# Properties

 Label 17.1.133...504.1 Degree $17$ Signature $[1, 8]$ Discriminant $1.332\times 10^{26}$ Root discriminant $34.41$ Ramified primes $2, 137$ Class number $15$ Class group $[15]$ Galois group $\PSL(2,16)$ (as 17T6)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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]);

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

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: $$133249137678121328919445504$$$$\medspace = 2^{30}\cdot 137^{8}$$ 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);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{8}\cdot 810656.541895 \cdot 15}{2\sqrt{133249137678121328919445504}}\approx 2.55879146886$

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

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$$0Trivial[\ ] 137.4.2.1x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
137.4.2.1$x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 137.4.2.1x^{4} + 1507 x^{2} + 675684$$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 Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
15.133...504.240.a.a$15$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133...504.240.a.b$15$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133...504.240.a.c$15$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133...504.240.a.d$15$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133...504.240.a.e$15$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133...504.240.a.f$15$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133...504.240.a.g$15$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.133...504.240.a.h$15$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
* 16.133...504.17t6.a.a$16$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $0$
17.133...504.51.a.a$17$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133...504.68.a.a$17$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133...504.68.a.b$17$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133...504.120.a.a$17$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133...504.120.a.b$17$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133...504.120.a.c$17$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.133...504.120.a.d$17$ $2^{30} \cdot 137^{8}$ 17.1.133249137678121328919445504.1 $\PSL(2,16)$ (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 *.