Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 85*x^15 + 2975*x^13 - 55250*x^11 + 584375*x^9 - 3506250*x^7 + 11156250*x^5 - 15937500*x^3 + 6640625*x - 1737500)
gp: K = bnfinit(y^17 - 85*y^15 + 2975*y^13 - 55250*y^11 + 584375*y^9 - 3506250*y^7 + 11156250*y^5 - 15937500*y^3 + 6640625*y - 1737500, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 85*x^15 + 2975*x^13 - 55250*x^11 + 584375*x^9 - 3506250*x^7 + 11156250*x^5 - 15937500*x^3 + 6640625*x - 1737500);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 85*x^15 + 2975*x^13 - 55250*x^11 + 584375*x^9 - 3506250*x^7 + 11156250*x^5 - 15937500*x^3 + 6640625*x - 1737500)
\( x^{17} - 85 x^{15} + 2975 x^{13} - 55250 x^{11} + 584375 x^{9} - 3506250 x^{7} + 11156250 x^{5} + \cdots - 1737500 \)
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: | $[17, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(8272402618863367641770000000000000000\)
\(\medspace = 2^{16}\cdot 5^{16}\cdot 17^{17}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(148.47\) | 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: | $2\cdot 5^{16/17}17^{287/272}\approx 180.79594174691047$ | ||
| Ramified primes: |
\(2\), \(5\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{25}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{725}a^{9}+\frac{11}{725}a^{8}+\frac{13}{725}a^{7}-\frac{1}{145}a^{6}-\frac{2}{29}a^{5}-\frac{2}{145}a^{4}-\frac{5}{29}a^{3}-\frac{10}{29}a^{2}-\frac{7}{29}a-\frac{1}{29}$, $\frac{1}{725}a^{10}+\frac{8}{725}a^{8}-\frac{3}{725}a^{7}+\frac{1}{145}a^{6}-\frac{8}{145}a^{5}-\frac{3}{145}a^{4}-\frac{13}{29}a^{3}-\frac{13}{29}a^{2}-\frac{11}{29}a+\frac{11}{29}$, $\frac{1}{3625}a^{11}+\frac{1}{145}a^{8}-\frac{14}{725}a^{7}-\frac{2}{145}a^{5}-\frac{4}{145}a^{4}-\frac{12}{29}a^{3}+\frac{8}{29}a^{2}-\frac{4}{29}a-\frac{10}{29}$, $\frac{1}{3625}a^{12}-\frac{11}{725}a^{8}-\frac{7}{725}a^{7}+\frac{3}{145}a^{6}-\frac{12}{145}a^{5}+\frac{8}{145}a^{4}+\frac{4}{29}a^{3}-\frac{12}{29}a^{2}-\frac{4}{29}a+\frac{5}{29}$, $\frac{1}{3625}a^{13}-\frac{2}{725}a^{8}+\frac{13}{725}a^{7}+\frac{6}{145}a^{6}+\frac{14}{145}a^{5}-\frac{2}{145}a^{4}-\frac{9}{29}a^{3}+\frac{2}{29}a^{2}-\frac{14}{29}a-\frac{11}{29}$, $\frac{1}{18125}a^{14}+\frac{7}{725}a^{8}-\frac{12}{725}a^{7}+\frac{14}{145}a^{6}+\frac{13}{145}a^{5}-\frac{4}{145}a^{4}+\frac{10}{29}a^{3}-\frac{1}{29}a^{2}-\frac{5}{29}a-\frac{12}{29}$, $\frac{1}{18125}a^{15}-\frac{2}{725}a^{8}+\frac{8}{725}a^{7}-\frac{9}{145}a^{6}+\frac{8}{145}a^{5}+\frac{6}{145}a^{4}+\frac{5}{29}a^{3}+\frac{7}{29}a^{2}+\frac{8}{29}a+\frac{7}{29}$, $\frac{1}{18125}a^{16}+\frac{1}{725}a^{8}+\frac{2}{145}a^{7}+\frac{6}{145}a^{6}-\frac{14}{145}a^{5}-\frac{8}{145}a^{4}-\frac{3}{29}a^{3}-\frac{12}{29}a^{2}-\frac{7}{29}a-\frac{2}{29}$
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
Trivial group, which has order $1$ (assuming GRH)
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: | $16$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{1}{18125}a^{16}-\frac{2}{18125}a^{15}-\frac{72}{18125}a^{14}+\frac{27}{3625}a^{13}+\frac{411}{3625}a^{12}-\frac{698}{3625}a^{11}-\frac{1184}{725}a^{10}+\frac{1718}{725}a^{9}+\frac{9069}{725}a^{8}-\frac{10119}{725}a^{7}-\frac{7102}{145}a^{6}+34a^{5}+\frac{2413}{29}a^{4}-\frac{568}{29}a^{3}-\frac{1213}{29}a^{2}+\frac{75}{29}a+\frac{151}{29}$, $\frac{1}{18125}a^{15}-\frac{4}{18125}a^{14}-\frac{14}{3625}a^{13}+\frac{56}{3625}a^{12}+\frac{77}{725}a^{11}-\frac{308}{725}a^{10}-\frac{1039}{725}a^{9}+\frac{144}{25}a^{8}+\frac{1371}{145}a^{7}-\frac{5688}{145}a^{6}-\frac{683}{29}a^{5}+\frac{17156}{145}a^{4}-\frac{384}{29}a^{3}-\frac{2848}{29}a^{2}+\frac{1610}{29}a-\frac{229}{29}$, $\frac{2}{18125}a^{15}-\frac{7}{18125}a^{14}-\frac{27}{3625}a^{13}+\frac{91}{3625}a^{12}+\frac{729}{3625}a^{11}-\frac{466}{725}a^{10}-\frac{2017}{725}a^{9}+\frac{5962}{725}a^{8}+\frac{3042}{145}a^{7}-\frac{7966}{145}a^{6}-\frac{12281}{145}a^{5}+\frac{26364}{145}a^{4}+\frac{4839}{29}a^{3}-\frac{7258}{29}a^{2}-\frac{3935}{29}a+\frac{2689}{29}$, $\frac{3}{18125}a^{15}-\frac{1}{3625}a^{14}-\frac{9}{725}a^{13}+\frac{72}{3625}a^{12}+\frac{1333}{3625}a^{11}-\frac{396}{725}a^{10}-\frac{137}{25}a^{9}+\frac{5219}{725}a^{8}+\frac{6286}{145}a^{7}-\frac{6782}{145}a^{6}-\frac{5096}{29}a^{5}+\frac{20079}{145}a^{4}+\frac{9124}{29}a^{3}-\frac{4372}{29}a^{2}-\frac{4935}{29}a+\frac{2129}{29}$, $\frac{2}{18125}a^{16}+\frac{3}{18125}a^{15}-\frac{167}{18125}a^{14}-\frac{9}{725}a^{13}+\frac{1142}{3625}a^{12}+\frac{1352}{3625}a^{11}-\frac{823}{145}a^{10}-\frac{832}{145}a^{9}+\frac{41861}{725}a^{8}+\frac{34857}{725}a^{7}-\frac{9557}{29}a^{6}-\frac{6298}{29}a^{5}+\frac{142139}{145}a^{4}+\frac{14432}{29}a^{3}-\frac{36276}{29}a^{2}-\frac{14736}{29}a+\frac{11369}{29}$, $\frac{1}{18125}a^{16}-\frac{3}{18125}a^{15}-\frac{16}{3625}a^{14}+\frac{43}{3625}a^{13}+\frac{523}{3625}a^{12}-\frac{244}{725}a^{11}-\frac{72}{29}a^{10}+\frac{3486}{725}a^{9}+\frac{17486}{725}a^{8}-\frac{26404}{725}a^{7}-\frac{18988}{145}a^{6}+\frac{4082}{29}a^{5}+\frac{52592}{145}a^{4}-\frac{6836}{29}a^{3}-\frac{11587}{29}a^{2}+\frac{2216}{29}a+\frac{1389}{29}$, $\frac{2}{3625}a^{13}+\frac{4}{3625}a^{12}-\frac{26}{725}a^{11}-\frac{44}{725}a^{10}+\frac{648}{725}a^{9}+\frac{887}{725}a^{8}-\frac{7751}{725}a^{7}-\frac{1596}{145}a^{6}+\frac{9076}{145}a^{5}+\frac{6083}{145}a^{4}-\frac{4619}{29}a^{3}-\frac{1192}{29}a^{2}+\frac{3210}{29}a-\frac{849}{29}$, $\frac{2}{18125}a^{16}+\frac{6}{18125}a^{15}-\frac{141}{18125}a^{14}-\frac{16}{725}a^{13}+\frac{794}{3625}a^{12}+\frac{2078}{3625}a^{11}-\frac{457}{145}a^{10}-\frac{212}{29}a^{9}+\frac{17822}{725}a^{8}+\frac{34148}{725}a^{7}-\frac{2923}{29}a^{6}-\frac{4014}{29}a^{5}+\frac{27114}{145}a^{4}+\frac{3590}{29}a^{3}-\frac{2159}{29}a^{2}+40a-\frac{511}{29}$, $\frac{4}{18125}a^{15}+\frac{8}{18125}a^{14}-\frac{11}{725}a^{13}-\frac{109}{3625}a^{12}+\frac{1486}{3625}a^{11}+\frac{586}{725}a^{10}-\frac{3992}{725}a^{9}-\frac{7904}{725}a^{8}+\frac{27664}{725}a^{7}+\frac{11218}{145}a^{6}-\frac{18251}{145}a^{5}-\frac{39783}{145}a^{4}+\frac{4094}{29}a^{3}+\frac{11797}{29}a^{2}+\frac{1498}{29}a-\frac{2951}{29}$, $\frac{2}{18125}a^{16}+\frac{6}{18125}a^{15}-\frac{137}{18125}a^{14}-\frac{82}{3625}a^{13}+\frac{742}{3625}a^{12}+\frac{2208}{3625}a^{11}-\frac{2022}{725}a^{10}-\frac{5961}{725}a^{9}+\frac{14576}{725}a^{8}+\frac{8481}{145}a^{7}-\frac{10642}{145}a^{6}-\frac{6086}{29}a^{5}+\frac{16781}{145}a^{4}+\frac{9422}{29}a^{3}-\frac{1372}{29}a^{2}-\frac{3645}{29}a+\frac{1051}{29}$, $\frac{1}{18125}a^{16}+\frac{2}{18125}a^{15}-\frac{99}{18125}a^{14}-\frac{19}{3625}a^{13}+\frac{767}{3625}a^{12}+\frac{214}{3625}a^{11}-\frac{603}{145}a^{10}+\frac{578}{725}a^{9}+\frac{6431}{145}a^{8}-\frac{17026}{725}a^{7}-\frac{36328}{145}a^{6}+\frac{5648}{29}a^{5}+\frac{94923}{145}a^{4}-\frac{18237}{29}a^{3}-\frac{14005}{29}a^{2}+\frac{15003}{29}a-\frac{3089}{29}$, $\frac{1}{18125}a^{16}+\frac{4}{18125}a^{15}-\frac{69}{18125}a^{14}-\frac{57}{3625}a^{13}+\frac{363}{3625}a^{12}+\frac{1577}{3625}a^{11}-\frac{892}{725}a^{10}-\frac{4233}{725}a^{9}+\frac{4917}{725}a^{8}+\frac{28021}{725}a^{7}-\frac{1722}{145}a^{6}-\frac{3318}{29}a^{5}-\frac{469}{145}a^{4}+\frac{3703}{29}a^{3}-\frac{164}{29}a^{2}-\frac{1485}{29}a+\frac{391}{29}$, $\frac{2}{18125}a^{14}+\frac{2}{3625}a^{13}-\frac{1}{125}a^{12}-\frac{117}{3625}a^{11}+\frac{162}{725}a^{10}+\frac{104}{145}a^{9}-\frac{2201}{725}a^{8}-\frac{5484}{725}a^{7}+\frac{3017}{145}a^{6}+\frac{5566}{145}a^{5}-\frac{9712}{145}a^{4}-\frac{2401}{29}a^{3}+\frac{2403}{29}a^{2}+\frac{1678}{29}a-\frac{949}{29}$, $\frac{2}{18125}a^{16}-\frac{4}{18125}a^{15}-\frac{6}{725}a^{14}+\frac{11}{725}a^{13}+\frac{36}{145}a^{12}-\frac{1468}{3625}a^{11}-\frac{2754}{725}a^{10}+\frac{3822}{725}a^{9}+\frac{4529}{145}a^{8}-\frac{24999}{725}a^{7}-\frac{3834}{29}a^{6}+\frac{15308}{145}a^{5}+\frac{35078}{145}a^{4}-\frac{3843}{29}a^{3}-\frac{3207}{29}a^{2}+\frac{2643}{29}a-\frac{491}{29}$, $\frac{1}{18125}a^{16}+\frac{1}{18125}a^{15}-\frac{3}{725}a^{14}-\frac{14}{3625}a^{13}+\frac{451}{3625}a^{12}+\frac{383}{3625}a^{11}-\frac{1387}{725}a^{10}-\frac{1039}{725}a^{9}+\frac{11544}{725}a^{8}+\frac{1481}{145}a^{7}-\frac{2039}{29}a^{6}-\frac{5618}{145}a^{5}+\frac{22271}{145}a^{4}+\frac{2296}{29}a^{3}-\frac{4248}{29}a^{2}-\frac{1839}{29}a+\frac{1129}{29}$, $\frac{1}{18125}a^{16}-\frac{4}{18125}a^{15}-\frac{86}{18125}a^{14}+\frac{64}{3625}a^{13}+\frac{592}{3625}a^{12}-\frac{81}{145}a^{11}-\frac{2073}{725}a^{10}+\frac{1283}{145}a^{9}+\frac{3862}{145}a^{8}-\frac{52944}{725}a^{7}-\frac{18104}{145}a^{6}+\frac{42956}{145}a^{5}+247a^{4}-\frac{14659}{29}a^{3}-\frac{3926}{29}a^{2}+\frac{8036}{29}a-\frac{1911}{29}$
| 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: | \( 48080253767000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{0}\cdot 48080253767000 \cdot 1}{2\cdot\sqrt{8272402618863367641770000000000000000}}\cr\approx \mathstrut & 1.09554658672590 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 85*x^15 + 2975*x^13 - 55250*x^11 + 584375*x^9 - 3506250*x^7 + 11156250*x^5 - 15937500*x^3 + 6640625*x - 1737500)
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 - 85*x^15 + 2975*x^13 - 55250*x^11 + 584375*x^9 - 3506250*x^7 + 11156250*x^5 - 15937500*x^3 + 6640625*x - 1737500, 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 - 85*x^15 + 2975*x^13 - 55250*x^11 + 584375*x^9 - 3506250*x^7 + 11156250*x^5 - 15937500*x^3 + 6640625*x - 1737500);
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 - 85*x^15 + 2975*x^13 - 55250*x^11 + 584375*x^9 - 3506250*x^7 + 11156250*x^5 - 15937500*x^3 + 6640625*x - 1737500);
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
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 solvable group of order 272 |
| The 17 conjugacy class representatives for $F_{17}$ |
| Character table for $F_{17}$ |
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 | $16{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | $16{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# 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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.8.8.4 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ | |
| 2.8.8.4 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ | |
|
\(5\)
| 5.17.16.1 | $x^{17} + 5$ | $17$ | $1$ | $16$ | $F_{17}$ | $[\ ]_{17}^{16}$ |
|
\(17\)
| 17.17.17.1 | $x^{17} + 17 x + 17$ | $17$ | $1$ | $17$ | $F_{17}$ | $[17/16]_{16}$ |