Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + x^15 + 2*x^14 - 3*x^13 + 3*x^11 - 3*x^10 - x^9 + 6*x^8 - 5*x^7 - 4*x^6 + 8*x^5 - 3*x^4 - 4*x^3 + 3*x^2 - 1)
gp: K = bnfinit(y^17 - 2*y^16 + y^15 + 2*y^14 - 3*y^13 + 3*y^11 - 3*y^10 - y^9 + 6*y^8 - 5*y^7 - 4*y^6 + 8*y^5 - 3*y^4 - 4*y^3 + 3*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 + x^15 + 2*x^14 - 3*x^13 + 3*x^11 - 3*x^10 - x^9 + 6*x^8 - 5*x^7 - 4*x^6 + 8*x^5 - 3*x^4 - 4*x^3 + 3*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 + x^15 + 2*x^14 - 3*x^13 + 3*x^11 - 3*x^10 - x^9 + 6*x^8 - 5*x^7 - 4*x^6 + 8*x^5 - 3*x^4 - 4*x^3 + 3*x^2 - 1)
\( x^{17} - 2 x^{16} + x^{15} + 2 x^{14} - 3 x^{13} + 3 x^{11} - 3 x^{10} - x^{9} + 6 x^{8} - 5 x^{7} + \cdots - 1 \)
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: | $[3, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-2074524434943857177144\)
\(\medspace = -\,2^{3}\cdot 17\cdot 20759\cdot 9318913\cdot 78851137\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.94\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(17\), \(20759\), \(9318913\), \(78851137\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-51863\!\cdots\!94286}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: | $9$ | 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: |
$a$, $2a^{16}-4a^{15}+3a^{14}+2a^{13}-5a^{12}+2a^{11}+3a^{10}-6a^{9}+a^{8}+9a^{7}-11a^{6}-2a^{5}+11a^{4}-10a^{3}+3a-3$, $a^{3}-a^{2}+a$, $7a^{16}-17a^{15}+15a^{14}+4a^{13}-19a^{12}+9a^{11}+12a^{10}-24a^{9}+7a^{8}+34a^{7}-48a^{6}-a^{5}+44a^{4}-39a^{3}+3a^{2}+12a-8$, $a^{16}-3a^{14}+3a^{13}+2a^{12}-5a^{11}+a^{10}+3a^{9}-5a^{8}+3a^{7}+7a^{6}-12a^{5}-4a^{4}+12a^{3}-3a^{2}-5a+2$, $a^{16}-4a^{15}+6a^{14}-2a^{13}-5a^{12}+6a^{11}+a^{10}-7a^{9}+5a^{8}+5a^{7}-16a^{6}+10a^{5}+10a^{4}-17a^{3}+5a^{2}+4a-3$, $8a^{16}-23a^{15}+27a^{14}-3a^{13}-26a^{12}+22a^{11}+11a^{10}-37a^{9}+20a^{8}+37a^{7}-75a^{6}+25a^{5}+57a^{4}-75a^{3}+18a^{2}+19a-16$, $7a^{16}-20a^{15}+23a^{14}-2a^{13}-23a^{12}+19a^{11}+10a^{10}-32a^{9}+17a^{8}+33a^{7}-65a^{6}+20a^{5}+50a^{4}-65a^{3}+14a^{2}+18a-13$, $3a^{16}-8a^{15}+7a^{14}+3a^{13}-10a^{12}+4a^{11}+7a^{10}-11a^{9}+2a^{8}+17a^{7}-24a^{6}-2a^{5}+25a^{4}-16a^{3}-3a^{2}+7a-2$
| 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: | \( 12720.6080668 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 12720.6080668 \cdot 1}{2\cdot\sqrt{2074524434943857177144}}\cr\approx \mathstrut & 0.431884546286 \end{aligned}\]
# 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 + 2*x^14 - 3*x^13 + 3*x^11 - 3*x^10 - x^9 + 6*x^8 - 5*x^7 - 4*x^6 + 8*x^5 - 3*x^4 - 4*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 + 2*x^14 - 3*x^13 + 3*x^11 - 3*x^10 - x^9 + 6*x^8 - 5*x^7 - 4*x^6 + 8*x^5 - 3*x^4 - 4*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 + 2*x^14 - 3*x^13 + 3*x^11 - 3*x^10 - x^9 + 6*x^8 - 5*x^7 - 4*x^6 + 8*x^5 - 3*x^4 - 4*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 + 2*x^14 - 3*x^13 + 3*x^11 - 3*x^10 - x^9 + 6*x^8 - 5*x^7 - 4*x^6 + 8*x^5 - 3*x^4 - 4*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 | $17$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.15.0.1 | $x^{15} + x^{5} + x^{4} + x^{2} + 1$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.7.0.1 | $x^{7} + 12 x + 14$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 17.7.0.1 | $x^{7} + 12 x + 14$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(20759\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
|
\(9318913\)
| $\Q_{9318913}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{9318913}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{9318913}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(78851137\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |