Properties

Label 17.1.12489025081...7441.1
Degree $17$
Signature $[1, 8]$
Discriminant $11^{8}\cdot 17^{12}$
Root discriminant $22.84$
Ramified primes $11, 17$
Class number $1$
Class group Trivial
Galois group $C_{17}:C_{4}$ (as 17T3)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^16 + 6*x^15 - 5*x^14 + 7*x^13 - 17*x^12 + 24*x^11 - 16*x^10 + 42*x^9 - 83*x^8 + 82*x^7 - 72*x^6 + 37*x^5 + 24*x^4 - 10*x^3 + 14*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^17 - 4*x^16 + 6*x^15 - 5*x^14 + 7*x^13 - 17*x^12 + 24*x^11 - 16*x^10 + 42*x^9 - 83*x^8 + 82*x^7 - 72*x^6 + 37*x^5 + 24*x^4 - 10*x^3 + 14*x^2 - 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 14, -10, 24, 37, -72, 82, -83, 42, -16, 24, -17, 7, -5, 6, -4, 1]);
 

Normalized defining polynomial

\( x^{17} - 4 x^{16} + 6 x^{15} - 5 x^{14} + 7 x^{13} - 17 x^{12} + 24 x^{11} - 16 x^{10} + 42 x^{9} - 83 x^{8} + 82 x^{7} - 72 x^{6} + 37 x^{5} + 24 x^{4} - 10 x^{3} + 14 x^{2} - 2 x + 1 \)

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:  \(124890250818288107857441=11^{8}\cdot 17^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $22.84$
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:  $11, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{8689217585} a^{16} - \frac{409389363}{8689217585} a^{15} + \frac{746341569}{8689217585} a^{14} + \frac{68674948}{8689217585} a^{13} + \frac{2982233818}{8689217585} a^{12} + \frac{804301212}{1737843517} a^{11} + \frac{81858777}{1737843517} a^{10} + \frac{1092336176}{8689217585} a^{9} + \frac{288687551}{8689217585} a^{8} + \frac{247093695}{1737843517} a^{7} + \frac{492792409}{1737843517} a^{6} - \frac{2557040536}{8689217585} a^{5} + \frac{833505687}{8689217585} a^{4} - \frac{67104075}{1737843517} a^{3} + \frac{2641511796}{8689217585} a^{2} - \frac{929864671}{8689217585} a + \frac{2495583072}{8689217585}$

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:  \( 83428.2679357 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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 $17$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R $17$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $17$ $17$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$