Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 + 4*x^15 + x^14 - 13*x^13 + 26*x^12 - 25*x^11 + 2*x^10 + 35*x^9 - 62*x^8 + 59*x^7 - 25*x^6 - 13*x^5 + 33*x^4 - 30*x^3 + 17*x^2 - 6*x + 1)
gp: K = bnfinit(y^17 - 3*y^16 + 4*y^15 + y^14 - 13*y^13 + 26*y^12 - 25*y^11 + 2*y^10 + 35*y^9 - 62*y^8 + 59*y^7 - 25*y^6 - 13*y^5 + 33*y^4 - 30*y^3 + 17*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 3*x^16 + 4*x^15 + x^14 - 13*x^13 + 26*x^12 - 25*x^11 + 2*x^10 + 35*x^9 - 62*x^8 + 59*x^7 - 25*x^6 - 13*x^5 + 33*x^4 - 30*x^3 + 17*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 3*x^16 + 4*x^15 + x^14 - 13*x^13 + 26*x^12 - 25*x^11 + 2*x^10 + 35*x^9 - 62*x^8 + 59*x^7 - 25*x^6 - 13*x^5 + 33*x^4 - 30*x^3 + 17*x^2 - 6*x + 1)
\( x^{17} - 3 x^{16} + 4 x^{15} + x^{14} - 13 x^{13} + 26 x^{12} - 25 x^{11} + 2 x^{10} + 35 x^{9} + \cdots + 1 \)
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: |
\(-8553979677514650635476\)
\(\medspace = -\,2^{2}\cdot 11\cdot 53\cdot 34759\cdot 105529138935877\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.50\) | 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: |
\(2\), \(11\), \(53\), \(34759\), \(105529138935877\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-21384\!\cdots\!58869}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11}a^{16}+\frac{5}{11}a^{15}+\frac{1}{11}a^{13}-\frac{5}{11}a^{12}-\frac{3}{11}a^{11}-\frac{5}{11}a^{10}-\frac{5}{11}a^{9}-\frac{5}{11}a^{8}-\frac{3}{11}a^{7}+\frac{2}{11}a^{6}+\frac{2}{11}a^{5}+\frac{3}{11}a^{4}+\frac{2}{11}a^{3}-\frac{3}{11}a^{2}+\frac{4}{11}a+\frac{4}{11}$
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: |
$a$, $a^{16}-2a^{15}+a^{14}+4a^{13}-10a^{12}+12a^{11}-3a^{10}-13a^{9}+25a^{8}-24a^{7}+10a^{6}+9a^{5}-14a^{4}+10a^{3}-6a^{2}+a$, $\frac{21}{11}a^{16}+\frac{17}{11}a^{15}-8a^{14}+\frac{153}{11}a^{13}+\frac{5}{11}a^{12}-\frac{316}{11}a^{11}+\frac{665}{11}a^{10}-\frac{501}{11}a^{9}-\frac{149}{11}a^{8}+\frac{960}{11}a^{7}-\frac{1278}{11}a^{6}+\frac{878}{11}a^{5}+\frac{151}{11}a^{4}-\frac{596}{11}a^{3}+\frac{597}{11}a^{2}-\frac{312}{11}a+\frac{62}{11}$, $\frac{1}{11}a^{16}-\frac{6}{11}a^{15}+a^{14}-\frac{10}{11}a^{13}-\frac{16}{11}a^{12}+\frac{63}{11}a^{11}-\frac{104}{11}a^{10}+\frac{72}{11}a^{9}+\frac{39}{11}a^{8}-\frac{190}{11}a^{7}+\frac{266}{11}a^{6}-\frac{196}{11}a^{5}+\frac{25}{11}a^{4}+\frac{123}{11}a^{3}-\frac{179}{11}a^{2}+\frac{103}{11}a-\frac{29}{11}$, $\frac{15}{11}a^{16}-\frac{24}{11}a^{15}+a^{14}+\frac{59}{11}a^{13}-\frac{130}{11}a^{12}+\frac{153}{11}a^{11}-\frac{20}{11}a^{10}-\frac{174}{11}a^{9}+\frac{332}{11}a^{8}-\frac{309}{11}a^{7}+\frac{129}{11}a^{6}+\frac{118}{11}a^{5}-\frac{175}{11}a^{4}+\frac{173}{11}a^{3}-\frac{89}{11}a^{2}+\frac{38}{11}a-\frac{6}{11}$, $\frac{45}{11}a^{16}-\frac{105}{11}a^{15}+9a^{14}+\frac{133}{11}a^{13}-\frac{511}{11}a^{12}+\frac{789}{11}a^{11}-\frac{489}{11}a^{10}-\frac{379}{11}a^{9}+\frac{1370}{11}a^{8}-\frac{1752}{11}a^{7}+\frac{1212}{11}a^{6}-\frac{42}{11}a^{5}-\frac{745}{11}a^{4}+\frac{915}{11}a^{3}-\frac{608}{11}a^{2}+\frac{246}{11}a-\frac{51}{11}$, $\frac{15}{11}a^{16}-\frac{46}{11}a^{15}+5a^{14}+\frac{37}{11}a^{13}-\frac{229}{11}a^{12}+\frac{384}{11}a^{11}-\frac{273}{11}a^{10}-\frac{163}{11}a^{9}+\frac{684}{11}a^{8}-\frac{881}{11}a^{7}+\frac{569}{11}a^{6}+\frac{63}{11}a^{5}-\frac{494}{11}a^{4}+\frac{492}{11}a^{3}-\frac{210}{11}a^{2}+\frac{27}{11}a+\frac{5}{11}$, $\frac{26}{11}a^{16}-\frac{68}{11}a^{15}+7a^{14}+\frac{59}{11}a^{13}-\frac{317}{11}a^{12}+\frac{549}{11}a^{11}-\frac{427}{11}a^{10}-\frac{119}{11}a^{9}+\frac{860}{11}a^{8}-\frac{1266}{11}a^{7}+\frac{1042}{11}a^{6}-\frac{256}{11}a^{5}-\frac{428}{11}a^{4}+\frac{690}{11}a^{3}-\frac{518}{11}a^{2}+\frac{258}{11}a-\frac{61}{11}$, $\frac{5}{11}a^{16}-\frac{30}{11}a^{15}+4a^{14}-\frac{6}{11}a^{13}-\frac{124}{11}a^{12}+\frac{260}{11}a^{11}-\frac{278}{11}a^{10}+\frac{30}{11}a^{9}+\frac{349}{11}a^{8}-\frac{642}{11}a^{7}+\frac{593}{11}a^{6}-\frac{243}{11}a^{5}-\frac{194}{11}a^{4}+\frac{329}{11}a^{3}-\frac{290}{11}a^{2}+\frac{141}{11}a-\frac{35}{11}$
| 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: | \( 30334.8896134 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 30334.8896134 \cdot 1}{2\cdot\sqrt{8553979677514650635476}}\cr\approx \mathstrut & 0.507197824041 \end{aligned}\]
# 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 + x^14 - 13*x^13 + 26*x^12 - 25*x^11 + 2*x^10 + 35*x^9 - 62*x^8 + 59*x^7 - 25*x^6 - 13*x^5 + 33*x^4 - 30*x^3 + 17*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 - 3*x^16 + 4*x^15 + x^14 - 13*x^13 + 26*x^12 - 25*x^11 + 2*x^10 + 35*x^9 - 62*x^8 + 59*x^7 - 25*x^6 - 13*x^5 + 33*x^4 - 30*x^3 + 17*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 - 3*x^16 + 4*x^15 + x^14 - 13*x^13 + 26*x^12 - 25*x^11 + 2*x^10 + 35*x^9 - 62*x^8 + 59*x^7 - 25*x^6 - 13*x^5 + 33*x^4 - 30*x^3 + 17*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 - 3*x^16 + 4*x^15 + x^14 - 13*x^13 + 26*x^12 - 25*x^11 + 2*x^10 + 35*x^9 - 62*x^8 + 59*x^7 - 25*x^6 - 13*x^5 + 33*x^4 - 30*x^3 + 17*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 | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $17$ | R | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $17$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $17$ | R | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.15.0.1 | $x^{15} + x^{5} + x^{4} + x^{2} + 1$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.7.0.1 | $x^{7} + 4 x + 9$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(53\)
| 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 53.15.0.1 | $x^{15} + 22 x^{6} + 31 x^{5} + 15 x^{4} + 11 x^{3} + 20 x^{2} + 4 x + 51$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
|
\(34759\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(105529138935877\)
| $\Q_{105529138935877}$ | $x$ | $1$ | $1$ | $0$ | 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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |