Properties

Label 17.1.92267897090...0049.1
Degree $17$
Signature $[1, 8]$
Discriminant $3^{8}\cdot 17^{18}$
Root discriminant $33.68$
Ramified primes $3, 17$
Class number $1$
Class group Trivial
Galois group $C_{17}:C_{8}$ (as 17T4)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 34*x^14 - 68*x^13 + 17*x^12 + 323*x^11 + 884*x^10 + 1241*x^9 + 1394*x^8 + 1003*x^7 + 935*x^6 + 663*x^5 + 901*x^4 + 578*x^3 + 493*x^2 + 136*x + 57)
 
gp: K = bnfinit(x^17 - 34*x^14 - 68*x^13 + 17*x^12 + 323*x^11 + 884*x^10 + 1241*x^9 + 1394*x^8 + 1003*x^7 + 935*x^6 + 663*x^5 + 901*x^4 + 578*x^3 + 493*x^2 + 136*x + 57, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![57, 136, 493, 578, 901, 663, 935, 1003, 1394, 1241, 884, 323, 17, -68, -34, 0, 0, 1]);
 

Normalized defining polynomial

\( x^{17} - 34 x^{14} - 68 x^{13} + 17 x^{12} + 323 x^{11} + 884 x^{10} + 1241 x^{9} + 1394 x^{8} + 1003 x^{7} + 935 x^{6} + 663 x^{5} + 901 x^{4} + 578 x^{3} + 493 x^{2} + 136 x + 57 \)

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:  \(92267897090016343666010049=3^{8}\cdot 17^{18}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $33.68$
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:  $3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{26} a^{15} - \frac{1}{26} a^{14} - \frac{1}{26} a^{13} - \frac{5}{26} a^{12} + \frac{2}{13} a^{11} + \frac{5}{13} a^{10} + \frac{3}{13} a^{9} - \frac{1}{2} a^{8} - \frac{3}{13} a^{7} - \frac{2}{13} a^{6} + \frac{5}{26} a^{5} - \frac{11}{26} a^{4} - \frac{6}{13} a^{3} - \frac{1}{26} a^{2} - \frac{4}{13} a - \frac{2}{13}$, $\frac{1}{29364921293260777978} a^{16} + \frac{4098531100523443}{2258840099481598306} a^{15} + \frac{2290759332112233059}{14682460646630388989} a^{14} - \frac{1170527218531311085}{14682460646630388989} a^{13} + \frac{4745795395826816265}{29364921293260777978} a^{12} + \frac{2119269114832081635}{14682460646630388989} a^{11} - \frac{6516938756224068970}{14682460646630388989} a^{10} + \frac{10822074663067952511}{29364921293260777978} a^{9} + \frac{10080944291125566983}{29364921293260777978} a^{8} + \frac{2626619353481275667}{14682460646630388989} a^{7} - \frac{5697224104791830259}{29364921293260777978} a^{6} - \frac{6506936825805858263}{14682460646630388989} a^{5} - \frac{7890559222185381011}{29364921293260777978} a^{4} + \frac{979884448016261775}{2258840099481598306} a^{3} - \frac{9844269148539855935}{29364921293260777978} a^{2} + \frac{1392516107826284591}{14682460646630388989} a + \frac{11582301900463138945}{29364921293260777978}$

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:  \( 4992205.48619 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_{17}:C_8$ (as 17T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 136
The 10 conjugacy class representatives for $C_{17}:C_{8}$
Character table for $C_{17}:C_{8}$

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
17Data not computed