Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^15 - x^14 - 5*x^13 + 4*x^12 + x^11 + 15*x^9 - 13*x^8 - 2*x^7 + 7*x^6 - 18*x^5 + 20*x^4 - 10*x^3 + 6*x^2 - 4*x + 1)
gp: K = bnfinit(y^17 - y^15 - y^14 - 5*y^13 + 4*y^12 + y^11 + 15*y^9 - 13*y^8 - 2*y^7 + 7*y^6 - 18*y^5 + 20*y^4 - 10*y^3 + 6*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^15 - x^14 - 5*x^13 + 4*x^12 + x^11 + 15*x^9 - 13*x^8 - 2*x^7 + 7*x^6 - 18*x^5 + 20*x^4 - 10*x^3 + 6*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^15 - x^14 - 5*x^13 + 4*x^12 + x^11 + 15*x^9 - 13*x^8 - 2*x^7 + 7*x^6 - 18*x^5 + 20*x^4 - 10*x^3 + 6*x^2 - 4*x + 1)
\( x^{17} - x^{15} - x^{14} - 5 x^{13} + 4 x^{12} + x^{11} + 15 x^{9} - 13 x^{8} - 2 x^{7} + 7 x^{6} + \cdots + 1 \)
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: |
\(-1586349738982991023\)
\(\medspace = -\,2131\cdot 546781\cdot 1361451193\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.77\) | 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: | $2131^{1/2}546781^{1/2}1361451193^{1/2}\approx 1259503766.9586349$ | ||
| Ramified primes: |
\(2131\), \(546781\), \(1361451193\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-15863\!\cdots\!91023}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{55463}a^{16}-\frac{19341}{55463}a^{15}-\frac{23655}{55463}a^{14}-\frac{2933}{55463}a^{13}-\frac{11501}{55463}a^{12}-\frac{21248}{55463}a^{11}-\frac{23261}{55463}a^{10}-\frac{24855}{55463}a^{9}+\frac{22749}{55463}a^{8}-\frac{443}{55463}a^{7}+\frac{26759}{55463}a^{6}-\frac{20559}{55463}a^{5}+\frac{17354}{55463}a^{4}+\frac{18382}{55463}a^{3}-\frac{8442}{55463}a^{2}-\frac{6344}{55463}a+\frac{15144}{55463}$
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: |
$a$, $\frac{3313505}{55463}a^{16}+\frac{1883703}{55463}a^{15}-\frac{2257602}{55463}a^{14}-\frac{4609419}{55463}a^{13}-\frac{19183903}{55463}a^{12}+\frac{2372962}{55463}a^{11}+\frac{4763612}{55463}a^{10}+\frac{2726001}{55463}a^{9}+\frac{51241702}{55463}a^{8}-\frac{13975633}{55463}a^{7}-\frac{14819665}{55463}a^{6}+\frac{14778150}{55463}a^{5}-\frac{51222941}{55463}a^{4}+\frac{37097150}{55463}a^{3}-\frac{11769705}{55463}a^{2}+\frac{13023326}{55463}a-\frac{5809441}{55463}$, $\frac{531044}{55463}a^{16}+\frac{381892}{55463}a^{15}-\frac{252802}{55463}a^{14}-\frac{705642}{55463}a^{13}-\frac{3168338}{55463}a^{12}-\frac{174729}{55463}a^{11}+\frac{382191}{55463}a^{10}+\frac{263435}{55463}a^{9}+\frac{8199672}{55463}a^{8}-\frac{976780}{55463}a^{7}-\frac{1743650}{55463}a^{6}+\frac{2542863}{55463}a^{5}-\frac{7814787}{55463}a^{4}+\frac{4932626}{55463}a^{3}-\frac{1718511}{55463}a^{2}+\frac{1876152}{55463}a-\frac{725683}{55463}$, $\frac{1690022}{55463}a^{16}+\frac{1013641}{55463}a^{15}-\frac{1071122}{55463}a^{14}-\frac{2324736}{55463}a^{13}-\frac{9862549}{55463}a^{12}+\frac{819302}{55463}a^{11}+\frac{2125622}{55463}a^{10}+\frac{1291982}{55463}a^{9}+\frac{26202970}{55463}a^{8}-\frac{6252028}{55463}a^{7}-\frac{6999654}{55463}a^{6}+\frac{7620724}{55463}a^{5}-\frac{26038370}{55463}a^{4}+\frac{18240708}{55463}a^{3}-\frac{6019997}{55463}a^{2}+\frac{6542133}{55463}a-\frac{2758647}{55463}$, $\frac{3709248}{55463}a^{16}+\frac{2156044}{55463}a^{15}-\frac{2457664}{55463}a^{14}-\frac{5148604}{55463}a^{13}-\frac{21548886}{55463}a^{12}+\frac{2315665}{55463}a^{11}+\frac{5078966}{55463}a^{10}+\frac{3007712}{55463}a^{9}+\frac{57401442}{55463}a^{8}-\frac{14858647}{55463}a^{7}-\frac{16069893}{55463}a^{6}+\frac{16506488}{55463}a^{5}-\frac{57194898}{55463}a^{4}+\frac{40944843}{55463}a^{3}-\frac{13315807}{55463}a^{2}+\frac{14570856}{55463}a-\frac{6397470}{55463}$, $\frac{1181405}{55463}a^{16}+\frac{647265}{55463}a^{15}-\frac{825410}{55463}a^{14}-\frac{1618367}{55463}a^{13}-\frac{6779651}{55463}a^{12}+\frac{1003231}{55463}a^{11}+\frac{1698499}{55463}a^{10}+\frac{842223}{55463}a^{9}+\frac{18157373}{55463}a^{8}-\frac{5393458}{55463}a^{7}-\frac{5297034}{55463}a^{6}+\frac{5564254}{55463}a^{5}-\frac{18153833}{55463}a^{4}+\frac{13637495}{55463}a^{3}-\frac{4279538}{55463}a^{2}+\frac{4596225}{55463}a-\frac{2164814}{55463}$, $\frac{46194}{1499}a^{16}+\frac{25806}{1499}a^{15}-\frac{32014}{1499}a^{14}-\frac{64344}{1499}a^{13}-\frac{266937}{1499}a^{12}+\frac{36074}{1499}a^{11}+\frac{67997}{1499}a^{10}+\frac{38659}{1499}a^{9}+\frac{714375}{1499}a^{8}-\frac{201959}{1499}a^{7}-\frac{208495}{1499}a^{6}+\frac{206359}{1499}a^{5}-\frac{716056}{1499}a^{4}+\frac{522729}{1499}a^{3}-\frac{166790}{1499}a^{2}+\frac{182642}{1499}a-\frac{82823}{1499}$, $\frac{1969398}{55463}a^{16}+\frac{1140663}{55463}a^{15}-\frac{1293952}{55463}a^{14}-\frac{2712423}{55463}a^{13}-\frac{11436373}{55463}a^{12}+\frac{1231185}{55463}a^{11}+\frac{2603900}{55463}a^{10}+\frac{1535491}{55463}a^{9}+\frac{30493075}{55463}a^{8}-\frac{7941533}{55463}a^{7}-\frac{8310689}{55463}a^{6}+\frac{8898006}{55463}a^{5}-\frac{30482451}{55463}a^{4}+\frac{21849876}{55463}a^{3}-\frac{7279226}{55463}a^{2}+\frac{7721140}{55463}a-\frac{3327162}{55463}$, $\frac{2378614}{55463}a^{16}+\frac{1341422}{55463}a^{15}-\frac{1618357}{55463}a^{14}-\frac{3278261}{55463}a^{13}-\frac{13735244}{55463}a^{12}+\frac{1754220}{55463}a^{11}+\frac{3333455}{55463}a^{10}+\frac{1831381}{55463}a^{9}+\frac{36717017}{55463}a^{8}-\frac{10189657}{55463}a^{7}-\frac{10489281}{55463}a^{6}+\frac{10839011}{55463}a^{5}-\frac{36697826}{55463}a^{4}+\frac{26825220}{55463}a^{3}-\frac{8587929}{55463}a^{2}+\frac{9375367}{55463}a-\frac{4205771}{55463}$
| 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: | \( 130.579387289 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{7}\cdot 130.579387289 \cdot 1}{2\cdot\sqrt{1586349738982991023}}\cr\approx \mathstrut & 0.160322407384 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - x^15 - x^14 - 5*x^13 + 4*x^12 + x^11 + 15*x^9 - 13*x^8 - 2*x^7 + 7*x^6 - 18*x^5 + 20*x^4 - 10*x^3 + 6*x^2 - 4*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^15 - x^14 - 5*x^13 + 4*x^12 + x^11 + 15*x^9 - 13*x^8 - 2*x^7 + 7*x^6 - 18*x^5 + 20*x^4 - 10*x^3 + 6*x^2 - 4*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^15 - x^14 - 5*x^13 + 4*x^12 + x^11 + 15*x^9 - 13*x^8 - 2*x^7 + 7*x^6 - 18*x^5 + 20*x^4 - 10*x^3 + 6*x^2 - 4*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^15 - x^14 - 5*x^13 + 4*x^12 + x^11 + 15*x^9 - 13*x^8 - 2*x^7 + 7*x^6 - 18*x^5 + 20*x^4 - 10*x^3 + 6*x^2 - 4*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 | $17$ | ${\href{/padicField/3.13.0.1}{13} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | $17$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | $17$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | $15{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $17$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $17$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2131\)
| $\Q_{2131}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2131}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2131}$ | $x$ | $1$ | $1$ | $0$ | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(546781\)
| $\Q_{546781}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
|
\(1361451193\)
| $\Q_{1361451193}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |