Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 4*x^15 - 2*x^14 + 6*x^12 - 5*x^11 + x^10 + 3*x^9 - 6*x^8 - 7*x^7 - x^6 - 5*x^5 - 9*x^4 - 3*x^3 - 2*x^2 - 3*x - 1)
gp: K = bnfinit(y^17 - 2*y^16 + 4*y^15 - 2*y^14 + 6*y^12 - 5*y^11 + y^10 + 3*y^9 - 6*y^8 - 7*y^7 - y^6 - 5*y^5 - 9*y^4 - 3*y^3 - 2*y^2 - 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 + 4*x^15 - 2*x^14 + 6*x^12 - 5*x^11 + x^10 + 3*x^9 - 6*x^8 - 7*x^7 - x^6 - 5*x^5 - 9*x^4 - 3*x^3 - 2*x^2 - 3*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 + 4*x^15 - 2*x^14 + 6*x^12 - 5*x^11 + x^10 + 3*x^9 - 6*x^8 - 7*x^7 - x^6 - 5*x^5 - 9*x^4 - 3*x^3 - 2*x^2 - 3*x - 1)
\( x^{17} - 2 x^{16} + 4 x^{15} - 2 x^{14} + 6 x^{12} - 5 x^{11} + x^{10} + 3 x^{9} - 6 x^{8} - 7 x^{7} + \cdots - 1 \)
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: |
\(14517428819890014793\)
\(\medspace = 10170343\cdot 1427427651151\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.40\) | 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: | $10170343^{1/2}1427427651151^{1/2}\approx 3810174381.821653$ | ||
| Ramified primes: |
\(10170343\), \(1427427651151\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{14517\!\cdots\!14793}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{307}a^{16}+\frac{68}{307}a^{15}-\frac{148}{307}a^{14}+\frac{76}{307}a^{13}+\frac{101}{307}a^{12}+\frac{15}{307}a^{11}+\frac{124}{307}a^{10}+\frac{85}{307}a^{9}+\frac{120}{307}a^{8}+\frac{105}{307}a^{7}-\frac{25}{307}a^{6}+\frac{91}{307}a^{5}-\frac{82}{307}a^{4}+\frac{84}{307}a^{3}+\frac{44}{307}a^{2}+\frac{8}{307}a-\frac{57}{307}$
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: |
$a$, $\frac{1141}{307}a^{16}-\frac{2539}{307}a^{15}+\frac{5201}{307}a^{14}-\frac{3542}{307}a^{13}+\frac{730}{307}a^{12}+\frac{7291}{307}a^{11}-\frac{8639}{307}a^{10}+\frac{4578}{307}a^{9}+\frac{1533}{307}a^{8}-\frac{7907}{307}a^{7}-\frac{4886}{307}a^{6}-\frac{856}{307}a^{5}-\frac{6067}{307}a^{4}-\frac{8229}{307}a^{3}-\frac{451}{307}a^{2}-\frac{2845}{307}a-\frac{2409}{307}$, $\frac{5200}{307}a^{16}-\frac{13879}{307}a^{15}+\frac{29828}{307}a^{14}-\frac{29688}{307}a^{13}+\frac{18650}{307}a^{12}+\frac{19670}{307}a^{11}-\frac{39196}{307}a^{10}+\frac{30006}{307}a^{9}-\frac{2587}{307}a^{8}-\frac{29932}{307}a^{7}-\frac{17024}{307}a^{6}+\frac{7481}{307}a^{5}-\frac{29756}{307}a^{4}-\frac{27998}{307}a^{3}+\frac{2848}{307}a^{2}-\frac{11204}{307}a-\frac{8434}{307}$, $\frac{819}{307}a^{16}-\frac{2331}{307}a^{15}+\frac{5272}{307}a^{14}-\frac{6217}{307}a^{13}+\frac{5355}{307}a^{12}+\frac{312}{307}a^{11}-\frac{4666}{307}a^{10}+\frac{5145}{307}a^{9}-\frac{2416}{307}a^{8}-\frac{3035}{307}a^{7}-\frac{2669}{307}a^{6}+\frac{1156}{307}a^{5}-\frac{4837}{307}a^{4}-\frac{2428}{307}a^{3}+\frac{424}{307}a^{2}-\frac{1737}{307}a-\frac{633}{307}$, $\frac{1022}{307}a^{16}-\frac{2956}{307}a^{15}+\frac{6542}{307}a^{14}-\frac{7367}{307}a^{13}+\frac{5596}{307}a^{12}+\frac{2129}{307}a^{11}-\frac{7431}{307}a^{10}+\frac{7050}{307}a^{9}-\frac{2002}{307}a^{8}-\frac{4745}{307}a^{7}-\frac{3139}{307}a^{6}+\frac{2437}{307}a^{5}-\frac{6440}{307}a^{4}-\frac{4717}{307}a^{3}+\frac{1067}{307}a^{2}-\frac{2569}{307}a-\frac{1459}{307}$, $\frac{2338}{307}a^{16}-\frac{5875}{307}a^{15}+\frac{12552}{307}a^{14}-\frac{11424}{307}a^{13}+\frac{6502}{307}a^{12}+\frac{10510}{307}a^{11}-\frac{17088}{307}a^{10}+\frac{12074}{307}a^{9}+\frac{269}{307}a^{8}-\frac{14232}{307}a^{7}-\frac{8716}{307}a^{6}+\frac{1235}{307}a^{5}-\frac{13963}{307}a^{4}-\frac{14824}{307}a^{3}-\frac{280}{307}a^{2}-\frac{6163}{307}a-\frac{4633}{307}$, $\frac{296}{307}a^{16}-\frac{748}{307}a^{15}+\frac{1628}{307}a^{14}-\frac{1450}{307}a^{13}+\frac{731}{307}a^{12}+\frac{1677}{307}a^{11}-\frac{2592}{307}a^{10}+\frac{1828}{307}a^{9}+\frac{215}{307}a^{8}-\frac{2383}{307}a^{7}-\frac{953}{307}a^{6}+\frac{534}{307}a^{5}-\frac{2782}{307}a^{4}-\frac{1845}{307}a^{3}+\frac{437}{307}a^{2}-\frac{1316}{307}a-\frac{601}{307}$, $\frac{4716}{307}a^{16}-\frac{12100}{307}a^{15}+\frac{25938}{307}a^{14}-\frac{24720}{307}a^{13}+\frac{15202}{307}a^{12}+\frac{18550}{307}a^{11}-\frac{33514}{307}a^{10}+\frac{24478}{307}a^{9}-\frac{1109}{307}a^{8}-\frac{27027}{307}a^{7}-\frac{17511}{307}a^{6}+\frac{4268}{307}a^{5}-\frac{27215}{307}a^{4}-\frac{26288}{307}a^{3}+\frac{586}{307}a^{2}-\frac{10778}{307}a-\frac{7862}{307}$, $\frac{4031}{307}a^{16}-\frac{10788}{307}a^{15}+\frac{23245}{307}a^{14}-\frac{23362}{307}a^{13}+\frac{15092}{307}a^{12}+\frac{14415}{307}a^{11}-\frac{29731}{307}a^{10}+\frac{23048}{307}a^{9}-\frac{2261}{307}a^{8}-\frac{22816}{307}a^{7}-\frac{13280}{307}a^{6}+\frac{5789}{307}a^{5}-\frac{23235}{307}a^{4}-\frac{20893}{307}a^{3}+\frac{1760}{307}a^{2}-\frac{8583}{307}a-\frac{6271}{307}$, $\frac{1738}{307}a^{16}-\frac{4616}{307}a^{15}+\frac{9866}{307}a^{14}-\frac{9746}{307}a^{13}+\frac{6074}{307}a^{12}+\frac{6422}{307}a^{11}-\frac{12589}{307}a^{10}+\frac{9273}{307}a^{9}-\frac{507}{307}a^{8}-\frac{9692}{307}a^{7}-\frac{6303}{307}a^{6}+\frac{2816}{307}a^{5}-\frac{9278}{307}a^{4}-\frac{9657}{307}a^{3}+\frac{950}{307}a^{2}-\frac{3288}{307}a-\frac{2975}{307}$
| 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: | \( 621.862699097 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{6}\cdot 621.862699097 \cdot 1}{2\cdot\sqrt{14517428819890014793}}\cr\approx \mathstrut & 0.160675199437 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 4*x^15 - 2*x^14 + 6*x^12 - 5*x^11 + x^10 + 3*x^9 - 6*x^8 - 7*x^7 - x^6 - 5*x^5 - 9*x^4 - 3*x^3 - 2*x^2 - 3*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 + 4*x^15 - 2*x^14 + 6*x^12 - 5*x^11 + x^10 + 3*x^9 - 6*x^8 - 7*x^7 - x^6 - 5*x^5 - 9*x^4 - 3*x^3 - 2*x^2 - 3*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 + 4*x^15 - 2*x^14 + 6*x^12 - 5*x^11 + x^10 + 3*x^9 - 6*x^8 - 7*x^7 - x^6 - 5*x^5 - 9*x^4 - 3*x^3 - 2*x^2 - 3*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 + 4*x^15 - 2*x^14 + 6*x^12 - 5*x^11 + x^10 + 3*x^9 - 6*x^8 - 7*x^7 - x^6 - 5*x^5 - 9*x^4 - 3*x^3 - 2*x^2 - 3*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 $S_{17}$ are not computed |
| Character table for $S_{17}$ is not computed |
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 | $17$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $15{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.8.0.1}{8} }$ | $16{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(10170343\)
| $\Q_{10170343}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(1427427651151\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |