Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 16*x^15 + 13*x^14 + 102*x^13 - 65*x^12 - 330*x^11 + 160*x^10 + 576*x^9 - 209*x^8 - 541*x^7 + 150*x^6 + 266*x^5 - 59*x^4 - 61*x^3 + 12*x^2 + 5*x - 1)
gp: K = bnfinit(y^17 - y^16 - 16*y^15 + 13*y^14 + 102*y^13 - 65*y^12 - 330*y^11 + 160*y^10 + 576*y^9 - 209*y^8 - 541*y^7 + 150*y^6 + 266*y^5 - 59*y^4 - 61*y^3 + 12*y^2 + 5*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^16 - 16*x^15 + 13*x^14 + 102*x^13 - 65*x^12 - 330*x^11 + 160*x^10 + 576*x^9 - 209*x^8 - 541*x^7 + 150*x^6 + 266*x^5 - 59*x^4 - 61*x^3 + 12*x^2 + 5*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^16 - 16*x^15 + 13*x^14 + 102*x^13 - 65*x^12 - 330*x^11 + 160*x^10 + 576*x^9 - 209*x^8 - 541*x^7 + 150*x^6 + 266*x^5 - 59*x^4 - 61*x^3 + 12*x^2 + 5*x - 1)
\( x^{17} - x^{16} - 16 x^{15} + 13 x^{14} + 102 x^{13} - 65 x^{12} - 330 x^{11} + 160 x^{10} + 576 x^{9} + \cdots - 1 \)
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: | $[11, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-4636997228978364408280160408\)
\(\medspace = -\,2^{3}\cdot 102293\cdot 5666317867520705728007\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.41\) | 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: | $2^{3/2}102293^{1/2}5666317867520705728007^{1/2}\approx 68095500798352.05$ | ||
| Ramified primes: |
\(2\), \(102293\), \(5666317867520705728007\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-11592\!\cdots\!40102}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 $1$ (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: | $13$ | 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: |
$a$, $a^{2}+a-1$, $a^{16}-a^{15}-16a^{14}+13a^{13}+102a^{12}-65a^{11}-330a^{10}+160a^{9}+576a^{8}-209a^{7}-541a^{6}+150a^{5}+266a^{4}-58a^{3}-61a^{2}+9a+5$, $4a^{16}-4a^{15}-63a^{14}+50a^{13}+394a^{12}-233a^{11}-1245a^{10}+500a^{9}+2114a^{8}-486a^{7}-1938a^{6}+166a^{5}+957a^{4}+13a^{3}-228a^{2}-10a+20$, $3a^{16}-3a^{15}-47a^{14}+38a^{13}+290a^{12}-182a^{11}-889a^{10}+416a^{9}+1410a^{8}-477a^{7}-1100a^{6}+276a^{5}+364a^{4}-79a^{3}-19a^{2}+11a-3$, $a^{16}-16a^{14}-3a^{13}+99a^{12}+34a^{11}-296a^{10}-136a^{9}+440a^{8}+231a^{7}-310a^{6}-160a^{5}+106a^{4}+47a^{3}-14a^{2}-2a+2$, $a^{16}-a^{15}-15a^{14}+12a^{13}+87a^{12}-53a^{11}-243a^{10}+107a^{9}+333a^{8}-102a^{7}-208a^{6}+48a^{5}+58a^{4}-11a^{3}-2a^{2}+a-1$, $11a^{16}-11a^{15}-174a^{14}+141a^{13}+1090a^{12}-689a^{11}-3427a^{10}+1631a^{9}+5689a^{8}-1989a^{7}-4862a^{6}+1267a^{5}+1984a^{4}-403a^{3}-285a^{2}+53a+1$, $a^{16}-16a^{14}-4a^{13}+100a^{12}+48a^{11}-306a^{10}-211a^{9}+471a^{8}+420a^{7}-332a^{6}-380a^{5}+74a^{4}+144a^{3}+13a^{2}-13a-2$, $4a^{16}-2a^{15}-65a^{14}+20a^{13}+417a^{12}-58a^{11}-1336a^{10}+3a^{9}+2242a^{8}+221a^{7}-1911a^{6}-308a^{5}+769a^{4}+152a^{3}-122a^{2}-18a+7$, $a^{15}-a^{14}-16a^{13}+13a^{12}+101a^{11}-64a^{10}-318a^{9}+152a^{8}+522a^{7}-189a^{6}-431a^{5}+131a^{4}+165a^{3}-48a^{2}-21a+6$, $6a^{16}-3a^{15}-97a^{14}+29a^{13}+619a^{12}-75a^{11}-1973a^{10}-47a^{9}+3300a^{8}+423a^{7}-2825a^{6}-512a^{5}+1165a^{4}+212a^{3}-185a^{2}-18a+8$, $4a^{16}+2a^{15}-68a^{14}-44a^{13}+453a^{12}+346a^{11}-1496a^{10}-1269a^{9}+2574a^{8}+2313a^{7}-2254a^{6}-2042a^{5}+957a^{4}+808a^{3}-173a^{2}-97a+17$
| 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: | \( 132494211.856 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{3}\cdot 132494211.856 \cdot 1}{2\cdot\sqrt{4636997228978364408280160408}}\cr\approx \mathstrut & 0.494217420489 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 16*x^15 + 13*x^14 + 102*x^13 - 65*x^12 - 330*x^11 + 160*x^10 + 576*x^9 - 209*x^8 - 541*x^7 + 150*x^6 + 266*x^5 - 59*x^4 - 61*x^3 + 12*x^2 + 5*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 - 16*x^15 + 13*x^14 + 102*x^13 - 65*x^12 - 330*x^11 + 160*x^10 + 576*x^9 - 209*x^8 - 541*x^7 + 150*x^6 + 266*x^5 - 59*x^4 - 61*x^3 + 12*x^2 + 5*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 - 16*x^15 + 13*x^14 + 102*x^13 - 65*x^12 - 330*x^11 + 160*x^10 + 576*x^9 - 209*x^8 - 541*x^7 + 150*x^6 + 266*x^5 - 59*x^4 - 61*x^3 + 12*x^2 + 5*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 - 16*x^15 + 13*x^14 + 102*x^13 - 65*x^12 - 330*x^11 + 160*x^10 + 576*x^9 - 209*x^8 - 541*x^7 + 150*x^6 + 266*x^5 - 59*x^4 - 61*x^3 + 12*x^2 + 5*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 | R | $17$ | ${\href{/padicField/5.13.0.1}{13} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | $17$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(102293\)
| $\Q_{102293}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(566\!\cdots\!007\)
| $\Q_{56\!\cdots\!07}$ | $x$ | $1$ | $1$ | $0$ | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |