Properties

Label 17.17.258...281.1
Degree $17$
Signature $[17, 0]$
Discriminant $2.589\times 10^{47}$
Root discriminant $615.18$
Ramified prime $919$
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 - 432*x^15 + 1911*x^14 + 56071*x^13 - 377127*x^12 - 2275999*x^11 + 22947072*x^10 - 5751373*x^9 - 395586237*x^8 + 1094097337*x^7 - 39485017*x^6 - 2920148551*x^5 + 2341974035*x^4 + 1284864535*x^3 - 1464500037*x^2 - 140787928*x + 238840843)
 
gp: K = bnfinit(x^17 - x^16 - 432*x^15 + 1911*x^14 + 56071*x^13 - 377127*x^12 - 2275999*x^11 + 22947072*x^10 - 5751373*x^9 - 395586237*x^8 + 1094097337*x^7 - 39485017*x^6 - 2920148551*x^5 + 2341974035*x^4 + 1284864535*x^3 - 1464500037*x^2 - 140787928*x + 238840843, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![238840843, -140787928, -1464500037, 1284864535, 2341974035, -2920148551, -39485017, 1094097337, -395586237, -5751373, 22947072, -2275999, -377127, 56071, 1911, -432, -1, 1]);
 

\(x^{17} - x^{16} - 432 x^{15} + 1911 x^{14} + 56071 x^{13} - 377127 x^{12} - 2275999 x^{11} + 22947072 x^{10} - 5751373 x^{9} - 395586237 x^{8} + 1094097337 x^{7} - 39485017 x^{6} - 2920148551 x^{5} + 2341974035 x^{4} + 1284864535 x^{3} - 1464500037 x^{2} - 140787928 x + 238840843\)  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:  \(258850007664814362506653464185428842369437873281\)\(\medspace = 919^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $615.18$
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:  $919$
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:  \(919\)
Dirichlet character group:    $\lbrace$$\chi_{919}(512,·)$, $\chi_{919}(1,·)$, $\chi_{919}(706,·)$, $\chi_{919}(206,·)$, $\chi_{919}(849,·)$, $\chi_{919}(338,·)$, $\chi_{919}(416,·)$, $\chi_{919}(535,·)$, $\chi_{919}(284,·)$, $\chi_{919}(607,·)$, $\chi_{919}(288,·)$, $\chi_{919}(162,·)$, $\chi_{919}(229,·)$, $\chi_{919}(234,·)$, $\chi_{919}(305,·)$, $\chi_{919}(58,·)$, $\chi_{919}(703,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{53} a^{15} - \frac{22}{53} a^{14} - \frac{23}{53} a^{13} + \frac{9}{53} a^{12} + \frac{20}{53} a^{11} + \frac{25}{53} a^{10} - \frac{15}{53} a^{9} - \frac{23}{53} a^{8} - \frac{19}{53} a^{7} + \frac{14}{53} a^{6} + \frac{23}{53} a^{5} - \frac{23}{53} a^{4} - \frac{19}{53} a^{3} - \frac{7}{53} a^{2} - \frac{8}{53} a$, $\frac{1}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{16} + \frac{417454015905576707931374622077248108323243949363974814588017539364895822}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{15} - \frac{9718194280290977804592393000882413045767897419046903375947063155603578380}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{14} - \frac{17258145176792693642790369466278531800611106484852013124275528666448771298}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{13} + \frac{588634623862484836495587874850655361067630441614704715639777892363760877}{1205878875739970169090451277767519256166586488483730215558724488996750211} a^{12} - \frac{31898291626316003379690781189884248967932414486487169729561196331951029682}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{11} - \frac{23641197807586306350952848556919492195624167057259564226221259065687466058}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{10} - \frac{12653627139058625506871255237636322531825531726238196696550382093698687276}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{9} + \frac{14189122928595243468731426717403635651715750404035922932896729622331775305}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{8} + \frac{1541820500219721729856484321977329338150583117656494543018252298179838810}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{7} + \frac{31327929920678712222644237181963602396996642389852960851880153813155357065}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{6} - \frac{23312512276374797261198662851052839779766164946152359326428728975146641132}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{5} - \frac{19834144198038018280109750381216612310530383537063987155804536304228987652}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{4} + \frac{9597765110741742682932269676042472790857813662458015740358355049700999331}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{3} - \frac{22272245719408462907950423060915185772433899232666979649705382698708828177}{63911580414218418961793917721678520576829083889637701424612397916827761183} a^{2} - \frac{4886110869208312975488931715982303298107574979617717249549020732080245795}{63911580414218418961793917721678520576829083889637701424612397916827761183} a - \frac{505170996438625176032099302029500797698080190401945237739039357599919025}{1205878875739970169090451277767519256166586488483730215558724488996750211}$  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:  \( 935275490448786200 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 935275490448786200 \cdot 1}{2\sqrt{258850007664814362506653464185428842369437873281}}\approx 0.120474572261678$ (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 $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{17}$ $17$

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
919Data not computed