Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 2*x + 1)
gp: K = bnfinit(y^17 - 4*y^16 + 12*y^15 - 26*y^14 + 48*y^13 - 75*y^12 + 105*y^11 - 126*y^10 + 136*y^9 - 129*y^8 + 107*y^7 - 80*y^6 + 47*y^5 - 23*y^4 + 11*y^3 - 2*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 2*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^17 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 2*x + 1)
\( x^{17} - 4 x^{16} + 12 x^{15} - 26 x^{14} + 48 x^{13} - 75 x^{12} + 105 x^{11} - 126 x^{10} + 136 x^{9} + \cdots + 1 \)
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: |
\(-3779800000525860359\)
\(\medspace = -\,137\cdot 1926157\cdot 14323744651\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.38\) | 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: | $137^{1/2}1926157^{1/2}14323744651^{1/2}\approx 1944170774.5272431$ | ||
| Ramified primes: |
\(137\), \(1926157\), \(14323744651\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-37798\!\cdots\!60359}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}$
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: |
$\frac{8}{3}a^{16}-\frac{25}{3}a^{15}+\frac{74}{3}a^{14}-\frac{142}{3}a^{13}+\frac{254}{3}a^{12}-\frac{362}{3}a^{11}+\frac{491}{3}a^{10}-\frac{527}{3}a^{9}+\frac{557}{3}a^{8}-\frac{463}{3}a^{7}+122a^{6}-82a^{5}+\frac{104}{3}a^{4}-\frac{55}{3}a^{3}+\frac{14}{3}a^{2}+\frac{11}{3}a-3$, $\frac{4}{3}a^{16}-4a^{15}+\frac{35}{3}a^{14}-\frac{67}{3}a^{13}+\frac{121}{3}a^{12}-59a^{11}+\frac{245}{3}a^{10}-90a^{9}+97a^{8}-\frac{254}{3}a^{7}+\frac{203}{3}a^{6}-\frac{145}{3}a^{5}+\frac{61}{3}a^{4}-\frac{34}{3}a^{3}+6a^{2}+3a-2$, $2a^{16}-6a^{15}+\frac{52}{3}a^{14}-\frac{97}{3}a^{13}+\frac{173}{3}a^{12}-82a^{11}+\frac{338}{3}a^{10}-\frac{364}{3}a^{9}+\frac{392}{3}a^{8}-\frac{332}{3}a^{7}+90a^{6}-\frac{185}{3}a^{5}+\frac{79}{3}a^{4}-15a^{3}+\frac{14}{3}a^{2}+\frac{7}{3}a-\frac{8}{3}$, $a$, $\frac{2}{3}a^{16}-\frac{7}{3}a^{15}+\frac{20}{3}a^{14}-\frac{40}{3}a^{13}+\frac{71}{3}a^{12}-\frac{104}{3}a^{11}+\frac{140}{3}a^{10}-\frac{155}{3}a^{9}+\frac{161}{3}a^{8}-\frac{142}{3}a^{7}+38a^{6}-27a^{5}+\frac{47}{3}a^{4}-\frac{25}{3}a^{3}+\frac{17}{3}a^{2}-\frac{1}{3}a-1$, $\frac{4}{3}a^{16}-\frac{13}{3}a^{15}+\frac{38}{3}a^{14}-24a^{13}+\frac{125}{3}a^{12}-\frac{172}{3}a^{11}+\frac{224}{3}a^{10}-\frac{227}{3}a^{9}+75a^{8}-\frac{167}{3}a^{7}+\frac{118}{3}a^{6}-22a^{5}+\frac{11}{3}a^{4}-\frac{2}{3}a^{3}-\frac{5}{3}a^{2}+3a-2$, $a^{16}-3a^{15}+\frac{25}{3}a^{14}-\frac{46}{3}a^{13}+\frac{77}{3}a^{12}-35a^{11}+\frac{134}{3}a^{10}-\frac{136}{3}a^{9}+\frac{134}{3}a^{8}-\frac{110}{3}a^{7}+25a^{6}-\frac{62}{3}a^{5}+\frac{16}{3}a^{4}-6a^{3}+\frac{8}{3}a^{2}+\frac{4}{3}a+\frac{1}{3}$, $\frac{2}{3}a^{16}-3a^{15}+\frac{26}{3}a^{14}-19a^{13}+\frac{100}{3}a^{12}-51a^{11}+68a^{10}-\frac{241}{3}a^{9}+\frac{245}{3}a^{8}-77a^{7}+\frac{178}{3}a^{6}-\frac{136}{3}a^{5}+26a^{4}-\frac{32}{3}a^{3}+\frac{20}{3}a^{2}-\frac{5}{3}a-\frac{2}{3}$, $\frac{7}{3}a^{16}-8a^{15}+\frac{71}{3}a^{14}-\frac{145}{3}a^{13}+\frac{265}{3}a^{12}-134a^{11}+\frac{560}{3}a^{10}-216a^{9}+233a^{8}-\frac{641}{3}a^{7}+\frac{524}{3}a^{6}-\frac{385}{3}a^{5}+\frac{202}{3}a^{4}-\frac{103}{3}a^{3}+17a^{2}+a-4$
| 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: | \( 206.585713852 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 206.585713852 \cdot 1}{2\cdot\sqrt{3779800000525860359}}\cr\approx \mathstrut & 0.164317925990 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 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 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 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 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 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 - 4*x^16 + 12*x^15 - 26*x^14 + 48*x^13 - 75*x^12 + 105*x^11 - 126*x^10 + 136*x^9 - 129*x^8 + 107*x^7 - 80*x^6 + 47*x^5 - 23*x^4 + 11*x^3 - 2*x^2 - 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 $S_{17}$ |
| 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 | $17$ | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | $17$ | $17$ | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | $17$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.7.0.1}{7} }$ | $17$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(137\)
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 137.3.0.1 | $x^{3} + 6 x + 134$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 137.5.0.1 | $x^{5} + 7 x + 134$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
|
\(1926157\)
| $\Q_{1926157}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1926157}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
|
\(14323744651\)
| $\Q_{14323744651}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{14323744651}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |