# Properties

 Label 17.1.200...521.1 Degree $17$ Signature $[1, 8]$ Discriminant $2.007\times 10^{24}$ Root discriminant $26.89$ Ramified prime $1091$ Class number $1$ (GRH) Class group trivial (GRH) Galois group $D_{17}$ (as 17T2)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64)

gp: K = bnfinit(x^17 - 2*x^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 80, 240, 220, 200, 275, 88, 134, 18, 72, 10, 44, -16, 10, -18, 4, -2, 1]);

$$x^{17} - 2 x^{16} + 4 x^{15} - 18 x^{14} + 10 x^{13} - 16 x^{12} + 44 x^{11} + 10 x^{10} + 72 x^{9} + 18 x^{8} + 134 x^{7} + 88 x^{6} + 275 x^{5} + 200 x^{4} + 220 x^{3} + 240 x^{2} + 80 x + 64$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $17$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[1, 8]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$2007233999721653909999521$$$$\medspace = 1091^{8}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $26.89$ 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: $1091$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{3}{16} a^{5} + \frac{1}{16} a^{4} + \frac{7}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} - \frac{1}{32} a^{10} + \frac{1}{16} a^{9} + \frac{1}{32} a^{8} + \frac{1}{16} a^{7} + \frac{3}{32} a^{6} + \frac{1}{16} a^{5} - \frac{5}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{16} a^{10} + \frac{1}{32} a^{9} + \frac{1}{16} a^{8} + \frac{3}{32} a^{7} + \frac{1}{16} a^{6} - \frac{5}{32} a^{5} + \frac{1}{4} a^{2}$, $\frac{1}{8286671488} a^{16} - \frac{68849395}{8286671488} a^{15} + \frac{120169979}{8286671488} a^{14} - \frac{31185489}{8286671488} a^{13} - \frac{231372529}{8286671488} a^{12} - \frac{155375939}{8286671488} a^{11} + \frac{626061315}{8286671488} a^{10} + \frac{5191329}{123681664} a^{9} - \frac{370481855}{8286671488} a^{8} + \frac{21699013}{8286671488} a^{7} + \frac{508174493}{8286671488} a^{6} + \frac{100883231}{487451264} a^{5} - \frac{61534825}{258958484} a^{4} + \frac{612237787}{2071667872} a^{3} - \frac{15242284}{64739621} a^{2} + \frac{225047811}{517916968} a - \frac{42857061}{129479242}$

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: $8$ 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: $$1757145.67579$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{8}\cdot 1757145.67579 \cdot 1}{2\sqrt{2007233999721653909999521}}\approx 3.01264329152$ (assuming GRH)

## Galois group

$D_{17}$ (as 17T2):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 34 The 10 conjugacy class representatives for $D_{17}$ Character table for $D_{17}$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Sibling fields

 Galois closure: data not computed

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.2.0.1}{2} }^{8}{,}\,{\href{/padicField/2.1.0.1}{1} }$ $17$ $17$ $17$ $17$ $17$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }$ $17$ $17$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }$ $17$ ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }$ $17$ ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }$ $17$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
1091Data not computed