Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^15 - 4*x^14 + 4*x^13 + 7*x^12 - 3*x^11 - 3*x^10 - 3*x^9 + 5*x^8 - 5*x^7 - 4*x^6 + 14*x^5 - 2*x^4 - 7*x^3 + 2*x + 1)
gp: K = bnfinit(y^17 - 2*y^15 - 4*y^14 + 4*y^13 + 7*y^12 - 3*y^11 - 3*y^10 - 3*y^9 + 5*y^8 - 5*y^7 - 4*y^6 + 14*y^5 - 2*y^4 - 7*y^3 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^15 - 4*x^14 + 4*x^13 + 7*x^12 - 3*x^11 - 3*x^10 - 3*x^9 + 5*x^8 - 5*x^7 - 4*x^6 + 14*x^5 - 2*x^4 - 7*x^3 + 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^15 - 4*x^14 + 4*x^13 + 7*x^12 - 3*x^11 - 3*x^10 - 3*x^9 + 5*x^8 - 5*x^7 - 4*x^6 + 14*x^5 - 2*x^4 - 7*x^3 + 2*x + 1)
\( x^{17} - 2 x^{15} - 4 x^{14} + 4 x^{13} + 7 x^{12} - 3 x^{11} - 3 x^{10} - 3 x^{9} + 5 x^{8} - 5 x^{7} + \cdots + 1 \)
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: |
\(-2151149236330118383\)
\(\medspace = -\,37^{5}\cdot 433\cdot 5531\cdot 12953\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.98\) | 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: | $37^{5/6}433^{1/2}5531^{1/2}12953^{1/2}\approx 3569979.5655382182$ | ||
| Ramified primes: |
\(37\), \(433\), \(5531\), \(12953\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-1147793191903}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{228469}a^{16}+\frac{13230}{228469}a^{15}+\frac{25644}{228469}a^{14}-\frac{6349}{228469}a^{13}+\frac{79326}{228469}a^{12}-\frac{103599}{228469}a^{11}-\frac{29242}{228469}a^{10}-\frac{73646}{228469}a^{9}+\frac{83702}{228469}a^{8}-\frac{11778}{228469}a^{7}-\frac{7087}{228469}a^{6}-\frac{88724}{228469}a^{5}+\frac{55216}{228469}a^{4}+\frac{92285}{228469}a^{3}-\frac{7793}{228469}a^{2}-\frac{61871}{228469}a+\frac{51099}{228469}$
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: |
$a$, $\frac{17939}{228469}a^{16}-\frac{46321}{228469}a^{15}-\frac{108850}{228469}a^{14}-\frac{117149}{228469}a^{13}+\frac{352651}{228469}a^{12}+\frac{361323}{228469}a^{11}-\frac{7414}{228469}a^{10}-\frac{584774}{228469}a^{9}-\frac{196559}{228469}a^{8}+\frac{48283}{228469}a^{7}-\frac{104929}{228469}a^{6}+\frac{123687}{228469}a^{5}+\frac{335178}{228469}a^{4}+\frac{471179}{228469}a^{3}-\frac{661006}{228469}a^{2}-\frac{1467}{228469}a+\frac{47333}{228469}$, $\frac{5264}{228469}a^{16}+\frac{188144}{228469}a^{15}-\frac{35163}{228469}a^{14}-\frac{293131}{228469}a^{13}-\frac{754675}{228469}a^{12}+\frac{695774}{228469}a^{11}+\frac{743625}{228469}a^{10}-\frac{189120}{228469}a^{9}-\frac{109373}{228469}a^{8}-\frac{541231}{228469}a^{7}+\frac{848355}{228469}a^{6}-\frac{1423314}{228469}a^{5}+\frac{44456}{228469}a^{4}+\frac{1662429}{228469}a^{3}-\frac{583339}{228469}a^{2}-\frac{349088}{228469}a-\frac{151346}{228469}$, $\frac{112961}{228469}a^{16}+\frac{58301}{228469}a^{15}-\frac{215036}{228469}a^{14}-\frac{482136}{228469}a^{13}+\frac{190106}{228469}a^{12}+\frac{906355}{228469}a^{11}-\frac{229229}{228469}a^{10}-\frac{112578}{228469}a^{9}-\frac{356412}{228469}a^{8}+\frac{377267}{228469}a^{7}-\frac{456169}{228469}a^{6}-\frac{787548}{228469}a^{5}+\frac{1650159}{228469}a^{4}-\frac{206116}{228469}a^{3}-\frac{470954}{228469}a^{2}+\frac{85148}{228469}a+\frac{153323}{228469}$, $\frac{42326}{228469}a^{16}-\frac{4539}{228469}a^{15}-\frac{48275}{228469}a^{14}-\frac{48230}{228469}a^{13}+\frac{200321}{228469}a^{12}+\frac{74243}{228469}a^{11}-\frac{537257}{228469}a^{10}+\frac{90440}{228469}a^{9}+\frac{130538}{228469}a^{8}+\frac{460668}{228469}a^{7}-\frac{213034}{228469}a^{6}-\frac{215540}{228469}a^{5}+\frac{748422}{228469}a^{4}-\frac{993459}{228469}a^{3}-\frac{165751}{228469}a^{2}+\frac{416670}{228469}a+\frac{128720}{228469}$, $\frac{53826}{228469}a^{16}-\frac{19893}{228469}a^{15}-\frac{95754}{228469}a^{14}-\frac{180119}{228469}a^{13}+\frac{172604}{228469}a^{12}+\frac{151578}{228469}a^{11}-\frac{285420}{228469}a^{10}+\frac{324492}{228469}a^{9}-\frac{64828}{228469}a^{8}+\frac{267316}{228469}a^{7}-\frac{607039}{228469}a^{6}-\frac{198986}{228469}a^{5}+\frac{817071}{228469}a^{4}-\frac{725995}{228469}a^{3}+\frac{231535}{228469}a^{2}+\frac{124167}{228469}a-\frac{83517}{228469}$, $\frac{40169}{228469}a^{16}+\frac{16976}{228469}a^{15}-\frac{72885}{228469}a^{14}-\frac{290046}{228469}a^{13}-\frac{11049}{228469}a^{12}+\frac{323073}{228469}a^{11}+\frac{394169}{228469}a^{10}-\frac{69562}{228469}a^{9}-\frac{381104}{228469}a^{8}+\frac{48817}{228469}a^{7}-\frac{233798}{228469}a^{6}-\frac{66425}{228469}a^{5}-\frac{5548}{228469}a^{4}+\frac{543578}{228469}a^{3}-\frac{34487}{228469}a^{2}-\frac{467355}{228469}a+\frac{30235}{228469}$, $\frac{57631}{228469}a^{16}+\frac{57077}{228469}a^{15}-\frac{76597}{228469}a^{14}-\frac{348819}{228469}a^{13}-\frac{27984}{228469}a^{12}+\frac{523346}{228469}a^{11}+\frac{398580}{228469}a^{10}-\frac{252482}{228469}a^{9}-\frac{521442}{228469}a^{8}+\frac{3481}{228469}a^{7}-\frac{156794}{228469}a^{6}-\frac{116624}{228469}a^{5}+\frac{494002}{228469}a^{4}+\frac{860860}{228469}a^{3}-\frac{176798}{228469}a^{2}-\frac{657325}{228469}a-\frac{78941}{228469}$, $\frac{181258}{228469}a^{16}+\frac{32716}{228469}a^{15}-\frac{250122}{228469}a^{14}-\frac{694096}{228469}a^{13}+\frac{461000}{228469}a^{12}+\frac{861913}{228469}a^{11}-\frac{322574}{228469}a^{10}+\frac{60064}{228469}a^{9}-\frac{512236}{228469}a^{8}+\frac{643019}{228469}a^{7}-\frac{1036604}{228469}a^{6}-\frac{458820}{228469}a^{5}+\frac{2084935}{228469}a^{4}-\frac{648712}{228469}a^{3}-\frac{376705}{228469}a^{2}+\frac{244085}{228469}a-\frac{30718}{228469}$
| 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: | \( 154.682742806 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 154.682742806 \cdot 1}{2\cdot\sqrt{2151149236330118383}}\cr\approx \mathstrut & 0.163089373269 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^15 - 4*x^14 + 4*x^13 + 7*x^12 - 3*x^11 - 3*x^10 - 3*x^9 + 5*x^8 - 5*x^7 - 4*x^6 + 14*x^5 - 2*x^4 - 7*x^3 + 2*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 - 2*x^15 - 4*x^14 + 4*x^13 + 7*x^12 - 3*x^11 - 3*x^10 - 3*x^9 + 5*x^8 - 5*x^7 - 4*x^6 + 14*x^5 - 2*x^4 - 7*x^3 + 2*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 - 2*x^15 - 4*x^14 + 4*x^13 + 7*x^12 - 3*x^11 - 3*x^10 - 3*x^9 + 5*x^8 - 5*x^7 - 4*x^6 + 14*x^5 - 2*x^4 - 7*x^3 + 2*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 - 2*x^15 - 4*x^14 + 4*x^13 + 7*x^12 - 3*x^11 - 3*x^10 - 3*x^9 + 5*x^8 - 5*x^7 - 4*x^6 + 14*x^5 - 2*x^4 - 7*x^3 + 2*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 | $17$ | ${\href{/padicField/3.13.0.1}{13} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | $17$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | $17$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | $17$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | $17$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 37.6.5.1 | $x^{6} + 37$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 37.10.0.1 | $x^{10} + 8 x^{5} + 29 x^{4} + 18 x^{3} + 11 x^{2} + 4 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(433\)
| $\Q_{433}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(5531\)
| $\Q_{5531}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5531}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{5531}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(12953\)
| $\Q_{12953}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |