# Properties

 Label 17.1.542...872.1 Degree $17$ Signature $[1, 8]$ Discriminant $5.421\times 10^{25}$ Root discriminant $32.64$ Ramified primes $2, 17$ Class number $1$ Class group trivial Galois group $F_{17}$ (as 17T5)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2)

gp: K = bnfinit(x^17 - 2, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);

$$x^{17} - 2$$

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: $$54214017802982966177103872$$$$\medspace = 2^{16}\cdot 17^{17}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $32.64$ 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: $2, 17$ 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 $$a$$)

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

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: $$a^{9} - a - 1$$,  $$a - 1$$,  $$a^{10} + a^{9} - a^{2} - a - 1$$,  $$a^{15} + a^{13} + a^{11} + a^{9} + a^{7} + a^{5} + a^{3} + a + 1$$,  $$a^{16} + a^{12} - a^{10} + a^{9} - a^{7} - a^{3} + a^{2} - 1$$,  $$a^{16} + a^{15} + a^{14} - a^{11} - a^{9} - a^{7} - a^{5} - a^{3} + 1$$,  $$a^{13} - a^{10} + a^{9} + a^{7} - a^{6} + a^{5} + a^{3} + a + 1$$,  $$a^{16} + a^{15} - a^{11} + a^{9} - a^{7} + a^{6} + a^{5} - 2 a - 1$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$2966240.41128$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{8}\cdot 2966240.41128 \cdot 1}{2\sqrt{54214017802982966177103872}}\approx 0.978564591531$

## Galois group

$F_{17}$ (as 17T5):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 272 The 17 conjugacy class representatives for $F_{17}$ Character table for $F_{17}$

## 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 R $16{,}\,{\href{/padicField/3.1.0.1}{1} }$ $16{,}\,{\href{/padicField/5.1.0.1}{1} }$ $16{,}\,{\href{/padicField/7.1.0.1}{1} }$ $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$ R ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $16{,}\,{\href{/padicField/23.1.0.1}{1} }$ $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ $16{,}\,{\href{/padicField/31.1.0.1}{1} }$ $16{,}\,{\href{/padicField/37.1.0.1}{1} }$ $16{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
2Data not computed
17Data not computed