Properties

Label 17.17.613...441.1
Degree $17$
Signature $[17, 0]$
Discriminant $6.132\times 10^{41}$
Root discriminant $287.14$
Ramified prime $409$
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 - 192*x^15 + 273*x^14 + 14752*x^13 - 28028*x^12 - 571107*x^11 + 1411675*x^10 + 11275657*x^9 - 36814399*x^8 - 91832077*x^7 + 461179352*x^6 - 109192148*x^5 - 1929139488*x^4 + 3679722325*x^3 - 2767754010*x^2 + 828153361*x - 45886883)
 
gp: K = bnfinit(x^17 - x^16 - 192*x^15 + 273*x^14 + 14752*x^13 - 28028*x^12 - 571107*x^11 + 1411675*x^10 + 11275657*x^9 - 36814399*x^8 - 91832077*x^7 + 461179352*x^6 - 109192148*x^5 - 1929139488*x^4 + 3679722325*x^3 - 2767754010*x^2 + 828153361*x - 45886883, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-45886883, 828153361, -2767754010, 3679722325, -1929139488, -109192148, 461179352, -91832077, -36814399, 11275657, 1411675, -571107, -28028, 14752, 273, -192, -1, 1]);
 

\(x^{17} - x^{16} - 192 x^{15} + 273 x^{14} + 14752 x^{13} - 28028 x^{12} - 571107 x^{11} + 1411675 x^{10} + 11275657 x^{9} - 36814399 x^{8} - 91832077 x^{7} + 461179352 x^{6} - 109192148 x^{5} - 1929139488 x^{4} + 3679722325 x^{3} - 2767754010 x^{2} + 828153361 x - 45886883\)  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:  \(613158747081871736694796283376344205805441\)\(\medspace = 409^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $287.14$
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:  $409$
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:  \(409\)
Dirichlet character group:    $\lbrace$$\chi_{409}(1,·)$, $\chi_{409}(5,·)$, $\chi_{409}(262,·)$, $\chi_{409}(82,·)$, $\chi_{409}(83,·)$, $\chi_{409}(341,·)$, $\chi_{409}(150,·)$, $\chi_{409}(89,·)$, $\chi_{409}(216,·)$, $\chi_{409}(345,·)$, $\chi_{409}(25,·)$, $\chi_{409}(30,·)$, $\chi_{409}(69,·)$, $\chi_{409}(36,·)$, $\chi_{409}(6,·)$, $\chi_{409}(180,·)$, $\chi_{409}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{49} a^{11} - \frac{2}{49} a^{10} + \frac{3}{49} a^{9} + \frac{3}{49} a^{7} - \frac{1}{49} a^{6} - \frac{24}{49} a^{5} + \frac{5}{49} a^{4} - \frac{4}{49} a^{3} - \frac{23}{49} a^{2}$, $\frac{1}{49} a^{12} - \frac{1}{49} a^{10} - \frac{1}{49} a^{9} + \frac{3}{49} a^{8} - \frac{2}{49} a^{7} + \frac{2}{49} a^{6} + \frac{13}{49} a^{5} + \frac{20}{49} a^{4} + \frac{4}{49} a^{3} + \frac{10}{49} a^{2} + \frac{3}{7} a$, $\frac{1}{2597} a^{13} + \frac{9}{2597} a^{12} + \frac{24}{2597} a^{11} - \frac{88}{2597} a^{10} - \frac{36}{2597} a^{9} + \frac{46}{2597} a^{8} - \frac{60}{2597} a^{7} + \frac{139}{2597} a^{6} + \frac{930}{2597} a^{5} - \frac{1287}{2597} a^{4} - \frac{208}{2597} a^{3} - \frac{268}{2597} a^{2} + \frac{92}{371} a - \frac{12}{53}$, $\frac{1}{18179} a^{14} - \frac{2}{18179} a^{13} - \frac{181}{18179} a^{12} - \frac{87}{18179} a^{11} + \frac{508}{18179} a^{10} + \frac{230}{18179} a^{9} - \frac{1255}{18179} a^{8} + \frac{99}{2597} a^{7} - \frac{705}{18179} a^{6} + \frac{2263}{18179} a^{5} - \frac{8735}{18179} a^{4} - \frac{577}{18179} a^{3} + \frac{3}{371} a^{2} + \frac{116}{371} a - \frac{19}{53}$, $\frac{1}{563549} a^{15} - \frac{9}{563549} a^{14} + \frac{85}{563549} a^{13} + \frac{1964}{563549} a^{12} - \frac{4336}{563549} a^{11} - \frac{16598}{563549} a^{10} - \frac{780}{18179} a^{9} + \frac{4229}{80507} a^{8} - \frac{13256}{563549} a^{7} + \frac{27386}{563549} a^{6} + \frac{201622}{563549} a^{5} + \frac{207414}{563549} a^{4} + \frac{55}{1519} a^{3} - \frac{38981}{80507} a^{2} + \frac{5335}{11501} a + \frac{443}{1643}$, $\frac{1}{282699425718873155283472558703117231069} a^{16} + \frac{3544656702612517850087507460521}{5333951428657984061952312428360702473} a^{15} + \frac{5397731127376526300667479784851404}{282699425718873155283472558703117231069} a^{14} + \frac{12266261588774916416845680363691109}{282699425718873155283472558703117231069} a^{13} + \frac{10641153046109385654558302222496531}{40385632245553307897638936957588175867} a^{12} + \frac{258773583121534663091721287352443145}{40385632245553307897638936957588175867} a^{11} - \frac{13076393434711740810791668573891579825}{282699425718873155283472558703117231069} a^{10} + \frac{1425914202246592967548147414844745074}{40385632245553307897638936957588175867} a^{9} + \frac{13723422051701087307874808136458185188}{282699425718873155283472558703117231069} a^{8} + \frac{9919500033399828362837361311197855770}{282699425718873155283472558703117231069} a^{7} - \frac{12762040566332437011325937303880339963}{282699425718873155283472558703117231069} a^{6} + \frac{70292445580843796500114508700554471722}{282699425718873155283472558703117231069} a^{5} - \frac{48046229527817108245110205994777379869}{282699425718873155283472558703117231069} a^{4} + \frac{63512512227237474720255830491298894958}{282699425718873155283472558703117231069} a^{3} - \frac{17213371481712910295918213340363931048}{40385632245553307897638936957588175867} a^{2} + \frac{28391457980761070100704265442831708}{108856151605264980856169641395116377} a + \frac{393524537959119858509142437334529947}{824196576439863426482427284848738283}$  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:  \( 37559948173498540 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 37559948173498540 \cdot 1}{2\sqrt{613158747081871736694796283376344205805441}}\approx 3.14353609133807$ (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 $17$ $17$ $17$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{17}$ $17$ $17$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{17}$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{17}$ $17$

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
409Data not computed