Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 - x^15 + 5*x^14 - x^13 - 8*x^12 + 6*x^11 + 8*x^10 - 11*x^9 - 3*x^8 + 13*x^7 - 3*x^6 - 8*x^5 + 5*x^4 + 3*x^3 - 3*x^2 + 1)
gp: K = bnfinit(y^17 - 2*y^16 - y^15 + 5*y^14 - y^13 - 8*y^12 + 6*y^11 + 8*y^10 - 11*y^9 - 3*y^8 + 13*y^7 - 3*y^6 - 8*y^5 + 5*y^4 + 3*y^3 - 3*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 - x^15 + 5*x^14 - x^13 - 8*x^12 + 6*x^11 + 8*x^10 - 11*x^9 - 3*x^8 + 13*x^7 - 3*x^6 - 8*x^5 + 5*x^4 + 3*x^3 - 3*x^2 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^17 - 2*x^16 - x^15 + 5*x^14 - x^13 - 8*x^12 + 6*x^11 + 8*x^10 - 11*x^9 - 3*x^8 + 13*x^7 - 3*x^6 - 8*x^5 + 5*x^4 + 3*x^3 - 3*x^2 + 1)
\( x^{17} - 2 x^{16} - x^{15} + 5 x^{14} - x^{13} - 8 x^{12} + 6 x^{11} + 8 x^{10} - 11 x^{9} - 3 x^{8} + \cdots + 1 \)
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: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(142063412679164600644\)
\(\medspace = 2^{2}\cdot 1697\cdot 20928611178427313\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.33\) | 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: | $2^{2/3}1697^{1/2}20928611178427313^{1/2}\approx 9460144926.331247$ | ||
| Ramified primes: |
\(2\), \(1697\), \(20928611178427313\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{35515\!\cdots\!50161}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: |
$a$, $a^{15}-2a^{14}-a^{13}+5a^{12}-a^{11}-8a^{10}+6a^{9}+8a^{8}-11a^{7}-3a^{6}+13a^{5}-3a^{4}-7a^{3}+4a^{2}+2a-2$, $a^{16}-a^{15}-3a^{14}+4a^{13}+3a^{12}-8a^{11}+11a^{9}-4a^{8}-9a^{7}+9a^{6}+4a^{5}-8a^{4}+2a^{2}-a-1$, $a^{15}-2a^{14}-a^{13}+5a^{12}-a^{11}-8a^{10}+6a^{9}+8a^{8}-11a^{7}-3a^{6}+13a^{5}-3a^{4}-8a^{3}+5a^{2}+2a-2$, $a^{16}-2a^{15}-a^{14}+5a^{13}-a^{12}-8a^{11}+6a^{10}+7a^{9}-9a^{8}-3a^{7}+11a^{6}-3a^{5}-5a^{4}+3a^{3}+a^{2}-a$, $a^{15}-2a^{14}+2a^{12}-3a^{10}+2a^{9}+2a^{8}-2a^{7}+a^{5}+a^{3}-2a^{2}+a$, $2a^{15}-3a^{14}-3a^{13}+7a^{12}+2a^{11}-12a^{10}+3a^{9}+14a^{8}-8a^{7}-9a^{6}+12a^{5}+4a^{4}-7a^{3}+4a+1$, $2a^{16}-2a^{15}-5a^{14}+6a^{13}+6a^{12}-12a^{11}-3a^{10}+17a^{9}-2a^{8}-14a^{7}+10a^{6}+9a^{5}-8a^{4}-3a^{3}+3a^{2}-1$
| 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: | \( 1990.43257715 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 1990.43257715 \cdot 1}{2\cdot\sqrt{142063412679164600644}}\cr\approx \mathstrut & 0.405644223981 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 - x^15 + 5*x^14 - x^13 - 8*x^12 + 6*x^11 + 8*x^10 - 11*x^9 - 3*x^8 + 13*x^7 - 3*x^6 - 8*x^5 + 5*x^4 + 3*x^3 - 3*x^2 + 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 - x^15 + 5*x^14 - x^13 - 8*x^12 + 6*x^11 + 8*x^10 - 11*x^9 - 3*x^8 + 13*x^7 - 3*x^6 - 8*x^5 + 5*x^4 + 3*x^3 - 3*x^2 + 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 - x^15 + 5*x^14 - x^13 - 8*x^12 + 6*x^11 + 8*x^10 - 11*x^9 - 3*x^8 + 13*x^7 - 3*x^6 - 8*x^5 + 5*x^4 + 3*x^3 - 3*x^2 + 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 - x^15 + 5*x^14 - x^13 - 8*x^12 + 6*x^11 + 8*x^10 - 11*x^9 - 3*x^8 + 13*x^7 - 3*x^6 - 8*x^5 + 5*x^4 + 3*x^3 - 3*x^2 + 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 | R | $17$ | $17$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $17$ | ${\href{/padicField/13.7.0.1}{7} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.14.0.1 | $x^{14} + x^{7} + x^{5} + x^{3} + 1$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(1697\)
| $\Q_{1697}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(20928611178427313\)
| $\Q_{20928611178427313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |