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Properties

Label 17.1.46300980897...3841.1
Degree $17$
Signature $[1, 8]$
Discriminant $383^{8}$
Root discriminant $16.43$
Ramified prime $383$
Class number $1$
Class group Trivial
Galois group $D_{17}$ (as 17T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 6, 24, 42, 31, 23, 7, 1, 4, 11, 7, 13, 1, 0, -1, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - x^15 - x^14 + x^12 + 13*x^11 + 7*x^10 + 11*x^9 + 4*x^8 + x^7 + 7*x^6 + 23*x^5 + 31*x^4 + 42*x^3 + 24*x^2 + 6*x - 1)
 
gp: K = bnfinit(x^17 - x^16 - x^15 - x^14 + x^12 + 13*x^11 + 7*x^10 + 11*x^9 + 4*x^8 + x^7 + 7*x^6 + 23*x^5 + 31*x^4 + 42*x^3 + 24*x^2 + 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{17} - x^{16} - x^{15} - x^{14} + x^{12} + 13 x^{11} + 7 x^{10} + 11 x^{9} + 4 x^{8} + x^{7} + 7 x^{6} + 23 x^{5} + 31 x^{4} + 42 x^{3} + 24 x^{2} + 6 x - 1 \)

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:  \(463009808974713123841=383^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.43$
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:  $383$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{65} a^{14} + \frac{11}{65} a^{13} - \frac{3}{13} a^{12} - \frac{31}{65} a^{11} + \frac{9}{65} a^{10} + \frac{1}{13} a^{9} + \frac{28}{65} a^{8} + \frac{12}{65} a^{7} - \frac{28}{65} a^{6} + \frac{24}{65} a^{5} - \frac{31}{65} a^{4} + \frac{3}{65} a^{3} - \frac{6}{65} a^{2} + \frac{2}{13} a + \frac{1}{65}$, $\frac{1}{715} a^{15} - \frac{2}{715} a^{14} - \frac{93}{715} a^{13} - \frac{226}{715} a^{12} - \frac{173}{715} a^{11} - \frac{307}{715} a^{10} + \frac{93}{715} a^{9} - \frac{92}{715} a^{8} + \frac{271}{715} a^{7} - \frac{67}{715} a^{6} - \frac{18}{715} a^{5} - \frac{244}{715} a^{4} - \frac{35}{143} a^{3} + \frac{218}{715} a^{2} - \frac{259}{715} a - \frac{1}{55}$, $\frac{1}{3754465} a^{16} + \frac{382}{750893} a^{15} - \frac{4687}{3754465} a^{14} + \frac{1760433}{3754465} a^{13} - \frac{328105}{750893} a^{12} + \frac{1358022}{3754465} a^{11} + \frac{1837179}{3754465} a^{10} + \frac{466309}{3754465} a^{9} + \frac{1345062}{3754465} a^{8} - \frac{301062}{750893} a^{7} - \frac{4218}{42185} a^{6} - \frac{205108}{750893} a^{5} + \frac{1173782}{3754465} a^{4} - \frac{654867}{3754465} a^{3} - \frac{940898}{3754465} a^{2} + \frac{1634929}{3754465} a + \frac{1364334}{3754465}$

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:  \( 2610.31631075 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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 $17$ $17$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $17$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
383Data not computed