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Properties

Label 17.17.1604706439...7921.1
Degree $17$
Signature $[17, 0]$
Discriminant $103^{16}$
Root discriminant $78.42$
Ramified prime $103$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{17}$ (as 17T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2837, 25763, -75722, 58507, 102976, -198615, 50288, 107759, -74838, -11031, 23311, -3653, -2579, 763, 105, -48, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 48*x^15 + 105*x^14 + 763*x^13 - 2579*x^12 - 3653*x^11 + 23311*x^10 - 11031*x^9 - 74838*x^8 + 107759*x^7 + 50288*x^6 - 198615*x^5 + 102976*x^4 + 58507*x^3 - 75722*x^2 + 25763*x - 2837)
 
gp: K = bnfinit(x^17 - x^16 - 48*x^15 + 105*x^14 + 763*x^13 - 2579*x^12 - 3653*x^11 + 23311*x^10 - 11031*x^9 - 74838*x^8 + 107759*x^7 + 50288*x^6 - 198615*x^5 + 102976*x^4 + 58507*x^3 - 75722*x^2 + 25763*x - 2837, 1)
 

Normalized defining polynomial

\( x^{17} - x^{16} - 48 x^{15} + 105 x^{14} + 763 x^{13} - 2579 x^{12} - 3653 x^{11} + 23311 x^{10} - 11031 x^{9} - 74838 x^{8} + 107759 x^{7} + 50288 x^{6} - 198615 x^{5} + 102976 x^{4} + 58507 x^{3} - 75722 x^{2} + 25763 x - 2837 \)

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

Invariants

Degree:  $17$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[17, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(160470643909878751793805444097921=103^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.42$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(103\)
Dirichlet character group:    $\lbrace$$\chi_{103}(64,·)$, $\chi_{103}(1,·)$, $\chi_{103}(66,·)$, $\chi_{103}(8,·)$, $\chi_{103}(9,·)$, $\chi_{103}(76,·)$, $\chi_{103}(13,·)$, $\chi_{103}(14,·)$, $\chi_{103}(79,·)$, $\chi_{103}(81,·)$, $\chi_{103}(23,·)$, $\chi_{103}(93,·)$, $\chi_{103}(30,·)$, $\chi_{103}(34,·)$, $\chi_{103}(100,·)$, $\chi_{103}(72,·)$, $\chi_{103}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7003} a^{15} + \frac{111}{7003} a^{14} - \frac{1548}{7003} a^{13} + \frac{3015}{7003} a^{12} - \frac{205}{7003} a^{11} + \frac{1487}{7003} a^{10} + \frac{14}{149} a^{9} - \frac{3045}{7003} a^{8} - \frac{2250}{7003} a^{7} + \frac{1069}{7003} a^{6} - \frac{2040}{7003} a^{5} - \frac{1044}{7003} a^{4} + \frac{2698}{7003} a^{3} - \frac{1245}{7003} a^{2} - \frac{332}{7003} a - \frac{1949}{7003}$, $\frac{1}{99752558263349} a^{16} + \frac{5380907932}{99752558263349} a^{15} - \frac{41954299176573}{99752558263349} a^{14} + \frac{22438092102485}{99752558263349} a^{13} + \frac{12225557661206}{99752558263349} a^{12} - \frac{46521131883065}{99752558263349} a^{11} - \frac{2461007305771}{99752558263349} a^{10} - \frac{7841565362988}{99752558263349} a^{9} + \frac{38136997848232}{99752558263349} a^{8} - \frac{3050863650300}{99752558263349} a^{7} + \frac{30138440774556}{99752558263349} a^{6} - \frac{48074873263192}{99752558263349} a^{5} - \frac{5871606948884}{99752558263349} a^{4} - \frac{489656773295}{2122394856667} a^{3} + \frac{28542809263483}{99752558263349} a^{2} - \frac{23350746439877}{99752558263349} a - \frac{3282112036992}{99752558263349}$

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $16$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 30768108227.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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 $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{17}$ $17$ $17$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
103Data not computed