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Properties

Label 17.17.113...161.1
Degree $17$
Signature $[17, 0]$
Discriminant $1.133\times 10^{38}$
Root discriminant $173.18$
Ramified prime $239$
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 - 112*x^15 + 47*x^14 + 3976*x^13 - 4314*x^12 - 64388*x^11 + 136247*x^10 + 422013*x^9 - 1631073*x^8 + 411840*x^7 + 5840196*x^6 - 11894369*x^5 + 10635750*x^4 - 4739804*x^3 + 938485*x^2 - 54850*x + 619)
 
gp: K = bnfinit(x^17 - x^16 - 112*x^15 + 47*x^14 + 3976*x^13 - 4314*x^12 - 64388*x^11 + 136247*x^10 + 422013*x^9 - 1631073*x^8 + 411840*x^7 + 5840196*x^6 - 11894369*x^5 + 10635750*x^4 - 4739804*x^3 + 938485*x^2 - 54850*x + 619, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![619, -54850, 938485, -4739804, 10635750, -11894369, 5840196, 411840, -1631073, 422013, 136247, -64388, -4314, 3976, 47, -112, -1, 1]);
 

\(x^{17} - x^{16} - 112 x^{15} + 47 x^{14} + 3976 x^{13} - 4314 x^{12} - 64388 x^{11} + 136247 x^{10} + 422013 x^{9} - 1631073 x^{8} + 411840 x^{7} + 5840196 x^{6} - 11894369 x^{5} + 10635750 x^{4} - 4739804 x^{3} + 938485 x^{2} - 54850 x + 619\)  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:  \(113335617496346216833223278514633468161\)\(\medspace = 239^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $173.18$
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:  $239$
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:  \(239\)
Dirichlet character group:    $\lbrace$$\chi_{239}(128,·)$, $\chi_{239}(1,·)$, $\chi_{239}(67,·)$, $\chi_{239}(132,·)$, $\chi_{239}(6,·)$, $\chi_{239}(71,·)$, $\chi_{239}(75,·)$, $\chi_{239}(211,·)$, $\chi_{239}(22,·)$, $\chi_{239}(216,·)$, $\chi_{239}(163,·)$, $\chi_{239}(36,·)$, $\chi_{239}(101,·)$, $\chi_{239}(166,·)$, $\chi_{239}(40,·)$, $\chi_{239}(51,·)$, $\chi_{239}(187,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17044241} a^{15} + \frac{7923170}{17044241} a^{14} + \frac{689862}{17044241} a^{13} + \frac{6847608}{17044241} a^{12} - \frac{7864577}{17044241} a^{11} + \frac{293236}{17044241} a^{10} - \frac{5493659}{17044241} a^{9} + \frac{7262608}{17044241} a^{8} - \frac{1203353}{17044241} a^{7} + \frac{1583552}{17044241} a^{6} - \frac{1066023}{17044241} a^{5} - \frac{510254}{17044241} a^{4} + \frac{638656}{17044241} a^{3} + \frac{5875424}{17044241} a^{2} - \frac{1483120}{17044241} a + \frac{1127039}{17044241}$, $\frac{1}{3146212685301157790197646909} a^{16} + \frac{321302530370239512}{11117359312018225407058823} a^{15} + \frac{931869356563958800550173360}{3146212685301157790197646909} a^{14} + \frac{552804684805408253390411164}{3146212685301157790197646909} a^{13} + \frac{1487953305986084086487087402}{3146212685301157790197646909} a^{12} - \frac{975115675359830725592604408}{3146212685301157790197646909} a^{11} + \frac{283672140826449234617724591}{3146212685301157790197646909} a^{10} + \frac{694304344494880202387970900}{3146212685301157790197646909} a^{9} - \frac{236539474078061516527332990}{3146212685301157790197646909} a^{8} + \frac{1341028456795626068941899775}{3146212685301157790197646909} a^{7} - \frac{168169734412620181297958224}{3146212685301157790197646909} a^{6} + \frac{1439773272590715438339647841}{3146212685301157790197646909} a^{5} + \frac{113389185701611144346840708}{3146212685301157790197646909} a^{4} + \frac{644557152152711011429553008}{3146212685301157790197646909} a^{3} + \frac{309396518439439996729185912}{3146212685301157790197646909} a^{2} + \frac{651624250772576365943634777}{3146212685301157790197646909} a - \frac{700795773717520913653501}{5082734548144035202257911}$  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:  \( 24055588816300 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 24055588816300 \cdot 1}{2\sqrt{113335617496346216833223278514633468161}}\approx 0.148085560941044$ (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
239Data not computed