# 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)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining 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]);

$$x^{17} - x^{16} - 144 x^{15} + 241 x^{14} + 6894 x^{13} - 14938 x^{12} - 127923 x^{11} + 323969 x^{10} + 847982 x^{9} - 2194186 x^{8} - 2617873 x^{7} + 6091397 x^{6} + 3745755 x^{5} - 7069429 x^{4} - 1600190 x^{3} + 3100257 x^{2} - 220118 x - 208777$$

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 $$a$$)

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

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 $2$) 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