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Properties

Label 17.1.23907243568...7153.1
Degree $17$
Signature $[1, 8]$
Discriminant $17^{19}$
Root discriminant $23.73$
Ramified prime $17$
Class number $1$
Class group Trivial
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![4, 17, 17, 68, 255, 255, -68, -272, -170, 102, 85, 17, -34, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 34*x^12 + 17*x^11 + 85*x^10 + 102*x^9 - 170*x^8 - 272*x^7 - 68*x^6 + 255*x^5 + 255*x^4 + 68*x^3 + 17*x^2 + 17*x + 4)
 
gp: K = bnfinit(x^17 - 34*x^12 + 17*x^11 + 85*x^10 + 102*x^9 - 170*x^8 - 272*x^7 - 68*x^6 + 255*x^5 + 255*x^4 + 68*x^3 + 17*x^2 + 17*x + 4, 1)
 

Normalized defining polynomial

\( x^{17} - 34 x^{12} + 17 x^{11} + 85 x^{10} + 102 x^{9} - 170 x^{8} - 272 x^{7} - 68 x^{6} + 255 x^{5} + 255 x^{4} + 68 x^{3} + 17 x^{2} + 17 x + 4 \)

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

Invariants

Degree:  $17$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(239072435685151324847153=17^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.73$
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:  $17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{88438805268825769} a^{16} + \frac{701881911652902}{88438805268825769} a^{15} + \frac{30517215125455540}{88438805268825769} a^{14} + \frac{33334804539175149}{88438805268825769} a^{13} + \frac{18524760416425501}{88438805268825769} a^{12} + \frac{40493661390712513}{88438805268825769} a^{11} - \frac{28980071630209230}{88438805268825769} a^{10} + \frac{6606287507116770}{88438805268825769} a^{9} - \frac{32047230171073133}{88438805268825769} a^{8} + \frac{21020098631402549}{88438805268825769} a^{7} + \frac{28865605374446275}{88438805268825769} a^{6} + \frac{14307358480140575}{88438805268825769} a^{5} + \frac{42797670772418278}{88438805268825769} a^{4} + \frac{10384855336836480}{88438805268825769} a^{3} - \frac{22008377977408818}{88438805268825769} a^{2} + \frac{21070310367492420}{88438805268825769} a + \frac{31429974412590759}{88438805268825769}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 195995.113671 \)
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ 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
17Data not computed

Additional information

The polynomial was contributed by Noam Elkies.