Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + x^15 - 3*x^14 + 4*x^13 - 4*x^12 - x^11 + 4*x^10 - 5*x^9 + 17*x^8 - 42*x^7 + 59*x^6 - 45*x^5 + 9*x^4 + 15*x^3 - 15*x^2 + 6*x - 1)
gp: K = bnfinit(y^17 - y^16 + y^15 - 3*y^14 + 4*y^13 - 4*y^12 - y^11 + 4*y^10 - 5*y^9 + 17*y^8 - 42*y^7 + 59*y^6 - 45*y^5 + 9*y^4 + 15*y^3 - 15*y^2 + 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^16 + x^15 - 3*x^14 + 4*x^13 - 4*x^12 - x^11 + 4*x^10 - 5*x^9 + 17*x^8 - 42*x^7 + 59*x^6 - 45*x^5 + 9*x^4 + 15*x^3 - 15*x^2 + 6*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^16 + x^15 - 3*x^14 + 4*x^13 - 4*x^12 - x^11 + 4*x^10 - 5*x^9 + 17*x^8 - 42*x^7 + 59*x^6 - 45*x^5 + 9*x^4 + 15*x^3 - 15*x^2 + 6*x - 1)
\( x^{17} - x^{16} + x^{15} - 3 x^{14} + 4 x^{13} - 4 x^{12} - x^{11} + 4 x^{10} - 5 x^{9} + 17 x^{8} + \cdots - 1 \)
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: | $[3, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-2311521177047925203\)
\(\medspace = -\,16843\cdot 137239279050521\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.03\) | 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: | $16843^{1/2}137239279050521^{1/2}\approx 1520368763.5070398$ | ||
| Ramified primes: |
\(16843\), \(137239279050521\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-23115\!\cdots\!25203}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{28117}a^{16}+\frac{11484}{28117}a^{15}-\frac{3106}{28117}a^{14}+\frac{260}{907}a^{13}+\frac{7940}{28117}a^{12}+\frac{7465}{28117}a^{11}+\frac{6791}{28117}a^{10}-\frac{1919}{28117}a^{9}+\frac{4008}{28117}a^{8}+\frac{4368}{28117}a^{7}+\frac{5710}{28117}a^{6}+\frac{10565}{28117}a^{5}-\frac{13992}{28117}a^{4}-\frac{9456}{28117}a^{3}+\frac{446}{907}a^{2}-\frac{13221}{28117}a-\frac{11379}{28117}$
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: | $9$ | 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: |
$a$, $\frac{312988}{28117}a^{16}+\frac{101848}{28117}a^{15}+\frac{201366}{28117}a^{14}-\frac{21835}{907}a^{13}+\frac{200494}{28117}a^{12}-\frac{404684}{28117}a^{11}-\frac{1184021}{28117}a^{10}+\frac{95733}{28117}a^{9}-\frac{630742}{28117}a^{8}+\frac{4160009}{28117}a^{7}-\frac{7040524}{28117}a^{6}+\frac{5669952}{28117}a^{5}+\frac{35239}{28117}a^{4}-\frac{3589967}{28117}a^{3}+\frac{58861}{907}a^{2}+\frac{20776}{28117}a-\frac{247466}{28117}$, $\frac{146434}{28117}a^{16}-\frac{282767}{28117}a^{15}+\frac{109056}{28117}a^{14}-\frac{16625}{907}a^{13}+\frac{891520}{28117}a^{12}-\frac{677724}{28117}a^{11}+\frac{19355}{28117}a^{10}+\frac{1091015}{28117}a^{9}-\frac{822179}{28117}a^{8}+\frac{2717428}{28117}a^{7}-\frac{7990434}{28117}a^{6}+\frac{11774542}{28117}a^{5}-\frac{9100529}{28117}a^{4}+\frac{1263260}{28117}a^{3}+\frac{126195}{907}a^{2}-\frac{3044515}{28117}a+\frac{812561}{28117}$, $\frac{273065}{28117}a^{16}-\frac{179252}{28117}a^{15}+\frac{150000}{28117}a^{14}-\frac{24828}{907}a^{13}+\frac{793389}{28117}a^{12}-\frac{670849}{28117}a^{11}-\frac{578426}{28117}a^{10}+\frac{988889}{28117}a^{9}-\frac{825098}{28117}a^{8}+\frac{4270447}{28117}a^{7}-\frac{9868035}{28117}a^{6}+\frac{11852314}{28117}a^{5}-\frac{6598196}{28117}a^{4}-\frac{1355678}{28117}a^{3}+\frac{129266}{907}a^{2}-\frac{2106457}{28117}a+\frac{368556}{28117}$, $\frac{165585}{28117}a^{16}+\frac{81664}{28117}a^{15}+\frac{93505}{28117}a^{14}-\frac{11353}{907}a^{13}+\frac{22097}{28117}a^{12}-\frac{156231}{28117}a^{11}-\frac{670254}{28117}a^{10}-\frac{7398}{28117}a^{9}-\frac{233924}{28117}a^{8}+\frac{2158581}{28117}a^{7}-\frac{3289698}{28117}a^{6}+\frac{2102677}{28117}a^{5}+\frac{987692}{28117}a^{4}-\frac{2297858}{28117}a^{3}+\frac{22924}{907}a^{2}+\frac{327739}{28117}a-\frac{240247}{28117}$, $\frac{74408}{28117}a^{16}-\frac{424030}{28117}a^{15}+\frac{66726}{28117}a^{14}-\frac{13835}{907}a^{13}+\frac{1129796}{28117}a^{12}-\frac{754774}{28117}a^{11}+\frac{492110}{28117}a^{10}+\frac{1451258}{28117}a^{9}-\frac{853265}{28117}a^{8}+\frac{2034165}{28117}a^{7}-\frac{8047769}{28117}a^{6}+\frac{13690296}{28117}a^{5}-\frac{12315706}{28117}a^{4}+\frac{3062513}{28117}a^{3}+\frac{147586}{907}a^{2}-\frac{4123771}{28117}a+\frac{1179503}{28117}$, $\frac{93379}{28117}a^{16}-\frac{158529}{28117}a^{15}+\frac{76032}{28117}a^{14}-\frac{10013}{907}a^{13}+\frac{518193}{28117}a^{12}-\frac{424184}{28117}a^{11}-\frac{42146}{28117}a^{10}+\frac{613914}{28117}a^{9}-\frac{508461}{28117}a^{8}+\frac{1701290}{28117}a^{7}-\frac{4768471}{28117}a^{6}+\frac{7037206}{28117}a^{5}-\frac{5304208}{28117}a^{4}+\frac{669252}{28117}a^{3}+\frac{74689}{907}a^{2}-\frac{1830128}{28117}a+\frac{487895}{28117}$, $\frac{385357}{28117}a^{16}-\frac{35427}{28117}a^{15}+\frac{218667}{28117}a^{14}-\frac{30680}{907}a^{13}+\frac{604980}{28117}a^{12}-\frac{663190}{28117}a^{11}-\frac{1183185}{28117}a^{10}+\frac{679942}{28117}a^{9}-\frac{883815}{28117}a^{8}+\frac{5526103}{28117}a^{7}-\frac{10872995}{28117}a^{6}+\frac{10948852}{28117}a^{5}-\frac{3685732}{28117}a^{4}-\frac{3487217}{28117}a^{3}+\frac{117888}{907}a^{2}-\frac{1129177}{28117}a+\frac{37549}{28117}$, $\frac{146434}{28117}a^{16}-\frac{282767}{28117}a^{15}+\frac{109056}{28117}a^{14}-\frac{16625}{907}a^{13}+\frac{891520}{28117}a^{12}-\frac{677724}{28117}a^{11}+\frac{19355}{28117}a^{10}+\frac{1091015}{28117}a^{9}-\frac{822179}{28117}a^{8}+\frac{2717428}{28117}a^{7}-\frac{7990434}{28117}a^{6}+\frac{11774542}{28117}a^{5}-\frac{9100529}{28117}a^{4}+\frac{1263260}{28117}a^{3}+\frac{125288}{907}a^{2}-\frac{3044515}{28117}a+\frac{812561}{28117}$
| 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: | \( 161.12597366 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 161.12597366 \cdot 1}{2\cdot\sqrt{2311521177047925203}}\cr\approx \mathstrut & 0.16388367201 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + x^15 - 3*x^14 + 4*x^13 - 4*x^12 - x^11 + 4*x^10 - 5*x^9 + 17*x^8 - 42*x^7 + 59*x^6 - 45*x^5 + 9*x^4 + 15*x^3 - 15*x^2 + 6*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 + x^15 - 3*x^14 + 4*x^13 - 4*x^12 - x^11 + 4*x^10 - 5*x^9 + 17*x^8 - 42*x^7 + 59*x^6 - 45*x^5 + 9*x^4 + 15*x^3 - 15*x^2 + 6*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 + x^15 - 3*x^14 + 4*x^13 - 4*x^12 - x^11 + 4*x^10 - 5*x^9 + 17*x^8 - 42*x^7 + 59*x^6 - 45*x^5 + 9*x^4 + 15*x^3 - 15*x^2 + 6*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 + x^15 - 3*x^14 + 4*x^13 - 4*x^12 - x^11 + 4*x^10 - 5*x^9 + 17*x^8 - 42*x^7 + 59*x^6 - 45*x^5 + 9*x^4 + 15*x^3 - 15*x^2 + 6*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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.8.0.1}{8} }$ | $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | $17$ | ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $17$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(16843\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(137239279050521\)
| $\Q_{137239279050521}$ | $x$ | $1$ | $1$ | $0$ | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |