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Properties

Label 17.1.37103038784...7761.1
Degree $17$
Signature $[1, 8]$
Discriminant $1571^{8}$
Root discriminant $31.92$
Ramified prime $1571$
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![-16, -88, -416, 166, -1041, 661, -1027, 181, -535, 55, -62, 56, 17, -13, 9, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 5*x^15 + 9*x^14 - 13*x^13 + 17*x^12 + 56*x^11 - 62*x^10 + 55*x^9 - 535*x^8 + 181*x^7 - 1027*x^6 + 661*x^5 - 1041*x^4 + 166*x^3 - 416*x^2 - 88*x - 16)
 
gp: K = bnfinit(x^17 - x^16 + 5*x^15 + 9*x^14 - 13*x^13 + 17*x^12 + 56*x^11 - 62*x^10 + 55*x^9 - 535*x^8 + 181*x^7 - 1027*x^6 + 661*x^5 - 1041*x^4 + 166*x^3 - 416*x^2 - 88*x - 16, 1)
 

Normalized defining polynomial

\( x^{17} - x^{16} + 5 x^{15} + 9 x^{14} - 13 x^{13} + 17 x^{12} + 56 x^{11} - 62 x^{10} + 55 x^{9} - 535 x^{8} + 181 x^{7} - 1027 x^{6} + 661 x^{5} - 1041 x^{4} + 166 x^{3} - 416 x^{2} - 88 x - 16 \)

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:  \(37103038784156796454937761=1571^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.92$
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:  $1571$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{11} - \frac{1}{16} a^{10} + \frac{3}{32} a^{9} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} - \frac{5}{32} a^{5} + \frac{1}{8} a^{4} - \frac{13}{32} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{32} a^{6} - \frac{3}{16} a^{5} - \frac{5}{32} a^{4} - \frac{7}{16} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{1}{64} a^{7} - \frac{5}{64} a^{6} + \frac{1}{64} a^{5} - \frac{9}{64} a^{4} - \frac{5}{32} a^{3} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{128} a^{14} - \frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{1}{128} a^{10} + \frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{7}{32} a^{6} + \frac{3}{16} a^{5} - \frac{19}{128} a^{4} + \frac{11}{64} a^{3} - \frac{7}{16} a^{2} + \frac{3}{16} a - \frac{1}{8}$, $\frac{1}{88064} a^{15} + \frac{135}{88064} a^{14} - \frac{59}{44032} a^{13} + \frac{333}{22016} a^{12} + \frac{917}{88064} a^{11} + \frac{3677}{88064} a^{10} + \frac{2673}{88064} a^{9} + \frac{9859}{88064} a^{8} + \frac{563}{22016} a^{7} - \frac{513}{5504} a^{6} + \frac{20753}{88064} a^{5} - \frac{11467}{88064} a^{4} + \frac{5677}{44032} a^{3} - \frac{2673}{5504} a^{2} - \frac{1817}{11008} a + \frac{1413}{5504}$, $\frac{1}{90806841344} a^{16} + \frac{161025}{45403420672} a^{15} + \frac{63666343}{90806841344} a^{14} - \frac{230362431}{45403420672} a^{13} - \frac{870664175}{90806841344} a^{12} - \frac{102756003}{22701710336} a^{11} + \frac{79105103}{2837713792} a^{10} + \frac{755216999}{45403420672} a^{9} - \frac{8022653443}{90806841344} a^{8} - \frac{4841956483}{22701710336} a^{7} + \frac{4677974369}{90806841344} a^{6} - \frac{16201311}{65993344} a^{5} + \frac{15286423313}{90806841344} a^{4} - \frac{15050594729}{45403420672} a^{3} + \frac{5596936081}{11350855168} a^{2} - \frac{617877433}{11350855168} a - \frac{1140399705}{5675427584}$

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:  \( 8867681.2165 \) (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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $17$ $17$ $17$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $17$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
1571Data not computed