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Properties

Label 17.1.10214350242...6241.1
Degree $17$
Signature $[1, 8]$
Discriminant $1783^{8}$
Root discriminant $33.88$
Ramified prime $1783$
Class number $1$
Class group Trivial
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![1, 175, -498, 446, -109, 314, -473, 465, -411, 268, -111, 94, -158, 115, -61, 30, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 7*x^16 + 30*x^15 - 61*x^14 + 115*x^13 - 158*x^12 + 94*x^11 - 111*x^10 + 268*x^9 - 411*x^8 + 465*x^7 - 473*x^6 + 314*x^5 - 109*x^4 + 446*x^3 - 498*x^2 + 175*x + 1)
 
gp: K = bnfinit(x^17 - 7*x^16 + 30*x^15 - 61*x^14 + 115*x^13 - 158*x^12 + 94*x^11 - 111*x^10 + 268*x^9 - 411*x^8 + 465*x^7 - 473*x^6 + 314*x^5 - 109*x^4 + 446*x^3 - 498*x^2 + 175*x + 1, 1)
 

Normalized defining polynomial

\( x^{17} - 7 x^{16} + 30 x^{15} - 61 x^{14} + 115 x^{13} - 158 x^{12} + 94 x^{11} - 111 x^{10} + 268 x^{9} - 411 x^{8} + 465 x^{7} - 473 x^{6} + 314 x^{5} - 109 x^{4} + 446 x^{3} - 498 x^{2} + 175 x + 1 \)

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:  \(102143502423134353014546241=1783^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.88$
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:  $1783$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{4}{9} a^{5} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{585} a^{14} + \frac{1}{117} a^{13} - \frac{7}{585} a^{12} - \frac{16}{585} a^{11} - \frac{11}{195} a^{10} + \frac{1}{39} a^{9} + \frac{86}{585} a^{8} - \frac{14}{65} a^{7} + \frac{209}{585} a^{6} + \frac{43}{585} a^{5} - \frac{131}{585} a^{4} - \frac{31}{117} a^{3} - \frac{89}{195} a^{2} - \frac{8}{65} a - \frac{74}{585}$, $\frac{1}{33345} a^{15} - \frac{11}{33345} a^{14} + \frac{1018}{33345} a^{13} + \frac{83}{1235} a^{12} - \frac{1724}{11115} a^{11} - \frac{4202}{33345} a^{10} - \frac{241}{3705} a^{9} + \frac{5258}{33345} a^{8} - \frac{296}{6669} a^{7} + \frac{599}{33345} a^{6} + \frac{137}{2565} a^{5} + \frac{58}{585} a^{4} - \frac{1733}{3705} a^{3} + \frac{1945}{6669} a^{2} + \frac{647}{3705} a - \frac{12011}{33345}$, $\frac{1}{11256194389635} a^{16} - \frac{355937}{3752064796545} a^{15} - \frac{54261082}{288620368965} a^{14} - \frac{594246340459}{11256194389635} a^{13} + \frac{36095198188}{750412959309} a^{12} + \frac{129922992988}{865861106895} a^{11} + \frac{308476329944}{11256194389635} a^{10} + \frac{972736546493}{11256194389635} a^{9} - \frac{158864838419}{1250688265515} a^{8} + \frac{149421815261}{750412959309} a^{7} - \frac{24117750937}{1250688265515} a^{6} + \frac{67069126507}{592431283665} a^{5} - \frac{1854497751497}{3752064796545} a^{4} + \frac{396662797030}{2251238877927} a^{3} - \frac{228569659978}{2251238877927} a^{2} - \frac{3077124090674}{11256194389635} a + \frac{10701895817}{45571637205}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4818947.77807 \)
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 $17$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $17$

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
1783Data not computed