Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*x^2 + 5*x - 1)
gp: K = bnfinit(y^17 - 3*y^16 + 7*y^15 - 10*y^14 + 12*y^13 - 11*y^12 + 10*y^11 - 7*y^10 + 2*y^9 + 5*y^8 - 13*y^7 + 11*y^6 - 5*y^5 - 6*y^4 + 11*y^3 - 9*y^2 + 5*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*x^2 + 5*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*x^2 + 5*x - 1)
\( x^{17} - 3 x^{16} + 7 x^{15} - 10 x^{14} + 12 x^{13} - 11 x^{12} + 10 x^{11} - 7 x^{10} + 2 x^{9} + 5 x^{8} - 13 x^{7} + 11 x^{6} - 5 x^{5} - 6 x^{4} + 11 x^{3} - 9 x^{2} + 5 x - 1 \)
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: |
\(-2757250718309135423\)
\(\medspace = -\,75653\cdot 3119681\cdot 11682611\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.15\) | 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))
| |
| Ramified primes: |
\(75653\), \(3119681\), \(11682611\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-27572\!\cdots\!35423}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{109}a^{16}+\frac{6}{109}a^{15}-\frac{48}{109}a^{14}-\frac{6}{109}a^{13}-\frac{42}{109}a^{12}+\frac{47}{109}a^{11}-\frac{3}{109}a^{10}-\frac{34}{109}a^{9}+\frac{23}{109}a^{8}-\frac{6}{109}a^{7}+\frac{42}{109}a^{6}-\frac{47}{109}a^{5}+\frac{8}{109}a^{4}-\frac{43}{109}a^{3}-\frac{49}{109}a^{2}-\frac{14}{109}a-\frac{12}{109}$
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: |
$a$, $\frac{49}{109}a^{16}-\frac{251}{109}a^{15}+\frac{482}{109}a^{14}-\frac{730}{109}a^{13}+\frac{667}{109}a^{12}-\frac{640}{109}a^{11}+\frac{507}{109}a^{10}-\frac{467}{109}a^{9}-\frac{72}{109}a^{8}+\frac{578}{109}a^{7}-\frac{885}{109}a^{6}+\frac{858}{109}a^{5}+\frac{174}{109}a^{4}-\frac{472}{109}a^{3}+\frac{760}{109}a^{2}-\frac{468}{109}a+\frac{175}{109}$, $\frac{175}{109}a^{16}-\frac{476}{109}a^{15}+\frac{865}{109}a^{14}-\frac{941}{109}a^{13}+\frac{716}{109}a^{12}-\frac{495}{109}a^{11}+\frac{456}{109}a^{10}-\frac{282}{109}a^{9}-\frac{553}{109}a^{8}+\frac{1130}{109}a^{7}-\frac{1479}{109}a^{6}+\frac{168}{109}a^{5}+\frac{1182}{109}a^{4}-\frac{1203}{109}a^{3}+\frac{799}{109}a^{2}+\frac{166}{109}a-\frac{247}{109}$, $\frac{22}{109}a^{16}-\frac{413}{109}a^{15}+\frac{1124}{109}a^{14}-\frac{2094}{109}a^{13}+\frac{2455}{109}a^{12}-\frac{2236}{109}a^{11}+\frac{1787}{109}a^{10}-\frac{1620}{109}a^{9}+\frac{942}{109}a^{8}+\frac{849}{109}a^{7}-\frac{2455}{109}a^{6}+\frac{3435}{109}a^{5}-\frac{1241}{109}a^{4}-\frac{1491}{109}a^{3}+\frac{2519}{109}a^{2}-\frac{2270}{109}a+\frac{608}{109}$, $\frac{246}{109}a^{16}-\frac{268}{109}a^{15}+\frac{400}{109}a^{14}+\frac{268}{109}a^{13}-\frac{413}{109}a^{12}+\frac{662}{109}a^{11}-\frac{193}{109}a^{10}+\frac{683}{109}a^{9}-\frac{991}{109}a^{8}+\frac{595}{109}a^{7}-\frac{241}{109}a^{6}-\frac{2297}{109}a^{5}+\frac{1205}{109}a^{4}-\frac{550}{109}a^{3}-\frac{827}{109}a^{2}+\frac{1243}{109}a-\frac{227}{109}$, $\frac{279}{109}a^{16}-\frac{506}{109}a^{15}+\frac{887}{109}a^{14}-\frac{584}{109}a^{13}+\frac{381}{109}a^{12}-\frac{76}{109}a^{11}+\frac{253}{109}a^{10}+\frac{215}{109}a^{9}-\frac{995}{109}a^{8}+\frac{1160}{109}a^{7}-\frac{1362}{109}a^{6}-\frac{1014}{109}a^{5}+\frac{1360}{109}a^{4}-\frac{1097}{109}a^{3}+\frac{390}{109}a^{2}+\frac{672}{109}a-\frac{187}{109}$, $\frac{106}{109}a^{16}-\frac{127}{109}a^{15}+\frac{144}{109}a^{14}+\frac{127}{109}a^{13}-\frac{310}{109}a^{12}+\frac{404}{109}a^{11}-\frac{318}{109}a^{10}+\frac{429}{109}a^{9}-\frac{614}{109}a^{8}+\frac{345}{109}a^{7}-\frac{235}{109}a^{6}-\frac{949}{109}a^{5}+\frac{848}{109}a^{4}-\frac{198}{109}a^{3}-\frac{180}{109}a^{2}+\frac{696}{109}a-\frac{291}{109}$, $\frac{155}{109}a^{16}-\frac{378}{109}a^{15}+\frac{626}{109}a^{14}-\frac{603}{109}a^{13}+\frac{357}{109}a^{12}-\frac{236}{109}a^{11}+\frac{189}{109}a^{10}-\frac{38}{109}a^{9}-\frac{686}{109}a^{8}+\frac{923}{109}a^{7}-\frac{1120}{109}a^{6}-\frac{91}{109}a^{5}+\frac{1022}{109}a^{4}-\frac{670}{109}a^{3}+\frac{471}{109}a^{2}+\frac{337}{109}a-\frac{225}{109}$, $\frac{211}{109}a^{16}-\frac{151}{109}a^{15}+\frac{118}{109}a^{14}+\frac{587}{109}a^{13}-\frac{687}{109}a^{12}+\frac{761}{109}a^{11}-\frac{306}{109}a^{10}+\frac{783}{109}a^{9}-\frac{924}{109}a^{8}+\frac{151}{109}a^{7}+\frac{360}{109}a^{6}-\frac{2396}{109}a^{5}+\frac{925}{109}a^{4}+\frac{192}{109}a^{3}-\frac{1074}{109}a^{2}+\frac{1188}{109}a-\frac{134}{109}$
| 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: | \( 178.334436956 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 178.334436956 \cdot 1}{2\cdot\sqrt{2757250718309135423}}\cr\approx \mathstrut & 0.166079548501 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*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 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*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 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*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 - 3*x^16 + 7*x^15 - 10*x^14 + 12*x^13 - 11*x^12 + 10*x^11 - 7*x^10 + 2*x^9 + 5*x^8 - 13*x^7 + 11*x^6 - 5*x^5 - 6*x^4 + 11*x^3 - 9*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 $S_{17}$ 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 | $17$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.13.0.1}{13} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | $15{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | $17$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(75653\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(3119681\)
| $\Q_{3119681}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(11682611\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |