Properties

Label 17.1.58497592625...5121.1
Degree $17$
Signature $[1, 8]$
Discriminant $1663^{8}$
Root discriminant $32.79$
Ramified prime $1663$
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 - 2*x^16 + 9*x^15 - 23*x^14 + 59*x^13 - 112*x^12 + 156*x^11 - 49*x^10 + 221*x^9 - 25*x^8 - 162*x^7 - 4*x^6 + 553*x^5 + 1435*x^4 + 1716*x^3 + 1462*x^2 + 615*x + 225)
 
gp: K = bnfinit(x^17 - 2*x^16 + 9*x^15 - 23*x^14 + 59*x^13 - 112*x^12 + 156*x^11 - 49*x^10 + 221*x^9 - 25*x^8 - 162*x^7 - 4*x^6 + 553*x^5 + 1435*x^4 + 1716*x^3 + 1462*x^2 + 615*x + 225, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![225, 615, 1462, 1716, 1435, 553, -4, -162, -25, 221, -49, 156, -112, 59, -23, 9, -2, 1]);
 

Normalized defining polynomial

\( x^{17} - 2 x^{16} + 9 x^{15} - 23 x^{14} + 59 x^{13} - 112 x^{12} + 156 x^{11} - 49 x^{10} + 221 x^{9} - 25 x^{8} - 162 x^{7} - 4 x^{6} + 553 x^{5} + 1435 x^{4} + 1716 x^{3} + 1462 x^{2} + 615 x + 225 \)

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:  \(58497592625273225470725121=1663^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $32.79$
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:  $1663$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{21} a^{11} + \frac{2}{21} a^{10} - \frac{2}{21} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{4}{21} a^{3} - \frac{5}{21} a^{2} - \frac{1}{21} a + \frac{3}{7}$, $\frac{1}{21} a^{12} + \frac{1}{21} a^{10} - \frac{1}{21} a^{9} + \frac{3}{7} a^{7} - \frac{1}{3} a^{4} + \frac{1}{7} a^{3} + \frac{2}{21} a^{2} + \frac{4}{21} a + \frac{1}{7}$, $\frac{1}{21} a^{13} - \frac{1}{7} a^{10} + \frac{2}{21} a^{9} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{8}{21} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{4}{21} a - \frac{3}{7}$, $\frac{1}{105} a^{14} + \frac{2}{105} a^{13} - \frac{1}{105} a^{12} + \frac{2}{105} a^{11} + \frac{1}{21} a^{10} + \frac{2}{105} a^{9} - \frac{12}{35} a^{8} + \frac{1}{5} a^{7} + \frac{38}{105} a^{6} - \frac{26}{105} a^{5} - \frac{10}{21} a^{4} + \frac{19}{105} a^{3} - \frac{26}{105} a^{2} + \frac{4}{105} a + \frac{3}{7}$, $\frac{1}{315} a^{15} - \frac{1}{315} a^{14} - \frac{1}{45} a^{13} - \frac{1}{315} a^{11} + \frac{52}{315} a^{10} - \frac{2}{315} a^{9} + \frac{8}{105} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{19}{45} a^{5} + \frac{11}{35} a^{4} - \frac{14}{45} a^{3} + \frac{2}{315} a^{2} + \frac{83}{315} a - \frac{10}{21}$, $\frac{1}{621500650885485} a^{16} + \frac{7512315506}{88785807269355} a^{15} + \frac{130614479333}{621500650885485} a^{14} - \frac{446711379343}{41433376725699} a^{13} + \frac{279502235714}{12683686752765} a^{12} + \frac{4266351704}{183063520143} a^{11} + \frac{40188942280804}{621500650885485} a^{10} - \frac{1216593563974}{69055627876165} a^{9} + \frac{75376096973189}{621500650885485} a^{8} + \frac{196803201298633}{621500650885485} a^{7} - \frac{208227430263023}{621500650885485} a^{6} + \frac{16245006096862}{41433376725699} a^{5} - \frac{35586115088516}{621500650885485} a^{4} + \frac{57728372863951}{124300130177097} a^{3} - \frac{232377545045389}{621500650885485} a^{2} - \frac{28320475462726}{69055627876165} a - \frac{805141343915}{13811125575233}$

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:  \( 3798575.81941 \)
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 $17$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $17$ $17$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $17$ $17$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $17$ $17$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $17$

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
1663Data not computed