Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 - 5*x^15 + 8*x^14 + 14*x^13 - 6*x^12 - 34*x^11 - 12*x^10 + 42*x^9 + 47*x^8 - 33*x^7 - 52*x^6 + 13*x^5 + 21*x^4 + 3*x^3 - 5*x^2 - 2*x + 1)
gp: K = bnfinit(y^17 - 2*y^16 - 5*y^15 + 8*y^14 + 14*y^13 - 6*y^12 - 34*y^11 - 12*y^10 + 42*y^9 + 47*y^8 - 33*y^7 - 52*y^6 + 13*y^5 + 21*y^4 + 3*y^3 - 5*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 - 5*x^15 + 8*x^14 + 14*x^13 - 6*x^12 - 34*x^11 - 12*x^10 + 42*x^9 + 47*x^8 - 33*x^7 - 52*x^6 + 13*x^5 + 21*x^4 + 3*x^3 - 5*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 - 5*x^15 + 8*x^14 + 14*x^13 - 6*x^12 - 34*x^11 - 12*x^10 + 42*x^9 + 47*x^8 - 33*x^7 - 52*x^6 + 13*x^5 + 21*x^4 + 3*x^3 - 5*x^2 - 2*x + 1)
\( x^{17} - 2 x^{16} - 5 x^{15} + 8 x^{14} + 14 x^{13} - 6 x^{12} - 34 x^{11} - 12 x^{10} + 42 x^{9} + \cdots + 1 \)
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: | $[5, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(89056089711131593861\)
\(\medspace = 946391\cdot 3063653\cdot 30715207\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.91\) | 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: | $946391^{1/2}3063653^{1/2}30715207^{1/2}\approx 9436953412.576094$ | ||
| Ramified primes: |
\(946391\), \(3063653\), \(30715207\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{89056\!\cdots\!93861}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{28162}a^{16}-\frac{3877}{28162}a^{15}-\frac{1057}{28162}a^{14}-\frac{844}{14081}a^{13}+\frac{3715}{14081}a^{12}+\frac{4389}{28162}a^{11}+\frac{2439}{28162}a^{10}+\frac{11295}{28162}a^{9}+\frac{4873}{14081}a^{8}+\frac{6810}{14081}a^{7}+\frac{6068}{14081}a^{6}-\frac{10593}{28162}a^{5}+\frac{1773}{28162}a^{4}+\frac{587}{14081}a^{3}+\frac{12997}{28162}a^{2}-\frac{4862}{14081}a+\frac{13823}{28162}$
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 $1$
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: | $10$ | 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: |
$a$, $\frac{10207}{14081}a^{16}-\frac{23939}{28162}a^{15}-\frac{59077}{14081}a^{14}+\frac{53699}{28162}a^{13}+\frac{319351}{28162}a^{12}+\frac{196777}{28162}a^{11}-\frac{253793}{14081}a^{10}-\frac{760619}{28162}a^{9}+\frac{51481}{14081}a^{8}+\frac{574948}{14081}a^{7}+\frac{269134}{14081}a^{6}-\frac{290453}{14081}a^{5}-\frac{571469}{28162}a^{4}-\frac{84399}{14081}a^{3}+\frac{87764}{14081}a^{2}+\frac{78607}{28162}a-\frac{14599}{28162}$, $\frac{45053}{14081}a^{16}-\frac{66081}{14081}a^{15}-\frac{519155}{28162}a^{14}+\frac{440545}{28162}a^{13}+\frac{742470}{14081}a^{12}+\frac{263807}{28162}a^{11}-\frac{1440119}{14081}a^{10}-\frac{1309157}{14081}a^{9}+\frac{2292633}{28162}a^{8}+\frac{5393107}{28162}a^{7}-\frac{18125}{28162}a^{6}-\frac{4518193}{28162}a^{5}-\frac{1258297}{28162}a^{4}+\frac{539064}{14081}a^{3}+\frac{403805}{14081}a^{2}+\frac{20862}{14081}a-\frac{168589}{28162}$, $\frac{36516}{14081}a^{16}-\frac{102883}{28162}a^{15}-\frac{439293}{28162}a^{14}+\frac{176582}{14081}a^{13}+\frac{1283715}{28162}a^{12}+\frac{111430}{14081}a^{11}-\frac{1238929}{14081}a^{10}-\frac{2291905}{28162}a^{9}+\frac{2045229}{28162}a^{8}+\frac{4731135}{28162}a^{7}+\frac{44131}{28162}a^{6}-\frac{4171731}{28162}a^{5}-\frac{592972}{14081}a^{4}+\frac{556379}{14081}a^{3}+\frac{378534}{14081}a^{2}+\frac{40229}{28162}a-\frac{85425}{14081}$, $\frac{67015}{28162}a^{16}-\frac{46555}{14081}a^{15}-\frac{193806}{14081}a^{14}+\frac{143527}{14081}a^{13}+\frac{1101527}{28162}a^{12}+\frac{314689}{28162}a^{11}-\frac{2058489}{28162}a^{10}-\frac{1050940}{14081}a^{9}+\frac{1417267}{28162}a^{8}+\frac{4026965}{28162}a^{7}+\frac{383829}{28162}a^{6}-\frac{1561131}{14081}a^{5}-\frac{1208849}{28162}a^{4}+\frac{305273}{14081}a^{3}+\frac{619183}{28162}a^{2}+\frac{28901}{28162}a-\frac{96969}{28162}$, $\frac{26677}{28162}a^{16}-\frac{14973}{14081}a^{15}-\frac{176399}{28162}a^{14}+\frac{98829}{28162}a^{13}+\frac{526951}{28162}a^{12}+\frac{71324}{14081}a^{11}-\frac{946525}{28162}a^{10}-\frac{494137}{14081}a^{9}+\frac{367335}{14081}a^{8}+\frac{954816}{14081}a^{7}+\frac{71265}{14081}a^{6}-\frac{1757997}{28162}a^{5}-\frac{253337}{14081}a^{4}+\frac{297028}{14081}a^{3}+\frac{215721}{28162}a^{2}-\frac{21047}{28162}a-\frac{19650}{14081}$, $\frac{15901}{28162}a^{16}-\frac{1559}{28162}a^{15}-\frac{60686}{14081}a^{14}-\frac{44745}{28162}a^{13}+\frac{171392}{14081}a^{12}+\frac{192120}{14081}a^{11}-\frac{362641}{28162}a^{10}-\frac{1057435}{28162}a^{9}-\frac{300041}{28162}a^{8}+\frac{1230887}{28162}a^{7}+\frac{1205397}{28162}a^{6}-\frac{247603}{14081}a^{5}-\frac{484639}{14081}a^{4}-\frac{128545}{14081}a^{3}+\frac{125189}{28162}a^{2}+\frac{78514}{14081}a+\frac{4597}{14081}$, $\frac{1}{14081}a^{16}-\frac{3877}{14081}a^{15}+\frac{11967}{28162}a^{14}+\frac{38867}{28162}a^{13}-\frac{20732}{14081}a^{12}-\frac{89789}{28162}a^{11}+\frac{2439}{14081}a^{10}+\frac{81700}{14081}a^{9}+\frac{89897}{28162}a^{8}-\frac{155813}{28162}a^{7}-\frac{186943}{28162}a^{6}+\frac{161867}{28162}a^{5}+\frac{158437}{28162}a^{4}-\frac{97393}{14081}a^{3}-\frac{15165}{14081}a^{2}+\frac{32519}{14081}a+\frac{13565}{28162}$, $\frac{38494}{14081}a^{16}-\frac{120167}{28162}a^{15}-\frac{219364}{14081}a^{14}+\frac{420235}{28162}a^{13}+\frac{1275667}{28162}a^{12}+\frac{111223}{28162}a^{11}-\frac{1300694}{14081}a^{10}-\frac{2104845}{28162}a^{9}+\frac{1157083}{14081}a^{8}+\frac{2361934}{14081}a^{7}-\frac{213368}{14081}a^{6}-\frac{2121494}{14081}a^{5}-\frac{944917}{28162}a^{4}+\frac{625591}{14081}a^{3}+\frac{374694}{14081}a^{2}-\frac{71271}{28162}a-\frac{135423}{28162}$, $\frac{113707}{28162}a^{16}-\frac{88572}{14081}a^{15}-\frac{327345}{14081}a^{14}+\frac{317170}{14081}a^{13}+\frac{1912107}{28162}a^{12}+\frac{113869}{28162}a^{11}-\frac{3922521}{28162}a^{10}-\frac{1545592}{14081}a^{9}+\frac{3632539}{28162}a^{8}+\frac{7087379}{28162}a^{7}-\frac{820627}{28162}a^{6}-\frac{3306750}{14081}a^{5}-\frac{1332861}{28162}a^{4}+\frac{1029982}{14081}a^{3}+\frac{1231733}{28162}a^{2}-\frac{88991}{28162}a-\frac{259141}{28162}$
| 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: | \( 1603.54336024 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{6}\cdot 1603.54336024 \cdot 1}{2\cdot\sqrt{89056089711131593861}}\cr\approx \mathstrut & 0.167281567593 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 - 5*x^15 + 8*x^14 + 14*x^13 - 6*x^12 - 34*x^11 - 12*x^10 + 42*x^9 + 47*x^8 - 33*x^7 - 52*x^6 + 13*x^5 + 21*x^4 + 3*x^3 - 5*x^2 - 2*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 - 2*x^16 - 5*x^15 + 8*x^14 + 14*x^13 - 6*x^12 - 34*x^11 - 12*x^10 + 42*x^9 + 47*x^8 - 33*x^7 - 52*x^6 + 13*x^5 + 21*x^4 + 3*x^3 - 5*x^2 - 2*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 - 2*x^16 - 5*x^15 + 8*x^14 + 14*x^13 - 6*x^12 - 34*x^11 - 12*x^10 + 42*x^9 + 47*x^8 - 33*x^7 - 52*x^6 + 13*x^5 + 21*x^4 + 3*x^3 - 5*x^2 - 2*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 - 2*x^16 - 5*x^15 + 8*x^14 + 14*x^13 - 6*x^12 - 34*x^11 - 12*x^10 + 42*x^9 + 47*x^8 - 33*x^7 - 52*x^6 + 13*x^5 + 21*x^4 + 3*x^3 - 5*x^2 - 2*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
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 355687428096000 |
| The 297 conjugacy class representatives for $S_{17}$ |
| Character table for $S_{17}$ |
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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | $17$ | $17$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $17$ | $17$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(946391\)
| $\Q_{946391}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{946391}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(3063653\)
| $\Q_{3063653}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(30715207\)
| $\Q_{30715207}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |