8} - 79260104 x^{7} + 441919082 x^{6} + 345306489 x^{5} - 1259087517 x^{4} - 718017711 x^{3} + 1025767171 x^{2} + 183044979 x - 202031659\)  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:  \(942906198449660107953222334097149309547713921\)\(\medspace = 647^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $442.14$
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:  $647$
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:  \(647\)
Dirichlet character group:    $\lbrace$$\chi_{647}(1,·)$, $\chi_{647}(67,·)$, $\chi_{647}(43,·)$, $\chi_{647}(338,·)$, $\chi_{647}(468,·)$, $\chi_{647}(218,·)$, $\chi_{647}(221,·)$, $\chi_{647}(607,·)$, $\chi_{647}(293,·)$, $\chi_{647}(555,·)$, $\chi_{647}(300,·)$, $\chi_{647}(445,·)$, $\chi_{647}(306,·)$, $\chi_{647}(372,·)$, $\chi_{647}(53,·)$, $\chi_{647}(316,·)$, $\chi_{647}(573,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{103} a^{15} + \frac{45}{103} a^{14} - \frac{44}{103} a^{13} + \frac{43}{103} a^{12} + \frac{26}{103} a^{11} + \frac{43}{103} a^{10} + \frac{42}{103} a^{9} - \frac{38}{103} a^{8} - \frac{50}{103} a^{7} - \frac{48}{103} a^{6} - \frac{38}{103} a^{5} + \frac{5}{103} a^{4} + \frac{19}{103} a^{3} - \frac{3}{103} a^{2} + \frac{4}{103} a + \frac{8}{103}$, $\frac{1}{256860718724382658659069918106208027102789930572168874892652530311353} a^{16} - \frac{496156101747711740983172947036840801004373889865550196133402660601}{256860718724382658659069918106208027102789930572168874892652530311353} a^{15} + \frac{119840133788028729650097465122796989434321789806489959071421541397329}{256860718724382658659069918106208027102789930572168874892652530311353} a^{14} + \frac{59565544128600009299174286056009340253429714400238813841997824641829}{256860718724382658659069918106208027102789930572168874892652530311353} a^{13} - \frac{38407772859810279891134275860971601990802549947438121342002077089927}{256860718724382658659069918106208027102789930572168874892652530311353} a^{12} + \frac{18665168850810421848465086581377401316666231326464006197893856144602}{256860718724382658659069918106208027102789930572168874892652530311353} a^{11} + \frac{77861906509275191632862970196698574135329982887681266730857003117388}{256860718724382658659069918106208027102789930572168874892652530311353} a^{10} - \frac{86382441131314450162937361313822463615688364197192660028290811634148}{256860718724382658659069918106208027102789930572168874892652530311353} a^{9} + \frac{91956963248189986842367444751714873668434464651431434976729909910300}{256860718724382658659069918106208027102789930572168874892652530311353} a^{8} - \frac{74757742596651988997167279586962097350749195581446310149026902705740}{256860718724382658659069918106208027102789930572168874892652530311353} a^{7} + \frac{46534035709489505122052958003314552571750621371702321245908779065865}{256860718724382658659069918106208027102789930572168874892652530311353} a^{6} - \frac{55771697231757247364509269110594803091867128802283566168528504382266}{256860718724382658659069918106208027102789930572168874892652530311353} a^{5} - \frac{12220234121382698425255886212043245138168776389504676852062817131287}{256860718724382658659069918106208027102789930572168874892652530311353} a^{4} + \frac{66337800212460881088829105723246624679473159706850557153949831028145}{256860718724382658659069918106208027102789930572168874892652530311353} a^{3} - \frac{77997832158892012037925266117373147568142471760309983181563864463868}{256860718724382658659069918106208027102789930572168874892652530311353} a^{2} + \frac{115480062230345227915278322696002774907561558360554575376789811752032}{256860718724382658659069918106208027102789930572168874892652530311353} a + \frac{46692290860360856853276351168765539760150890798518272608106086814458}{256860718724382658659069918106208027102789930572168874892652530311353}$  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:  \( 53927069890502120 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 53927069890502120 \cdot 1}{2\sqrt{942906198449660107953222334097149309547713921}}\approx 0.115093953641942$ (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
647Data not computed