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Properties

Label 17.1.34895459505...2321.1
Degree $17$
Signature $[1, 8]$
Discriminant $1559^{8}$
Root discriminant $31.81$
Ramified prime $1559$
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![337, 874, 1015, 493, 132, 384, 588, 352, -75, -203, -138, 0, 18, 24, -3, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 - 3*x^14 + 24*x^13 + 18*x^12 - 138*x^10 - 203*x^9 - 75*x^8 + 352*x^7 + 588*x^6 + 384*x^5 + 132*x^4 + 493*x^3 + 1015*x^2 + 874*x + 337)
 
gp: K = bnfinit(x^17 - 2*x^16 - 3*x^14 + 24*x^13 + 18*x^12 - 138*x^10 - 203*x^9 - 75*x^8 + 352*x^7 + 588*x^6 + 384*x^5 + 132*x^4 + 493*x^3 + 1015*x^2 + 874*x + 337, 1)
 

Normalized defining polynomial

\( x^{17} - 2 x^{16} - 3 x^{14} + 24 x^{13} + 18 x^{12} - 138 x^{10} - 203 x^{9} - 75 x^{8} + 352 x^{7} + 588 x^{6} + 384 x^{5} + 132 x^{4} + 493 x^{3} + 1015 x^{2} + 874 x + 337 \)

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:  \(34895459505131153638432321=1559^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.81$
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:  $1559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{443} a^{15} - \frac{22}{443} a^{14} + \frac{5}{443} a^{13} + \frac{164}{443} a^{12} - \frac{115}{443} a^{11} + \frac{86}{443} a^{10} + \frac{18}{443} a^{9} + \frac{190}{443} a^{8} + \frac{128}{443} a^{7} + \frac{214}{443} a^{6} + \frac{197}{443} a^{5} + \frac{132}{443} a^{4} + \frac{206}{443} a^{3} + \frac{169}{443} a^{2} + \frac{90}{443} a + \frac{124}{443}$, $\frac{1}{6248448141901847587} a^{16} - \frac{5917373005460816}{6248448141901847587} a^{15} + \frac{2591849632023124683}{6248448141901847587} a^{14} - \frac{1956235081860935964}{6248448141901847587} a^{13} - \frac{162440894757254489}{6248448141901847587} a^{12} - \frac{2713326455039835886}{6248448141901847587} a^{11} - \frac{475841400320734068}{6248448141901847587} a^{10} + \frac{329526275621675768}{6248448141901847587} a^{9} + \frac{1853850797561825028}{6248448141901847587} a^{8} + \frac{667163196984398742}{6248448141901847587} a^{7} - \frac{1631723818809708979}{6248448141901847587} a^{6} - \frac{13397811352199677}{60664545067008229} a^{5} + \frac{1483113705698191815}{6248448141901847587} a^{4} - \frac{39899010751413071}{6248448141901847587} a^{3} - \frac{2192669092600096439}{6248448141901847587} a^{2} + \frac{30175447154931428}{60664545067008229} a + \frac{327744773698944580}{6248448141901847587}$

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:  \( 432952.23952 \)
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 $17$ $17$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1559Data not computed