Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*x + 1)
gp: K = bnfinit(y^17 - y^16 - 17*y^15 + 17*y^14 + 112*y^13 - 115*y^12 - 356*y^11 + 389*y^10 + 547*y^9 - 675*y^8 - 342*y^7 + 560*y^6 + 27*y^5 - 205*y^4 + 42*y^3 + 24*y^2 - 10*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*x + 1)
\( x^{17} - x^{16} - 17 x^{15} + 17 x^{14} + 112 x^{13} - 115 x^{12} - 356 x^{11} + 389 x^{10} + 547 x^{9} + \cdots + 1 \)
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: | $[13, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2232673506822932495146063557\)
\(\medspace = 3^{3}\cdot 205542871\cdot 402308340646912121\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(40.62\) | 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: | $3^{3/4}205542871^{1/2}402308340646912121^{1/2}\approx 20728681017507.81$ | ||
| Ramified primes: |
\(3\), \(205542871\), \(402308340646912121\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{24807\!\cdots\!18173}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{12}+\frac{1}{3}a^{9}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{21}a^{16}-\frac{1}{7}a^{15}+\frac{1}{7}a^{14}-\frac{10}{21}a^{13}+\frac{2}{7}a^{12}+\frac{2}{7}a^{11}+\frac{10}{21}a^{10}-\frac{3}{7}a^{9}-\frac{3}{7}a^{8}-\frac{2}{7}a^{7}+\frac{2}{7}a^{6}+\frac{2}{21}a^{5}+\frac{2}{21}a^{4}+\frac{8}{21}a^{3}-\frac{3}{7}a^{2}+\frac{1}{3}a-\frac{1}{7}$
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: | $14$ | 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: |
$a$, $a-1$, $\frac{101}{21}a^{16}-\frac{58}{21}a^{15}-\frac{578}{7}a^{14}+\frac{964}{21}a^{13}+\frac{11596}{21}a^{12}-\frac{2143}{7}a^{11}-\frac{37903}{21}a^{10}+\frac{21512}{21}a^{9}+\frac{20711}{7}a^{8}-\frac{12158}{7}a^{7}-\frac{15786}{7}a^{6}+\frac{28195}{21}a^{5}+\frac{14405}{21}a^{4}-\frac{9118}{21}a^{3}-\frac{776}{21}a^{2}+\frac{140}{3}a-\frac{142}{21}$, $\frac{80}{21}a^{16}-\frac{37}{21}a^{15}-\frac{459}{7}a^{14}+\frac{607}{21}a^{13}+\frac{9244}{21}a^{12}-\frac{1338}{7}a^{11}-\frac{30427}{21}a^{10}+\frac{13343}{21}a^{9}+\frac{16882}{7}a^{8}-\frac{7433}{7}a^{7}-\frac{13392}{7}a^{6}+\frac{16435}{21}a^{5}+\frac{13838}{21}a^{4}-\frac{4813}{21}a^{3}-\frac{1637}{21}a^{2}+\frac{68}{3}a+\frac{5}{21}$, $\frac{239}{21}a^{16}-\frac{122}{21}a^{15}-\frac{1392}{7}a^{14}+\frac{2083}{21}a^{13}+\frac{28664}{21}a^{12}-\frac{4842}{7}a^{11}-\frac{97675}{21}a^{10}+\frac{52183}{21}a^{9}+\frac{57376}{7}a^{8}-\frac{33175}{7}a^{7}-\frac{50027}{7}a^{6}+\frac{95377}{21}a^{5}+\frac{19260}{7}a^{4}-\frac{14499}{7}a^{3}-\frac{5686}{21}a^{2}+\frac{1055}{3}a-\frac{1004}{21}$, $\frac{23}{21}a^{16}+\frac{12}{7}a^{15}-\frac{145}{7}a^{14}-\frac{608}{21}a^{13}+\frac{1103}{7}a^{12}+\frac{1299}{7}a^{11}-\frac{13042}{21}a^{10}-\frac{3898}{7}a^{9}+\frac{9507}{7}a^{8}+\frac{5365}{7}a^{7}-\frac{11322}{7}a^{6}-\frac{7829}{21}a^{5}+\frac{19450}{21}a^{4}-\frac{404}{21}a^{3}-\frac{1399}{7}a^{2}+\frac{95}{3}a+\frac{26}{7}$, $\frac{34}{21}a^{16}-\frac{11}{21}a^{15}-\frac{204}{7}a^{14}+\frac{206}{21}a^{13}+\frac{4376}{21}a^{12}-\frac{548}{7}a^{11}-\frac{15809}{21}a^{10}+\frac{6988}{21}a^{9}+\frac{10104}{7}a^{8}-\frac{5395}{7}a^{7}-\frac{9844}{7}a^{6}+\frac{19115}{21}a^{5}+\frac{12346}{21}a^{4}-\frac{9983}{21}a^{3}-\frac{1153}{21}a^{2}+\frac{250}{3}a-\frac{242}{21}$, $\frac{39}{7}a^{16}-\frac{113}{21}a^{15}-\frac{660}{7}a^{14}+\frac{632}{7}a^{13}+\frac{12959}{21}a^{12}-\frac{4197}{7}a^{11}-\frac{13596}{7}a^{10}+\frac{41395}{21}a^{9}+\frac{20600}{7}a^{8}-\frac{22781}{7}a^{7}-\frac{12821}{7}a^{6}+\frac{17109}{7}a^{5}+\frac{5477}{21}a^{4}-\frac{15724}{21}a^{3}+\frac{1733}{21}a^{2}+67a-\frac{281}{21}$, $\frac{83}{21}a^{16}-\frac{109}{21}a^{15}-\frac{470}{7}a^{14}+\frac{1858}{21}a^{13}+\frac{9283}{21}a^{12}-\frac{4181}{7}a^{11}-\frac{29431}{21}a^{10}+\frac{42191}{21}a^{9}+\frac{14920}{7}a^{8}-\frac{24281}{7}a^{7}-\frac{8948}{7}a^{6}+\frac{60751}{21}a^{5}+\frac{1034}{21}a^{4}-\frac{22849}{21}a^{3}+\frac{3712}{21}a^{2}+\frac{443}{3}a-\frac{781}{21}$, $\frac{179}{21}a^{16}-\frac{61}{21}a^{15}-\frac{1032}{7}a^{14}+\frac{1003}{21}a^{13}+\frac{20926}{21}a^{12}-\frac{2267}{7}a^{11}-\frac{69631}{21}a^{10}+\frac{23792}{21}a^{9}+\frac{39377}{7}a^{8}-\frac{14400}{7}a^{7}-\frac{32283}{7}a^{6}+\frac{36919}{21}a^{5}+\frac{34280}{21}a^{4}-\frac{14500}{21}a^{3}-\frac{3368}{21}a^{2}+\frac{317}{3}a-\frac{271}{21}$, $\frac{136}{21}a^{16}-\frac{38}{7}a^{15}-\frac{767}{7}a^{14}+\frac{1916}{21}a^{13}+\frac{5018}{7}a^{12}-\frac{4271}{7}a^{11}-\frac{47402}{21}a^{10}+\frac{14229}{7}a^{9}+\frac{24050}{7}a^{8}-\frac{24002}{7}a^{7}-\frac{15317}{7}a^{6}+\frac{56195}{21}a^{5}+\frac{7706}{21}a^{4}-\frac{18862}{21}a^{3}+\frac{530}{7}a^{2}+\frac{310}{3}a-\frac{129}{7}$, $\frac{10}{21}a^{16}-\frac{23}{21}a^{15}-\frac{60}{7}a^{14}+\frac{404}{21}a^{13}+\frac{1292}{21}a^{12}-\frac{939}{7}a^{11}-\frac{4667}{21}a^{10}+\frac{9892}{21}a^{9}+\frac{2917}{7}a^{8}-\frac{6117}{7}a^{7}-\frac{2598}{7}a^{6}+\frac{17429}{21}a^{5}+\frac{2281}{21}a^{4}-\frac{7781}{21}a^{3}+\frac{575}{21}a^{2}+\frac{187}{3}a-\frac{275}{21}$, $\frac{53}{21}a^{16}-\frac{26}{21}a^{15}-\frac{297}{7}a^{14}+\frac{415}{21}a^{13}+\frac{5771}{21}a^{12}-\frac{874}{7}a^{11}-\frac{17908}{21}a^{10}+\frac{8035}{21}a^{9}+\frac{8920}{7}a^{8}-\frac{3753}{7}a^{7}-\frac{5683}{7}a^{6}+\frac{4495}{21}a^{5}+\frac{1377}{7}a^{4}+\frac{461}{7}a^{3}-\frac{841}{21}a^{2}-\frac{121}{3}a+\frac{268}{21}$, $\frac{14}{3}a^{16}-\frac{2}{3}a^{15}-79a^{14}+\frac{28}{3}a^{13}+\frac{1550}{3}a^{12}-56a^{11}-\frac{4891}{3}a^{10}+\frac{529}{3}a^{9}+2522a^{8}-251a^{7}-1737a^{6}+\frac{187}{3}a^{5}+452a^{4}+33a^{3}-\frac{97}{3}a^{2}-\frac{40}{3}a+\frac{10}{3}$
| 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: | \( 129255854.893 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{2}\cdot 129255854.893 \cdot 1}{2\cdot\sqrt{2232673506822932495146063557}}\cr\approx \mathstrut & 0.442341106108 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*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 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*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 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*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 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*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 | ${\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.8.0.1}{8} }$ | R | ${\href{/padicField/5.13.0.1}{13} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $17$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $17$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 3.13.0.1 | $x^{13} + 2 x + 1$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
|
\(205542871\)
| $\Q_{205542871}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(402308340646912121\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |