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Properties

Label 17.17.8272402618...0000.1
Degree $17$
Signature $[17, 0]$
Discriminant $2^{16}\cdot 5^{16}\cdot 17^{17}$
Root discriminant $148.47$
Ramified primes $2, 5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_{17}$ (as 17T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1737500, 6640625, 0, -15937500, 0, 11156250, 0, -3506250, 0, 584375, 0, -55250, 0, 2975, 0, -85, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 85*x^15 + 2975*x^13 - 55250*x^11 + 584375*x^9 - 3506250*x^7 + 11156250*x^5 - 15937500*x^3 + 6640625*x - 1737500)
 
gp: K = bnfinit(x^17 - 85*x^15 + 2975*x^13 - 55250*x^11 + 584375*x^9 - 3506250*x^7 + 11156250*x^5 - 15937500*x^3 + 6640625*x - 1737500, 1)
 

Normalized defining polynomial

\( x^{17} - 85 x^{15} + 2975 x^{13} - 55250 x^{11} + 584375 x^{9} - 3506250 x^{7} + 11156250 x^{5} - 15937500 x^{3} + 6640625 x - 1737500 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $17$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[17, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8272402618863367641770000000000000000=2^{16}\cdot 5^{16}\cdot 17^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $148.47$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{25} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{725} a^{9} + \frac{11}{725} a^{8} + \frac{13}{725} a^{7} - \frac{1}{145} a^{6} - \frac{2}{29} a^{5} - \frac{2}{145} a^{4} - \frac{5}{29} a^{3} - \frac{10}{29} a^{2} - \frac{7}{29} a - \frac{1}{29}$, $\frac{1}{725} a^{10} + \frac{8}{725} a^{8} - \frac{3}{725} a^{7} + \frac{1}{145} a^{6} - \frac{8}{145} a^{5} - \frac{3}{145} a^{4} - \frac{13}{29} a^{3} - \frac{13}{29} a^{2} - \frac{11}{29} a + \frac{11}{29}$, $\frac{1}{3625} a^{11} + \frac{1}{145} a^{8} - \frac{14}{725} a^{7} - \frac{2}{145} a^{5} - \frac{4}{145} a^{4} - \frac{12}{29} a^{3} + \frac{8}{29} a^{2} - \frac{4}{29} a - \frac{10}{29}$, $\frac{1}{3625} a^{12} - \frac{11}{725} a^{8} - \frac{7}{725} a^{7} + \frac{3}{145} a^{6} - \frac{12}{145} a^{5} + \frac{8}{145} a^{4} + \frac{4}{29} a^{3} - \frac{12}{29} a^{2} - \frac{4}{29} a + \frac{5}{29}$, $\frac{1}{3625} a^{13} - \frac{2}{725} a^{8} + \frac{13}{725} a^{7} + \frac{6}{145} a^{6} + \frac{14}{145} a^{5} - \frac{2}{145} a^{4} - \frac{9}{29} a^{3} + \frac{2}{29} a^{2} - \frac{14}{29} a - \frac{11}{29}$, $\frac{1}{18125} a^{14} + \frac{7}{725} a^{8} - \frac{12}{725} a^{7} + \frac{14}{145} a^{6} + \frac{13}{145} a^{5} - \frac{4}{145} a^{4} + \frac{10}{29} a^{3} - \frac{1}{29} a^{2} - \frac{5}{29} a - \frac{12}{29}$, $\frac{1}{18125} a^{15} - \frac{2}{725} a^{8} + \frac{8}{725} a^{7} - \frac{9}{145} a^{6} + \frac{8}{145} a^{5} + \frac{6}{145} a^{4} + \frac{5}{29} a^{3} + \frac{7}{29} a^{2} + \frac{8}{29} a + \frac{7}{29}$, $\frac{1}{18125} a^{16} + \frac{1}{725} a^{8} + \frac{2}{145} a^{7} + \frac{6}{145} a^{6} - \frac{14}{145} a^{5} - \frac{8}{145} a^{4} - \frac{3}{29} a^{3} - \frac{12}{29} a^{2} - \frac{7}{29} a - \frac{2}{29}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $16$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 48080253767000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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 $16{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R $16{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
5Data not computed
17Data not computed