Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^9 - 18*x^6 - 12*x^3 + 4*x - 2)
gp: K = bnfinit(y^17 - 4*y^9 - 18*y^6 - 12*y^3 + 4*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 4*x^9 - 18*x^6 - 12*x^3 + 4*x - 2);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^17 - 4*x^9 - 18*x^6 - 12*x^3 + 4*x - 2)
\( x^{17} - 4x^{9} - 18x^{6} - 12x^{3} + 4x - 2 \)
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: |
\(44542611246065932779454464\)
\(\medspace = 2^{16}\cdot 3\cdot 271\cdot 835997920414096373\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.27\) | 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: | $2^{16/17}3^{1/2}271^{1/2}835997920414096373^{1/2}\approx 50057622659.30103$ | ||
| Ramified primes: |
\(2\), \(3\), \(271\), \(835997920414096373\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{67966\!\cdots\!51249}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{472383}a^{16}+\frac{208976}{472383}a^{15}+\frac{104992}{472383}a^{14}+\frac{34991}{472383}a^{13}-\frac{209624}{472383}a^{12}+\frac{52481}{472383}a^{11}-\frac{46655}{472383}a^{10}+\frac{209840}{472383}a^{9}+\frac{69982}{157461}a^{8}+\frac{53135}{157461}a^{7}-\frac{52499}{157461}a^{6}+\frac{64145}{157461}a^{5}-\frac{46871}{157461}a^{4}-\frac{52591}{157461}a^{3}+\frac{16199}{52487}a^{2}+\frac{672}{52487}a-\frac{210056}{472383}$
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 $1$ (assuming GRH)
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: |
$\frac{59048}{472383}a^{16}+\frac{26122}{472383}a^{15}+\frac{13124}{472383}a^{14}-\frac{54674}{472383}a^{13}-\frac{26203}{472383}a^{12}+\frac{65608}{472383}a^{11}+\frac{53216}{472383}a^{10}+\frac{26230}{472383}a^{9}-\frac{109348}{157461}a^{8}-\frac{52406}{157461}a^{7}-\frac{26245}{157461}a^{6}-\frac{247856}{157461}a^{5}-\frac{104272}{157461}a^{4}-\frac{104987}{157461}a^{3}-\frac{4536}{52487}a^{2}+\frac{84}{52487}a-\frac{26257}{472383}$, $\frac{25474}{157461}a^{16}+\frac{13136}{157461}a^{15}-\frac{66338}{157461}a^{14}-\frac{25987}{157461}a^{13}+\frac{13117}{157461}a^{12}+\frac{57104}{157461}a^{11}+\frac{26158}{157461}a^{10}-\frac{21868}{157461}a^{9}-\frac{51974}{52487}a^{8}-\frac{26253}{52487}a^{7}+\frac{114208}{52487}a^{6}-\frac{100528}{52487}a^{5}-\frac{122552}{52487}a^{4}+\frac{289976}{52487}a^{3}+\frac{54083}{52487}a^{2}-\frac{81676}{52487}a+\frac{30619}{157461}$, $\frac{59048}{472383}a^{16}+\frac{26122}{472383}a^{15}+\frac{13124}{472383}a^{14}-\frac{54674}{472383}a^{13}-\frac{26203}{472383}a^{12}+\frac{65608}{472383}a^{11}+\frac{53216}{472383}a^{10}+\frac{26230}{472383}a^{9}-\frac{109348}{157461}a^{8}-\frac{52406}{157461}a^{7}-\frac{26245}{157461}a^{6}-\frac{247856}{157461}a^{5}-\frac{104272}{157461}a^{4}-\frac{104987}{157461}a^{3}+\frac{47951}{52487}a^{2}+\frac{52571}{52487}a-\frac{26257}{472383}$, $\frac{8748}{52487}a^{16}-\frac{162}{52487}a^{15}+\frac{3}{52487}a^{14}-\frac{2916}{52487}a^{13}+\frac{54}{52487}a^{12}-\frac{1}{52487}a^{11}+\frac{972}{52487}a^{10}-\frac{18}{52487}a^{9}-\frac{17496}{52487}a^{8}+\frac{324}{52487}a^{7}-\frac{6}{52487}a^{6}-\frac{151632}{52487}a^{5}+\frac{55295}{52487}a^{4}-\frac{52}{52487}a^{3}-\frac{54432}{52487}a^{2}+\frac{53495}{52487}a+\frac{52469}{52487}$, $\frac{2}{3}a^{16}+\frac{1}{3}a^{15}-\frac{1}{3}a^{14}-\frac{2}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{2}{3}a^{10}+\frac{1}{3}a^{9}-3a^{8}-2a^{7}+a^{6}-9a^{5}-4a^{4}+5a^{3}+a^{2}+\frac{5}{3}$, $\frac{26248}{472383}a^{16}-\frac{109348}{472383}a^{15}-\frac{52406}{472383}a^{14}+\frac{131216}{472383}a^{13}+\frac{106432}{472383}a^{12}+\frac{52460}{472383}a^{11}-\frac{183704}{472383}a^{10}-\frac{105460}{472383}a^{9}-\frac{52490}{157461}a^{8}+\frac{212864}{157461}a^{7}+\frac{104920}{157461}a^{6}-\frac{367435}{157461}a^{5}+\frac{445168}{157461}a^{4}+\frac{209480}{157461}a^{3}-\frac{163296}{52487}a^{2}+\frac{3024}{52487}a+\frac{576871}{472383}$, $\frac{95540}{157461}a^{16}-\frac{15377}{157461}a^{15}+\frac{40136}{157461}a^{14}-\frac{14351}{157461}a^{13}-\frac{12370}{157461}a^{12}+\frac{4117}{157461}a^{11}-\frac{12712}{157461}a^{10}+\frac{21619}{157461}a^{9}-\frac{133676}{52487}a^{8}+\frac{27747}{52487}a^{7}-\frac{44253}{52487}a^{6}-\frac{546177}{52487}a^{5}+\frac{135500}{52487}a^{4}-\frac{226065}{52487}a^{3}-\frac{305075}{52487}a^{2}+\frac{86324}{52487}a-\frac{188329}{157461}$, $\frac{1589914}{472383}a^{16}+\frac{923099}{472383}a^{15}+\frac{380446}{472383}a^{14}+\frac{134864}{472383}a^{13}+\frac{24718}{472383}a^{12}-\frac{39337}{472383}a^{11}-\frac{79946}{472383}a^{10}-\frac{25735}{472383}a^{9}-\frac{2092187}{157461}a^{8}-\frac{1210252}{157461}a^{7}-\frac{551057}{157461}a^{6}-\frac{9699376}{157461}a^{5}-\frac{5590064}{157461}a^{4}-\frac{2273947}{157461}a^{3}-\frac{2348520}{52487}a^{2}-\frac{1262852}{52487}a+\frac{2388667}{472383}$
| 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: | \( 1877308.99113 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 1877308.99113 \cdot 1}{2\cdot\sqrt{44542611246065932779454464}}\cr\approx \mathstrut & 0.683261374612 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^9 - 18*x^6 - 12*x^3 + 4*x - 2)
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 - 4*x^9 - 18*x^6 - 12*x^3 + 4*x - 2, 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 - 4*x^9 - 18*x^6 - 12*x^3 + 4*x - 2);
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 - 4*x^9 - 18*x^6 - 12*x^3 + 4*x - 2);
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 | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $17$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.17.16.1 | $x^{17} + 2$ | $17$ | $1$ | $16$ | $C_{17}:C_{8}$ | $[\ ]_{17}^{8}$ |
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.5.0.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 3.7.0.1 | $x^{7} + 2 x^{2} + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(271\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(835997920414096373\)
| $\Q_{835997920414096373}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |