Properties

Label 17.1.10582983069...0401.1
Degree $17$
Signature $[1, 8]$
Discriminant $17^{8}\cdot 79^{8}$
Root discriminant $29.65$
Ramified primes $17, 79$
Class number $1$
Class group Trivial
Galois group $D_{17}$ (as 17T2)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 - 2*x^15 + 24*x^14 - 20*x^13 - 80*x^12 + 115*x^11 + 27*x^10 + 132*x^9 - 302*x^8 - 219*x^7 - 172*x^6 + 41*x^5 - 25*x^4 + 44*x^3 + 39*x^2 - 2*x - 23)
 
gp: K = bnfinit(x^17 - 3*x^16 - 2*x^15 + 24*x^14 - 20*x^13 - 80*x^12 + 115*x^11 + 27*x^10 + 132*x^9 - 302*x^8 - 219*x^7 - 172*x^6 + 41*x^5 - 25*x^4 + 44*x^3 + 39*x^2 - 2*x - 23, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, -2, 39, 44, -25, 41, -172, -219, -302, 132, 27, 115, -80, -20, 24, -2, -3, 1]);
 

Normalized defining polynomial

\( x^{17} - 3 x^{16} - 2 x^{15} + 24 x^{14} - 20 x^{13} - 80 x^{12} + 115 x^{11} + 27 x^{10} + 132 x^{9} - 302 x^{8} - 219 x^{7} - 172 x^{6} + 41 x^{5} - 25 x^{4} + 44 x^{3} + 39 x^{2} - 2 x - 23 \)

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:  \(10582983069512347410470401=17^{8}\cdot 79^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $29.65$
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:  $17, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{85} a^{13} - \frac{1}{17} a^{12} + \frac{33}{85} a^{11} - \frac{2}{85} a^{10} - \frac{6}{85} a^{9} - \frac{2}{17} a^{8} - \frac{3}{17} a^{7} + \frac{21}{85} a^{6} - \frac{6}{85} a^{5} - \frac{28}{85} a^{4} + \frac{27}{85} a^{3} - \frac{6}{85} a^{2} + \frac{26}{85} a - \frac{6}{17}$, $\frac{1}{85} a^{14} + \frac{8}{85} a^{12} - \frac{7}{85} a^{11} - \frac{16}{85} a^{10} - \frac{8}{17} a^{9} + \frac{4}{17} a^{8} + \frac{31}{85} a^{7} + \frac{14}{85} a^{6} + \frac{27}{85} a^{5} - \frac{28}{85} a^{4} - \frac{41}{85} a^{3} - \frac{4}{85} a^{2} + \frac{3}{17} a + \frac{4}{17}$, $\frac{1}{85} a^{15} - \frac{1}{85} a^{12} - \frac{5}{17} a^{11} - \frac{41}{85} a^{10} - \frac{2}{5} a^{9} - \frac{5}{17} a^{8} - \frac{36}{85} a^{7} + \frac{29}{85} a^{6} - \frac{14}{85} a^{5} - \frac{38}{85} a^{4} - \frac{33}{85} a^{3} - \frac{1}{17} a^{2} + \frac{16}{85} a + \frac{36}{85}$, $\frac{1}{5986908182190965} a^{16} + \frac{19008147736664}{5986908182190965} a^{15} + \frac{1447083794532}{5986908182190965} a^{14} + \frac{719896244228}{1197381636438193} a^{13} - \frac{119627042156754}{1197381636438193} a^{12} - \frac{1809467236520317}{5986908182190965} a^{11} - \frac{766118369873373}{5986908182190965} a^{10} + \frac{2400907771710747}{5986908182190965} a^{9} - \frac{1354382065349224}{5986908182190965} a^{8} - \frac{131590314109049}{352171069540645} a^{7} - \frac{283626651046264}{5986908182190965} a^{6} - \frac{255764972524883}{5986908182190965} a^{5} + \frac{1156853177485618}{5986908182190965} a^{4} + \frac{1006256765537949}{5986908182190965} a^{3} - \frac{2918673005694162}{5986908182190965} a^{2} - \frac{2835855545095372}{5986908182190965} a + \frac{12100959331581}{352171069540645}$

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:  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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 663161.006215 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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 $17$ $17$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $17$ R $17$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $17$ $17$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.1$x^{2} - 79$$2$$1$$1$$C_2$$[\ ]_{2}$