Properties

Label 17.17.622...001.1
Degree $17$
Signature $[17, 0]$
Discriminant $6.226\times 10^{39}$
Root discriminant $219.20$
Ramified prime $307$
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 - 144*x^15 + 241*x^14 + 6894*x^13 - 14938*x^12 - 127923*x^11 + 323969*x^10 + 847982*x^9 - 2194186*x^8 - 2617873*x^7 + 6091397*x^6 + 3745755*x^5 - 7069429*x^4 - 1600190*x^3 + 3100257*x^2 - 220118*x - 208777)
 
gp: K = bnfinit(x^17 - x^16 - 144*x^15 + 241*x^14 + 6894*x^13 - 14938*x^12 - 127923*x^11 + 323969*x^10 + 847982*x^9 - 2194186*x^8 - 2617873*x^7 + 6091397*x^6 + 3745755*x^5 - 7069429*x^4 - 1600190*x^3 + 3100257*x^2 - 220118*x - 208777, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-208777, -220118, 3100257, -1600190, -7069429, 3745755, 6091397, -2617873, -2194186, 847982, 323969, -127923, -14938, 6894, 241, -144, -1, 1]);
 

\(x^{17} - x^{16} - 144 x^{15} + 241 x^{14} + 6894 x^{13} - 14938 x^{12} - 127923 x^{11} + 323969 x^{10} + 847982 x^{9} - 2194186 x^{8} - 2617873 x^{7} + 6091397 x^{6} + 3745755 x^{5} - 7069429 x^{4} - 1600190 x^{3} + 3100257 x^{2} - 220118 x - 208777\)  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:  \(6226070121392010397563990173530787496001\)\(\medspace = 307^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $219.20$
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:  $307$
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:  \(307\)
Dirichlet character group:    $\lbrace$$\chi_{307}(64,·)$, $\chi_{307}(1,·)$, $\chi_{307}(235,·)$, $\chi_{307}(9,·)$, $\chi_{307}(269,·)$, $\chi_{307}(272,·)$, $\chi_{307}(81,·)$, $\chi_{307}(24,·)$, $\chi_{307}(216,·)$, $\chi_{307}(280,·)$, $\chi_{307}(102,·)$, $\chi_{307}(273,·)$, $\chi_{307}(105,·)$, $\chi_{307}(299,·)$, $\chi_{307}(304,·)$, $\chi_{307}(114,·)$, $\chi_{307}(115,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} - \frac{8}{17} a^{10} + \frac{6}{17} a^{9} - \frac{1}{17} a^{8} - \frac{6}{17} a^{7} + \frac{3}{17} a^{6} + \frac{3}{17} a^{5} + \frac{1}{17} a^{4} + \frac{5}{17} a^{3} + \frac{6}{17} a^{2} + \frac{7}{17} a$, $\frac{1}{17} a^{13} - \frac{8}{17} a^{11} + \frac{6}{17} a^{10} - \frac{1}{17} a^{9} - \frac{6}{17} a^{8} + \frac{3}{17} a^{7} + \frac{3}{17} a^{6} + \frac{1}{17} a^{5} + \frac{5}{17} a^{4} + \frac{6}{17} a^{3} + \frac{7}{17} a^{2}$, $\frac{1}{901} a^{14} + \frac{25}{901} a^{13} - \frac{22}{901} a^{12} - \frac{7}{901} a^{11} - \frac{368}{901} a^{10} - \frac{285}{901} a^{9} + \frac{343}{901} a^{8} - \frac{178}{901} a^{7} + \frac{9}{53} a^{6} - \frac{114}{901} a^{5} - \frac{223}{901} a^{4} - \frac{355}{901} a^{3} - \frac{334}{901} a^{2} - \frac{30}{901} a + \frac{16}{53}$, $\frac{1}{901} a^{15} - \frac{11}{901} a^{13} + \frac{13}{901} a^{12} + \frac{125}{901} a^{11} - \frac{148}{901} a^{10} + \frac{48}{901} a^{9} - \frac{326}{901} a^{8} - \frac{220}{901} a^{7} - \frac{1}{53} a^{6} - \frac{129}{901} a^{5} - \frac{239}{901} a^{4} - \frac{12}{53} a^{3} - \frac{319}{901} a^{2} + \frac{15}{901} a + \frac{24}{53}$, $\frac{1}{6816521125182246159892694805163172837636879} a^{16} + \frac{2231368591320755466041376900885797066639}{6816521125182246159892694805163172837636879} a^{15} - \frac{500333047000286329477108482550244335860}{6816521125182246159892694805163172837636879} a^{14} + \frac{55820581008783809266382689734368151936425}{6816521125182246159892694805163172837636879} a^{13} + \frac{2894858803101190061009160635631828899164}{400971830893073303523099694421363108096287} a^{12} - \frac{2598196033489025939295318846091865418199883}{6816521125182246159892694805163172837636879} a^{11} + \frac{1807219679477676899241858792787053082720252}{6816521125182246159892694805163172837636879} a^{10} + \frac{1018014402928191491463464901067634783503503}{6816521125182246159892694805163172837636879} a^{9} + \frac{2723057339438563782143475190023639129651}{23586578287827841383711746730668418123311} a^{8} + \frac{339518509285383410198726778518476145810290}{6816521125182246159892694805163172837636879} a^{7} - \frac{1696497716458566781620601737999050018967381}{6816521125182246159892694805163172837636879} a^{6} + \frac{1173820640763274430634014690606032972730154}{6816521125182246159892694805163172837636879} a^{5} - \frac{182205103760111242159031364634130275701569}{400971830893073303523099694421363108096287} a^{4} + \frac{3169872267193071251956574700156738488118130}{6816521125182246159892694805163172837636879} a^{3} + \frac{642101989083470476722169554445476659623315}{6816521125182246159892694805163172837636879} a^{2} + \frac{58769020800320775578643900216631338524192}{6816521125182246159892694805163172837636879} a + \frac{107407058929814357173394846538108238027632}{400971830893073303523099694421363108096287}$  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:  \( 501600232257889.1 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 501600232257889.1 \cdot 1}{2\sqrt{6226070121392010397563990173530787496001}}\approx 0.416610926185679$ (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 $17$ $17$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{17}$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{17}$ $17$

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
307Data not computed