↑

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

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining 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(y^17 - 3*y^16 - 4*y^14 + 12*y^13 + 24*y^12 + 12*y^11 - 28*y^10 - 90*y^9 - 74*y^8 + 116*y^6 + 132*y^5 + 72*y^4 + 28*y^3 + 12*y^2 + 5*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(133249137678121328919445504\) \(\medspace = 2^{30}\cdot 137^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(34.41\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(137\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

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

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

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

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a$, $\frac{3253}{4422}a^{16}-\frac{22855}{8844}a^{15}+\frac{5611}{4422}a^{14}-\frac{4903}{1474}a^{13}+\frac{7519}{737}a^{12}+\frac{9441}{737}a^{11}+\frac{3075}{2948}a^{10}-\frac{93817}{4422}a^{9}-\frac{21629}{402}a^{8}-\frac{73387}{2948}a^{7}+\frac{11241}{737}a^{6}+\frac{29473}{402}a^{5}+\frac{10835}{201}a^{4}+\frac{93515}{4422}a^{3}+\frac{40273}{2948}a^{2}+\frac{11601}{1474}a+\frac{5131}{2211}$, $\frac{6977}{8844}a^{16}-\frac{20005}{8844}a^{15}-\frac{3499}{8844}a^{14}-\frac{8693}{2948}a^{13}+\frac{27059}{2948}a^{12}+\frac{15096}{737}a^{11}+\frac{32821}{2948}a^{10}-\frac{103489}{4422}a^{9}-\frac{60853}{804}a^{8}-\frac{193553}{2948}a^{7}+\frac{4699}{2948}a^{6}+\frac{81521}{804}a^{5}+\frac{95339}{804}a^{4}+\frac{262733}{4422}a^{3}+\frac{35221}{2948}a^{2}+\frac{485}{737}a-\frac{371}{2211}$, $\frac{2293}{8844}a^{16}-\frac{7301}{8844}a^{15}+\frac{2521}{8844}a^{14}-\frac{1104}{737}a^{13}+\frac{2492}{737}a^{12}+\frac{3787}{737}a^{11}+\frac{5855}{1474}a^{10}-\frac{22625}{4422}a^{9}-\frac{4256}{201}a^{8}-\frac{59725}{2948}a^{7}-\frac{22631}{2948}a^{6}+\frac{4783}{201}a^{5}+\frac{6688}{201}a^{4}+\frac{124909}{4422}a^{3}+\frac{12403}{737}a^{2}+\frac{10215}{1474}a+\frac{10547}{8844}$, $\frac{125}{1474}a^{16}-\frac{1731}{2948}a^{15}+\frac{1623}{1474}a^{14}-\frac{1509}{2948}a^{13}+\frac{5481}{2948}a^{12}-\frac{6065}{2948}a^{11}-\frac{19175}{2948}a^{10}-\frac{5531}{2948}a^{9}+\frac{687}{134}a^{8}+\frac{62317}{2948}a^{7}+\frac{8529}{737}a^{6}-\frac{2175}{268}a^{5}-\frac{8639}{268}a^{4}-\frac{67305}{2948}a^{3}+\frac{22839}{2948}a^{2}+\frac{28225}{2948}a+\frac{2623}{1474}$, $\frac{15}{737}a^{16}-\frac{1269}{1474}a^{15}+\frac{8945}{2948}a^{14}-\frac{7231}{2948}a^{13}+\frac{7615}{1474}a^{12}-\frac{9429}{737}a^{11}-\frac{25975}{2948}a^{10}-\frac{3199}{1474}a^{9}+\frac{5805}{268}a^{8}+\frac{76815}{1474}a^{7}+\frac{47013}{2948}a^{6}-\frac{3135}{268}a^{5}-\frac{5274}{67}a^{4}-\frac{29907}{737}a^{3}-\frac{64003}{2948}a^{2}-\frac{3834}{737}a-\frac{8263}{2948}$, $\frac{39583}{8844}a^{16}-\frac{127619}{8844}a^{15}+\frac{5176}{2211}a^{14}-\frac{44775}{2948}a^{13}+\frac{40706}{737}a^{12}+\frac{73109}{737}a^{11}+\frac{54753}{2948}a^{10}-\frac{1269649}{8844}a^{9}-\frac{148831}{402}a^{8}-\frac{652603}{2948}a^{7}+\frac{166153}{1474}a^{6}+\frac{415093}{804}a^{5}+\frac{90622}{201}a^{4}+\frac{575095}{4422}a^{3}+\frac{113687}{2948}a^{2}+\frac{91281}{2948}a+\frac{68363}{8844}$, $\frac{2413}{4422}a^{16}-\frac{13699}{8844}a^{15}-\frac{4135}{8844}a^{14}-\frac{1179}{737}a^{13}+\frac{19361}{2948}a^{12}+\frac{20221}{1474}a^{11}+\frac{22443}{2948}a^{10}-\frac{45974}{2211}a^{9}-\frac{20489}{402}a^{8}-\frac{115811}{2948}a^{7}+\frac{21107}{2948}a^{6}+\frac{15275}{201}a^{5}+\frac{57359}{804}a^{4}+\frac{60556}{2211}a^{3}+\frac{12167}{2948}a^{2}+\frac{3629}{737}a+\frac{10235}{4422}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 810656.541895 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 810656.541895 \cdot 15}{2\cdot\sqrt{133249137678121328919445504}}\cr\approx \mathstrut & 2.55879146886 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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 $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
Deg $16$$16$$1$$30$
\(137\) Copy content Toggle raw display $\Q_{137}$$x + 134$$1$$1$$0$Trivial$[\ ]$
137.4.2.1$x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
137.4.2.1$x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
137.4.2.1$x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
137.4.2.1$x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$$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 *.