# Properties

 Label 17.13.223...557.1 Degree $17$ Signature $[13, 2]$ Discriminant $2.233\times 10^{27}$ Root discriminant $40.62$ Ramified primes see page Class number $1$ (GRH) Class group trivial (GRH) Galois group $S_{17}$ (as 17T10)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*x + 1)

gp: K = bnfinit(x^17 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 24, 42, -205, 27, 560, -342, -675, 547, 389, -356, -115, 112, 17, -17, -1, 1]);

$$x^{17} - x^{16} - 17 x^{15} + 17 x^{14} + 112 x^{13} - 115 x^{12} - 356 x^{11} + 389 x^{10} + 547 x^{9} - 675 x^{8} - 342 x^{7} + 560 x^{6} + 27 x^{5} - 205 x^{4} + 42 x^{3} + 24 x^{2} - 10 x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $17$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[13, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$2232673506822932495146063557$$$$\medspace = 3^{3}\cdot 205542871\cdot 402308340646912121$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $40.62$ 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: $3, 205542871, 402308340646912121$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{21} a^{16} - \frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{10}{21} a^{13} + \frac{2}{7} a^{12} + \frac{2}{7} a^{11} + \frac{10}{21} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{21} a^{5} + \frac{2}{21} a^{4} + \frac{8}{21} a^{3} - \frac{3}{7} a^{2} + \frac{1}{3} a - \frac{1}{7}$

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: $14$ 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: $$129255854.893$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{13}\cdot(2\pi)^{2}\cdot 129255854.893 \cdot 1}{2\sqrt{2232673506822932495146063557}}\approx 0.442341106108$ (assuming GRH)

## Galois group

$S_{17}$ (as 17T10):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 355687428096000 The 297 conjugacy class representatives for $S_{17}$ are not computed Character table for $S_{17}$ is not computed

## 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 ${\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.8.0.1}{8} }$ R ${\href{/padicField/5.13.0.1}{13} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.8.0.1}{8} }$ ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ $17$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ $17$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$

In the table, R denotes a ramified prime. 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
$3$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2} 3.13.0.1x^{13} - x + 1$$1$$13$$0$$C_{13}$$[\ ]^{13}$
205542871Data not computed
402308340646912121Data not computed