Properties

Label 17.17.154...041.1
Degree $17$
Signature $[17, 0]$
Discriminant $1.540\times 10^{34}$
Root discriminant $102.57$
Ramified prime $137$
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 - 64*x^15 + 43*x^14 + 1478*x^13 - 932*x^12 - 16008*x^11 + 12183*x^10 + 86347*x^9 - 84507*x^8 - 213223*x^7 + 271237*x^6 + 152800*x^5 - 314540*x^4 + 100605*x^3 + 20132*x^2 - 13981*x + 1681)
 
gp: K = bnfinit(x^17 - x^16 - 64*x^15 + 43*x^14 + 1478*x^13 - 932*x^12 - 16008*x^11 + 12183*x^10 + 86347*x^9 - 84507*x^8 - 213223*x^7 + 271237*x^6 + 152800*x^5 - 314540*x^4 + 100605*x^3 + 20132*x^2 - 13981*x + 1681, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1681, -13981, 20132, 100605, -314540, 152800, 271237, -213223, -84507, 86347, 12183, -16008, -932, 1478, 43, -64, -1, 1]);
 

\(x^{17} - x^{16} - 64 x^{15} + 43 x^{14} + 1478 x^{13} - 932 x^{12} - 16008 x^{11} + 12183 x^{10} + 86347 x^{9} - 84507 x^{8} - 213223 x^{7} + 271237 x^{6} + 152800 x^{5} - 314540 x^{4} + 100605 x^{3} + 20132 x^{2} - 13981 x + 1681\)  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:  \(15400296222263289476715621650663041\)\(\medspace = 137^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $102.57$
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:  $137$
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:  \(137\)
Dirichlet character group:    $\lbrace$$\chi_{137}(1,·)$, $\chi_{137}(133,·)$, $\chi_{137}(72,·)$, $\chi_{137}(73,·)$, $\chi_{137}(74,·)$, $\chi_{137}(16,·)$, $\chi_{137}(88,·)$, $\chi_{137}(34,·)$, $\chi_{137}(59,·)$, $\chi_{137}(38,·)$, $\chi_{137}(50,·)$, $\chi_{137}(115,·)$, $\chi_{137}(119,·)$, $\chi_{137}(56,·)$, $\chi_{137}(122,·)$, $\chi_{137}(123,·)$, $\chi_{137}(60,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{37} a^{14} + \frac{2}{37} a^{12} + \frac{4}{37} a^{11} - \frac{14}{37} a^{10} + \frac{13}{37} a^{9} + \frac{11}{37} a^{8} + \frac{10}{37} a^{7} + \frac{7}{37} a^{6} - \frac{5}{37} a^{5} + \frac{18}{37} a^{4} - \frac{17}{37} a^{3} - \frac{3}{37} a^{2} - \frac{16}{37} a + \frac{4}{37}$, $\frac{1}{192659} a^{15} + \frac{2389}{192659} a^{14} - \frac{69632}{192659} a^{13} + \frac{82408}{192659} a^{12} + \frac{93458}{192659} a^{11} + \frac{17664}{192659} a^{10} + \frac{48606}{192659} a^{9} + \frac{49007}{192659} a^{8} + \frac{53978}{192659} a^{7} - \frac{35822}{192659} a^{6} - \frac{56623}{192659} a^{5} + \frac{86016}{192659} a^{4} - \frac{81908}{192659} a^{3} + \frac{70406}{192659} a^{2} + \frac{89356}{192659} a - \frac{53}{4699}$, $\frac{1}{458898274030885187123794199} a^{16} + \frac{572440381749563032811}{458898274030885187123794199} a^{15} + \frac{2592881780188347973655989}{458898274030885187123794199} a^{14} + \frac{142598118890034976961805060}{458898274030885187123794199} a^{13} + \frac{201155069052146647609332577}{458898274030885187123794199} a^{12} + \frac{226640544427017209931643033}{458898274030885187123794199} a^{11} + \frac{53208007998762935330093268}{458898274030885187123794199} a^{10} + \frac{36630608149021303226057121}{458898274030885187123794199} a^{9} + \frac{4539730493443987031589180}{12402656054888788841183627} a^{8} - \frac{166557196393640177095302289}{458898274030885187123794199} a^{7} + \frac{75327462266327090333963102}{458898274030885187123794199} a^{6} + \frac{50447121900504534961971255}{458898274030885187123794199} a^{5} - \frac{69845522159501561050478729}{458898274030885187123794199} a^{4} - \frac{62293975850374962848445315}{458898274030885187123794199} a^{3} + \frac{145289994810479287329603563}{458898274030885187123794199} a^{2} - \frac{211104461552108828961525495}{458898274030885187123794199} a + \frac{2708266844777972937222048}{11192640830021589929848639}$  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:  \( 559957546560 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 559957546560 \cdot 1}{2\sqrt{15400296222263289476715621650663041}}\approx 0.295713053238789$ (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 $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{17}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{17}$ $17$ $17$ $17$ $17$

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
137Data not computed