Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 + 4*x^15 - 3*x^14 + x^13 + 4*x^12 - 8*x^11 + 9*x^10 - 2*x^9 - 9*x^8 + 12*x^7 - x^6 - 4*x^5 + 2*x^4 - 3*x^3 + x^2 - x - 1)
gp: K = bnfinit(y^17 - 3*y^16 + 4*y^15 - 3*y^14 + y^13 + 4*y^12 - 8*y^11 + 9*y^10 - 2*y^9 - 9*y^8 + 12*y^7 - y^6 - 4*y^5 + 2*y^4 - 3*y^3 + y^2 - y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 3*x^16 + 4*x^15 - 3*x^14 + x^13 + 4*x^12 - 8*x^11 + 9*x^10 - 2*x^9 - 9*x^8 + 12*x^7 - x^6 - 4*x^5 + 2*x^4 - 3*x^3 + x^2 - x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 3*x^16 + 4*x^15 - 3*x^14 + x^13 + 4*x^12 - 8*x^11 + 9*x^10 - 2*x^9 - 9*x^8 + 12*x^7 - x^6 - 4*x^5 + 2*x^4 - 3*x^3 + x^2 - x - 1)
\( x^{17} - 3 x^{16} + 4 x^{15} - 3 x^{14} + x^{13} + 4 x^{12} - 8 x^{11} + 9 x^{10} - 2 x^{9} - 9 x^{8} + \cdots - 1 \)
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: |
\(-1912319465404851491\)
\(\medspace = -\,227\cdot 2411\cdot 3533\cdot 988994191\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.90\) | 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: | $227^{1/2}2411^{1/2}3533^{1/2}988994191^{1/2}\approx 1382866394.632848$ | ||
| Ramified primes: |
\(227\), \(2411\), \(3533\), \(988994191\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-19123\!\cdots\!51491}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{6939479}a^{16}+\frac{451058}{6939479}a^{15}+\frac{3027220}{6939479}a^{14}+\frac{416024}{6939479}a^{13}+\frac{1749826}{6939479}a^{12}+\frac{2742367}{6939479}a^{11}-\frac{1209329}{6939479}a^{10}-\frac{3401265}{6939479}a^{9}+\frac{2025153}{6939479}a^{8}+\frac{3098117}{6939479}a^{7}+\frac{2168524}{6939479}a^{6}+\frac{3159955}{6939479}a^{5}-\frac{1826954}{6939479}a^{4}+\frac{2372537}{6939479}a^{3}+\frac{1036727}{6939479}a^{2}+\frac{3385454}{6939479}a+\frac{33785}{6939479}$
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: |
$\frac{668821}{6939479}a^{16}-\frac{2907949}{6939479}a^{15}+\frac{5914580}{6939479}a^{14}-\frac{6701759}{6939479}a^{13}+\frac{4999712}{6939479}a^{12}-\frac{236746}{6939479}a^{11}-\frac{7535222}{6939479}a^{10}+\frac{13970862}{6939479}a^{9}-\frac{10414523}{6939479}a^{8}-\frac{1082469}{6939479}a^{7}+\frac{17158162}{6939479}a^{6}-\frac{15703269}{6939479}a^{5}+\frac{5200565}{6939479}a^{4}+\frac{482300}{6939479}a^{3}-\frac{1013334}{6939479}a^{2}+\frac{7884740}{6939479}a-\frac{5765618}{6939479}$, $a$, $\frac{2120656}{6939479}a^{16}-\frac{5870791}{6939479}a^{15}+\frac{7991336}{6939479}a^{14}-\frac{6870921}{6939479}a^{13}+\frac{3642270}{6939479}a^{12}+\frac{7475239}{6939479}a^{11}-\frac{16940584}{6939479}a^{10}+\frac{23079634}{6939479}a^{9}-\frac{10266278}{6939479}a^{8}-\frac{8491204}{6939479}a^{7}+\frac{18669587}{6939479}a^{6}-\frac{6700139}{6939479}a^{5}+\frac{4861271}{6939479}a^{4}-\frac{2574577}{6939479}a^{3}-\frac{10524910}{6939479}a^{2}+\frac{6548794}{6939479}a-\frac{3757715}{6939479}$, $\frac{1367013}{6939479}a^{16}-\frac{5256791}{6939479}a^{15}+\frac{8763353}{6939479}a^{14}-\frac{7845654}{6939479}a^{13}+\frac{3417917}{6939479}a^{12}+\frac{5994391}{6939479}a^{11}-\frac{18018981}{6939479}a^{10}+\frac{23187614}{6939479}a^{9}-\frac{11395313}{6939479}a^{8}-\frac{11698137}{6939479}a^{7}+\frac{28556987}{6939479}a^{6}-\frac{15081421}{6939479}a^{5}-\frac{6892134}{6939479}a^{4}+\frac{12379667}{6939479}a^{3}-\frac{9691282}{6939479}a^{2}+\frac{7204844}{6939479}a+\frac{2301460}{6939479}$, $\frac{100030}{6939479}a^{16}-\frac{1160718}{6939479}a^{15}+\frac{1710956}{6939479}a^{14}-\frac{1174843}{6939479}a^{13}+\frac{615963}{6939479}a^{12}+\frac{1366140}{6939479}a^{11}-\frac{7121421}{6939479}a^{10}+\frac{7177941}{6939479}a^{9}-\frac{8155857}{6939479}a^{8}+\frac{1390328}{6939479}a^{7}+\frac{3221138}{6939479}a^{6}-\frac{2969800}{6939479}a^{5}+\frac{970845}{6939479}a^{4}-\frac{5305690}{6939479}a^{3}-\frac{6711845}{6939479}a^{2}+\frac{7327899}{6939479}a-\frac{12723}{6939479}$, $\frac{186232}{6939479}a^{16}-\frac{959839}{6939479}a^{15}+\frac{1961080}{6939479}a^{14}-\frac{2301467}{6939479}a^{13}+\frac{2601271}{6939479}a^{12}-\frac{1405340}{6939479}a^{11}-\frac{1906862}{6939479}a^{10}+\frac{4320161}{6939479}a^{9}-\frac{5450675}{6939479}a^{8}+\frac{6362126}{6939479}a^{7}-\frac{1358316}{6939479}a^{6}+\frac{3041402}{6939479}a^{5}-\frac{1581437}{6939479}a^{4}-\frac{1256825}{6939479}a^{3}+\frac{1557926}{6939479}a^{2}+\frac{444262}{6939479}a+\frac{4680146}{6939479}$, $\frac{842332}{6939479}a^{16}-\frac{2827473}{6939479}a^{15}+\frac{5779011}{6939479}a^{14}-\frac{6422053}{6939479}a^{13}+\frac{2973590}{6939479}a^{12}+\frac{4407719}{6939479}a^{11}-\frac{10392818}{6939479}a^{10}+\frac{18131523}{6939479}a^{9}-\frac{11550984}{6939479}a^{8}-\frac{3504938}{6939479}a^{7}+\frac{21374546}{6939479}a^{6}-\frac{16987054}{6939479}a^{5}+\frac{3985791}{6939479}a^{4}+\frac{11855427}{6939479}a^{3}-\frac{2649475}{6939479}a^{2}+\frac{1435863}{6939479}a-\frac{7556238}{6939479}$, $\frac{1529449}{6939479}a^{16}-\frac{4218785}{6939479}a^{15}+\frac{6789333}{6939479}a^{14}-\frac{7217692}{6939479}a^{13}+\frac{4033692}{6939479}a^{12}+\frac{6084435}{6939479}a^{11}-\frac{13812893}{6939479}a^{10}+\frac{21926659}{6939479}a^{9}-\frac{12705121}{6939479}a^{8}-\frac{3103247}{6939479}a^{7}+\frac{20027932}{6939479}a^{6}-\frac{17074234}{6939479}a^{5}+\frac{10706315}{6939479}a^{4}+\frac{6894055}{6939479}a^{3}-\frac{8241203}{6939479}a^{2}+\frac{7797433}{6939479}a-\frac{5865648}{6939479}$, $\frac{2196622}{6939479}a^{16}-\frac{7946065}{6939479}a^{15}+\frac{13330754}{6939479}a^{14}-\frac{12518582}{6939479}a^{13}+\frac{5203941}{6939479}a^{12}+\frac{10967181}{6939479}a^{11}-\frac{26943875}{6939479}a^{10}+\frac{34083209}{6939479}a^{9}-\frac{15742910}{6939479}a^{8}-\frac{23244425}{6939479}a^{7}+\frac{51169185}{6939479}a^{6}-\frac{28834614}{6939479}a^{5}-\frac{5825251}{6939479}a^{4}+\frac{25180493}{6939479}a^{3}-\frac{17608278}{6939479}a^{2}+\frac{1916139}{6939479}a+\frac{2085844}{6939479}$
| 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: | \( 144.902374432 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 144.902374432 \cdot 1}{2\cdot\sqrt{1912319465404851491}}\cr\approx \mathstrut & 0.162037057871 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 + 4*x^15 - 3*x^14 + x^13 + 4*x^12 - 8*x^11 + 9*x^10 - 2*x^9 - 9*x^8 + 12*x^7 - x^6 - 4*x^5 + 2*x^4 - 3*x^3 + x^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 - 3*x^16 + 4*x^15 - 3*x^14 + x^13 + 4*x^12 - 8*x^11 + 9*x^10 - 2*x^9 - 9*x^8 + 12*x^7 - x^6 - 4*x^5 + 2*x^4 - 3*x^3 + x^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 - 3*x^16 + 4*x^15 - 3*x^14 + x^13 + 4*x^12 - 8*x^11 + 9*x^10 - 2*x^9 - 9*x^8 + 12*x^7 - x^6 - 4*x^5 + 2*x^4 - 3*x^3 + x^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 - 3*x^16 + 4*x^15 - 3*x^14 + x^13 + 4*x^12 - 8*x^11 + 9*x^10 - 2*x^9 - 9*x^8 + 12*x^7 - x^6 - 4*x^5 + 2*x^4 - 3*x^3 + x^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 $S_{17}$ 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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | $17$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $17$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $17$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(227\)
| 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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(2411\)
| $\Q_{2411}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(3533\)
| $\Q_{3533}$ | $x$ | $1$ | $1$ | $0$ | 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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(988994191\)
| $\Q_{988994191}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |