Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 2*x^13 + x^12 - 5*x^11 + x^10 + 3*x^9 + 3*x^8 - x^7 - 7*x^6 + 2*x^5 + 6*x^4 + x^3 - 3*x^2 - 3*x - 1)
gp: K = bnfinit(y^17 - y^16 + 2*y^13 + y^12 - 5*y^11 + y^10 + 3*y^9 + 3*y^8 - y^7 - 7*y^6 + 2*y^5 + 6*y^4 + y^3 - 3*y^2 - 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^16 + 2*x^13 + x^12 - 5*x^11 + x^10 + 3*x^9 + 3*x^8 - x^7 - 7*x^6 + 2*x^5 + 6*x^4 + x^3 - 3*x^2 - 3*x - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^17 - x^16 + 2*x^13 + x^12 - 5*x^11 + x^10 + 3*x^9 + 3*x^8 - x^7 - 7*x^6 + 2*x^5 + 6*x^4 + x^3 - 3*x^2 - 3*x - 1)
\( x^{17} - x^{16} + 2 x^{13} + x^{12} - 5 x^{11} + x^{10} + 3 x^{9} + 3 x^{8} - x^{7} - 7 x^{6} + 2 x^{5} + \cdots - 1 \)
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: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(231494845017984433\)
\(\medspace = 59^{2}\cdot 1217\cdot 1579\cdot 34607051\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.51\) | 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: | $59^{1/2}1217^{1/2}1579^{1/2}34607051^{1/2}\approx 62638977.010985635$ | ||
| Ramified primes: |
\(59\), \(1217\), \(1579\), \(34607051\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{66502397304793}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{82223}a^{16}+\frac{32015}{82223}a^{15}+\frac{322}{82223}a^{14}+\frac{31277}{82223}a^{13}-\frac{29483}{82223}a^{12}-\frac{7687}{82223}a^{11}-\frac{13558}{82223}a^{10}-\frac{17710}{82223}a^{9}+\frac{6451}{82223}a^{8}-\frac{8957}{82223}a^{7}+\frac{26511}{82223}a^{6}-\frac{11860}{82223}a^{5}-\frac{3944}{82223}a^{4}+\frac{23430}{82223}a^{3}+\frac{14452}{82223}a^{2}+\frac{26408}{82223}a-\frac{20584}{82223}$
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: | $8$ | 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: |
$a$, $\frac{64784}{82223}a^{16}-\frac{97638}{82223}a^{15}+\frac{58029}{82223}a^{14}-\frac{54444}{82223}a^{13}+\frac{178064}{82223}a^{12}-\frac{52120}{82223}a^{11}-\frac{282055}{82223}a^{10}+\frac{179548}{82223}a^{9}+\frac{64298}{82223}a^{8}+\frac{224092}{82223}a^{7}-\frac{232069}{82223}a^{6}-\frac{293197}{82223}a^{5}+\frac{287657}{82223}a^{4}+\frac{216986}{82223}a^{3}-\frac{14933}{82223}a^{2}-\frac{244758}{82223}a-\frac{103465}{82223}$, $\frac{90089}{82223}a^{16}-\frac{183505}{82223}a^{15}+\frac{148385}{82223}a^{14}-\frac{68557}{82223}a^{13}+\frac{202251}{82223}a^{12}-\frac{114260}{82223}a^{11}-\frac{415112}{82223}a^{10}+\frac{554463}{82223}a^{9}-\frac{70248}{82223}a^{8}+\frac{91572}{82223}a^{7}-\frac{228671}{82223}a^{6}-\frac{378770}{82223}a^{5}+\frac{714574}{82223}a^{4}+\frac{38637}{82223}a^{3}-\frac{281646}{82223}a^{2}-\frac{52193}{82223}a-\frac{16657}{82223}$, $\frac{85003}{82223}a^{16}-\frac{128032}{82223}a^{15}+\frac{72930}{82223}a^{14}-\frac{41874}{82223}a^{13}+\frac{178037}{82223}a^{12}+\frac{8120}{82223}a^{11}-\frac{444221}{82223}a^{10}+\frac{346669}{82223}a^{9}+\frac{9166}{82223}a^{8}+\frac{259778}{82223}a^{7}-\frac{217897}{82223}a^{6}-\frac{492715}{82223}a^{5}+\frac{464677}{82223}a^{4}+\frac{179230}{82223}a^{3}-\frac{30487}{82223}a^{2}-\frac{257568}{82223}a-\frac{160758}{82223}$, $\frac{54533}{82223}a^{16}-\frac{131410}{82223}a^{15}+\frac{128350}{82223}a^{14}-\frac{87494}{82223}a^{13}+\frac{156549}{82223}a^{12}-\frac{104540}{82223}a^{11}-\frac{255867}{82223}a^{10}+\frac{423043}{82223}a^{9}-\frac{204280}{82223}a^{8}+\frac{199208}{82223}a^{7}-\frac{249315}{82223}a^{6}-\frac{241931}{82223}a^{5}+\frac{510554}{82223}a^{4}-\frac{119453}{82223}a^{3}-\frac{78762}{82223}a^{2}-\frac{110604}{82223}a-\frac{81099}{82223}$, $\frac{3613}{82223}a^{16}-\frac{17566}{82223}a^{15}+\frac{12264}{82223}a^{14}+\frac{29399}{82223}a^{13}-\frac{43294}{82223}a^{12}+\frac{18243}{82223}a^{11}-\frac{62369}{82223}a^{10}+\frac{147710}{82223}a^{9}-\frac{43869}{82223}a^{8}-\frac{130225}{82223}a^{7}+\frac{76671}{82223}a^{6}-\frac{11997}{82223}a^{5}+\frac{139353}{82223}a^{4}-\frac{37100}{82223}a^{3}-\frac{160975}{82223}a^{2}+\frac{115647}{82223}a+\frac{41823}{82223}$, $\frac{61240}{82223}a^{16}-\frac{91058}{82223}a^{15}+\frac{67983}{82223}a^{14}-\frac{63528}{82223}a^{13}+\frac{160383}{82223}a^{12}-\frac{25205}{82223}a^{11}-\frac{250735}{82223}a^{10}+\frac{207639}{82223}a^{9}-\frac{22275}{82223}a^{8}+\frac{229622}{82223}a^{7}-\frac{123941}{82223}a^{6}-\frac{277310}{82223}a^{5}+\frac{287283}{82223}a^{4}+\frac{61850}{82223}a^{3}+\frac{74331}{82223}a^{2}-\frac{100490}{82223}a-\frac{85570}{82223}$, $\frac{24715}{82223}a^{16}-\frac{63427}{82223}a^{15}+\frac{64822}{82223}a^{14}-\frac{49591}{82223}a^{13}+\frac{70104}{82223}a^{12}-\frac{49075}{82223}a^{11}-\frac{109468}{82223}a^{10}+\frac{217048}{82223}a^{9}-\frac{158378}{82223}a^{8}+\frac{136507}{82223}a^{7}-\frac{97945}{82223}a^{6}-\frac{77128}{82223}a^{5}+\frac{204964}{82223}a^{4}-\frac{106362}{82223}a^{3}+\frac{86691}{82223}a^{2}-\frac{12454}{82223}a-\frac{19859}{82223}$
| 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: | \( 32.0498639783 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 32.0498639783 \cdot 1}{2\cdot\sqrt{231494845017984433}}\cr\approx \mathstrut & 0.161805945405 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 2*x^13 + x^12 - 5*x^11 + x^10 + 3*x^9 + 3*x^8 - x^7 - 7*x^6 + 2*x^5 + 6*x^4 + x^3 - 3*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 - x^16 + 2*x^13 + x^12 - 5*x^11 + x^10 + 3*x^9 + 3*x^8 - x^7 - 7*x^6 + 2*x^5 + 6*x^4 + x^3 - 3*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 - x^16 + 2*x^13 + x^12 - 5*x^11 + x^10 + 3*x^9 + 3*x^8 - x^7 - 7*x^6 + 2*x^5 + 6*x^4 + x^3 - 3*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 - x^16 + 2*x^13 + x^12 - 5*x^11 + x^10 + 3*x^9 + 3*x^8 - x^7 - 7*x^6 + 2*x^5 + 6*x^4 + x^3 - 3*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 $S_{17}$ |
| 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 | $17$ | $17$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $17$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.7.0.1}{7} }$ | $15{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $17$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | $16{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(59\)
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 59.10.0.1 | $x^{10} + x^{6} + 28 x^{5} + 25 x^{4} + 4 x^{3} + 39 x^{2} + 15 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(1217\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(1579\)
| $\Q_{1579}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(34607051\)
| $\Q_{34607051}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |