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Properties

Label 17.1.26635346946...3136.1
Degree $17$
Signature $[1, 8]$
Discriminant $2^{16}\cdot 17^{20}$
Root discriminant $53.82$
Ramified primes $2, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{17}:C_{4}$ (as 17T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4096, 0, -4352, 10880, -7616, 9248, -816, -952, -4420, 3638, -1496, 595, -68, 85, 0, 17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 + 17*x^15 + 85*x^13 - 68*x^12 + 595*x^11 - 1496*x^10 + 3638*x^9 - 4420*x^8 - 952*x^7 - 816*x^6 + 9248*x^5 - 7616*x^4 + 10880*x^3 - 4352*x^2 - 4096)
 
gp: K = bnfinit(x^17 + 17*x^15 + 85*x^13 - 68*x^12 + 595*x^11 - 1496*x^10 + 3638*x^9 - 4420*x^8 - 952*x^7 - 816*x^6 + 9248*x^5 - 7616*x^4 + 10880*x^3 - 4352*x^2 - 4096, 1)
 

Normalized defining polynomial

\( x^{17} + 17 x^{15} + 85 x^{13} - 68 x^{12} + 595 x^{11} - 1496 x^{10} + 3638 x^{9} - 4420 x^{8} - 952 x^{7} - 816 x^{6} + 9248 x^{5} - 7616 x^{4} + 10880 x^{3} - 4352 x^{2} - 4096 \)

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:  \(266353469466055312828111323136=2^{16}\cdot 17^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.82$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{8} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{32} a^{5} - \frac{3}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{10} - \frac{1}{16} a^{8} - \frac{7}{64} a^{6} - \frac{1}{16} a^{5} - \frac{5}{32} a^{4} - \frac{1}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{11} + \frac{1}{64} a^{7} + \frac{1}{16} a^{6} + \frac{1}{32} a^{5} + \frac{1}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{256} a^{12} + \frac{1}{256} a^{10} - \frac{1}{64} a^{9} + \frac{13}{256} a^{8} + \frac{3}{64} a^{7} - \frac{13}{256} a^{6} - \frac{7}{64} a^{5} + \frac{31}{128} a^{4} - \frac{23}{64} a^{3} + \frac{11}{32} a^{2} - \frac{7}{16} a - \frac{1}{4}$, $\frac{1}{512} a^{13} + \frac{1}{512} a^{11} - \frac{3}{512} a^{9} - \frac{1}{128} a^{8} + \frac{19}{512} a^{7} - \frac{3}{64} a^{6} + \frac{31}{256} a^{5} + \frac{7}{128} a^{4} - \frac{11}{64} a^{3} - \frac{1}{32} a^{2} - \frac{1}{2}$, $\frac{1}{2048} a^{14} - \frac{3}{2048} a^{12} - \frac{1}{128} a^{11} - \frac{15}{2048} a^{10} - \frac{5}{512} a^{9} + \frac{63}{2048} a^{8} - \frac{11}{256} a^{7} + \frac{117}{1024} a^{6} - \frac{5}{512} a^{5} + \frac{1}{8} a^{4} - \frac{13}{32} a^{3} - \frac{9}{64} a^{2} + \frac{1}{32} a - \frac{1}{8}$, $\frac{1}{196608} a^{15} - \frac{1}{6144} a^{14} - \frac{131}{196608} a^{13} - \frac{5}{6144} a^{12} + \frac{529}{196608} a^{11} + \frac{295}{49152} a^{10} + \frac{597}{65536} a^{9} + \frac{403}{24576} a^{8} - \frac{1051}{98304} a^{7} + \frac{1607}{49152} a^{6} - \frac{29}{384} a^{5} + \frac{1517}{6144} a^{4} - \frac{2159}{6144} a^{3} - \frac{809}{3072} a^{2} - \frac{35}{96} a + \frac{43}{192}$, $\frac{1}{7401923607134208} a^{16} - \frac{6311857681}{3700961803567104} a^{15} + \frac{1496202275965}{7401923607134208} a^{14} + \frac{1581660772531}{3700961803567104} a^{13} - \frac{13203271962863}{7401923607134208} a^{12} - \frac{20433318272323}{3700961803567104} a^{11} + \frac{5683434227757}{2467307869044736} a^{10} - \frac{41383079973299}{3700961803567104} a^{9} - \frac{49921422246931}{3700961803567104} a^{8} + \frac{7898100639697}{925240450891776} a^{7} - \frac{78417294247895}{925240450891776} a^{6} + \frac{9845341056773}{231310112722944} a^{5} + \frac{50812334425207}{231310112722944} a^{4} - \frac{24856911424093}{57827528180736} a^{3} + \frac{18106481364361}{57827528180736} a^{2} - \frac{1327001594645}{7228441022592} a + \frac{147572582639}{1204740170432}$

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:  \( 42372140421.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{17}.C_2$ (as 17T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 68
The 8 conjugacy class representatives for $C_{17}:C_{4}$
Character table for $C_{17}:C_{4}$

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $17$ R $17$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $17$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
17Data not computed