Properties

Label 17.1.13324913767...5504.1
Degree $17$
Signature $[1, 8]$
Discriminant $2^{30}\cdot 137^{8}$
Root discriminant $34.41$
Ramified primes $2, 137$
Class number $15$
Class group $[15]$
Galois group $\PSL(2,16)$ (as 17T6)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 - 4*x^14 + 12*x^13 + 24*x^12 + 12*x^11 - 28*x^10 - 90*x^9 - 74*x^8 + 116*x^6 + 132*x^5 + 72*x^4 + 28*x^3 + 12*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^17 - 3*x^16 - 4*x^14 + 12*x^13 + 24*x^12 + 12*x^11 - 28*x^10 - 90*x^9 - 74*x^8 + 116*x^6 + 132*x^5 + 72*x^4 + 28*x^3 + 12*x^2 + 5*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 12, 28, 72, 132, 116, 0, -74, -90, -28, 12, 24, 12, -4, 0, -3, 1]);
 

Normalized defining polynomial

\( x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1 \)

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:  \(133249137678121328919445504=2^{30}\cdot 137^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $34.41$
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:  $2, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{8} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{8844} a^{16} - \frac{29}{2211} a^{15} - \frac{79}{4422} a^{14} + \frac{27}{1474} a^{13} - \frac{101}{1474} a^{12} - \frac{13}{2948} a^{11} - \frac{1}{2948} a^{10} + \frac{311}{8844} a^{9} + \frac{13}{804} a^{8} + \frac{611}{1474} a^{7} - \frac{251}{737} a^{6} + \frac{100}{201} a^{5} - \frac{41}{201} a^{4} + \frac{2723}{8844} a^{3} - \frac{851}{2948} a^{2} - \frac{1117}{2948} a - \frac{406}{2211}$

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

Class group and class number

$C_{15}$, which has order $15$

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:  \( 810656.541895 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$\PSL(2,16)$ (as 17T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 4080
The 17 conjugacy class representatives for $\PSL(2,16)$
Character table for $\PSL(2,16)$

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 $15{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $17$ $15{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $17$ $17$ $17$ $15{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $17$ $17$ $17$ $15{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $17$ $17$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$137$$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.4.2.1$x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
137.4.2.1$x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
137.4.2.1$x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
137.4.2.1$x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
15.133249137678121328919445504.240.a.a$15$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133249137678121328919445504.240.a.b$15$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133249137678121328919445504.240.a.c$15$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133249137678121328919445504.240.a.d$15$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133249137678121328919445504.240.a.e$15$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133249137678121328919445504.240.a.f$15$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133249137678121328919445504.240.a.g$15$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133249137678121328919445504.240.a.h$15$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
* 16.133249137678121328919445504.17t6.a.a$16$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $0$
17.133249137678121328919445504.51.a.a$17$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133249137678121328919445504.68.a.a$17$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133249137678121328919445504.68.a.b$17$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133249137678121328919445504.120.a.a$17$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133249137678121328919445504.120.a.b$17$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133249137678121328919445504.120.a.c$17$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133249137678121328919445504.120.a.d$17$ $ 2^{30} \cdot 137^{8}$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$ $\PSL(2,16)$ (as 17T6) $1$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.