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Properties

Label 17.1.16071318903...0625.1
Degree $17$
Signature $[1, 8]$
Discriminant $5^{8}\cdot 283^{8}$
Root discriminant $30.39$
Ramified primes $5, 283$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![125, 550, 1505, 2855, 2972, 1252, -68, -803, -515, 133, 143, 88, -2, -30, 16, 1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^16 + x^15 + 16*x^14 - 30*x^13 - 2*x^12 + 88*x^11 + 143*x^10 + 133*x^9 - 515*x^8 - 803*x^7 - 68*x^6 + 1252*x^5 + 2972*x^4 + 2855*x^3 + 1505*x^2 + 550*x + 125)
 
gp: K = bnfinit(x^17 - 4*x^16 + x^15 + 16*x^14 - 30*x^13 - 2*x^12 + 88*x^11 + 143*x^10 + 133*x^9 - 515*x^8 - 803*x^7 - 68*x^6 + 1252*x^5 + 2972*x^4 + 2855*x^3 + 1505*x^2 + 550*x + 125, 1)
 

Normalized defining polynomial

\( x^{17} - 4 x^{16} + x^{15} + 16 x^{14} - 30 x^{13} - 2 x^{12} + 88 x^{11} + 143 x^{10} + 133 x^{9} - 515 x^{8} - 803 x^{7} - 68 x^{6} + 1252 x^{5} + 2972 x^{4} + 2855 x^{3} + 1505 x^{2} + 550 x + 125 \)

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:  \(16071318903145633687890625=5^{8}\cdot 283^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.39$
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:  $5, 283$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{10} + \frac{2}{25} a^{9} + \frac{2}{25} a^{8} + \frac{2}{25} a^{7} + \frac{4}{25} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{9}{25} a^{3} + \frac{2}{5} a$, $\frac{1}{25} a^{12} - \frac{2}{25} a^{10} + \frac{1}{25} a^{9} + \frac{1}{25} a^{8} - \frac{2}{25} a^{7} + \frac{3}{25} a^{6} + \frac{1}{5} a^{5} - \frac{11}{25} a^{4} - \frac{2}{25} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{125} a^{13} - \frac{2}{125} a^{11} + \frac{11}{125} a^{10} + \frac{6}{125} a^{9} - \frac{2}{125} a^{8} - \frac{2}{125} a^{7} - \frac{1}{5} a^{6} - \frac{11}{125} a^{5} + \frac{3}{125} a^{4} + \frac{8}{25} a^{3} - \frac{1}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{125} a^{14} - \frac{2}{125} a^{12} + \frac{1}{125} a^{11} + \frac{1}{125} a^{10} + \frac{3}{125} a^{9} + \frac{3}{125} a^{8} + \frac{1}{25} a^{7} - \frac{1}{125} a^{6} + \frac{53}{125} a^{5} + \frac{3}{25} a^{4} + \frac{1}{25} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{875} a^{15} + \frac{2}{875} a^{14} + \frac{3}{875} a^{13} + \frac{1}{125} a^{12} + \frac{3}{875} a^{11} - \frac{1}{175} a^{10} - \frac{8}{125} a^{9} + \frac{81}{875} a^{8} + \frac{24}{875} a^{7} - \frac{229}{875} a^{6} - \frac{184}{875} a^{5} + \frac{8}{175} a^{4} + \frac{8}{25} a^{3} - \frac{1}{35} a^{2} - \frac{1}{35} a - \frac{2}{7}$, $\frac{1}{2615944069375} a^{16} + \frac{218937184}{2615944069375} a^{15} + \frac{368743109}{373706295625} a^{14} + \frac{737489092}{523188813875} a^{13} + \frac{7919246736}{523188813875} a^{12} - \frac{35805577492}{2615944069375} a^{11} + \frac{196162750347}{2615944069375} a^{10} - \frac{26640288041}{2615944069375} a^{9} - \frac{46749711864}{523188813875} a^{8} + \frac{47954499897}{523188813875} a^{7} + \frac{710493631017}{2615944069375} a^{6} - \frac{229449530337}{2615944069375} a^{5} + \frac{171732759551}{2615944069375} a^{4} + \frac{244902547024}{523188813875} a^{3} + \frac{33135306798}{523188813875} a^{2} - \frac{52235576909}{104637762775} a - \frac{4281699832}{20927552555}$

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:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2534195.05633 \) (assuming GRH)
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 $17$ $17$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $17$ $17$ $17$ $17$ $17$ $17$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
283Data not computed