Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 4*x^15 - 18*x^14 + 11*x^13 + 14*x^12 + 21*x^11 - 62*x^10 + 14*x^9 + 50*x^8 + 54*x^7 - 121*x^6 - 7*x^5 + 36*x^4 + 64*x^3 - 23*x^2 + 43*x + 1)
gp: K = bnfinit(y^17 - y^16 + 4*y^15 - 18*y^14 + 11*y^13 + 14*y^12 + 21*y^11 - 62*y^10 + 14*y^9 + 50*y^8 + 54*y^7 - 121*y^6 - 7*y^5 + 36*y^4 + 64*y^3 - 23*y^2 + 43*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^16 + 4*x^15 - 18*x^14 + 11*x^13 + 14*x^12 + 21*x^11 - 62*x^10 + 14*x^9 + 50*x^8 + 54*x^7 - 121*x^6 - 7*x^5 + 36*x^4 + 64*x^3 - 23*x^2 + 43*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^16 + 4*x^15 - 18*x^14 + 11*x^13 + 14*x^12 + 21*x^11 - 62*x^10 + 14*x^9 + 50*x^8 + 54*x^7 - 121*x^6 - 7*x^5 + 36*x^4 + 64*x^3 - 23*x^2 + 43*x + 1)
\( x^{17} - x^{16} + 4 x^{15} - 18 x^{14} + 11 x^{13} + 14 x^{12} + 21 x^{11} - 62 x^{10} + 14 x^{9} + \cdots + 1 \)
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: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(930227631978098127294721\)
\(\medspace = 991^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.70\) | 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: | $991^{1/2}\approx 31.480152477394387$ | ||
| Ramified primes: |
\(991\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{2}{9}a^{3}+\frac{2}{9}a^{2}-\frac{2}{9}a+\frac{1}{9}$, $\frac{1}{63}a^{13}-\frac{1}{63}a^{12}+\frac{1}{7}a^{11}-\frac{2}{21}a^{10}-\frac{5}{63}a^{9}-\frac{1}{63}a^{8}+\frac{1}{7}a^{7}+\frac{17}{63}a^{5}-\frac{26}{63}a^{4}+\frac{1}{7}a^{3}+\frac{8}{21}a^{2}-\frac{4}{63}a-\frac{17}{63}$, $\frac{1}{63}a^{14}+\frac{1}{63}a^{12}-\frac{4}{63}a^{11}-\frac{4}{63}a^{10}+\frac{8}{63}a^{9}+\frac{8}{63}a^{8}-\frac{5}{63}a^{7}+\frac{31}{63}a^{6}-\frac{23}{63}a^{5}+\frac{4}{63}a^{4}+\frac{26}{63}a^{3}+\frac{3}{7}a^{2}+\frac{2}{9}a-\frac{1}{21}$, $\frac{1}{63}a^{15}-\frac{1}{21}a^{12}+\frac{8}{63}a^{11}-\frac{1}{9}a^{10}-\frac{8}{63}a^{9}-\frac{4}{63}a^{8}+\frac{1}{63}a^{7}-\frac{2}{63}a^{6}+\frac{29}{63}a^{5}+\frac{10}{63}a^{4}-\frac{8}{21}a^{3}-\frac{31}{63}a^{2}+\frac{1}{63}a-\frac{25}{63}$, $\frac{1}{929061441}a^{16}+\frac{674213}{309687147}a^{15}+\frac{106045}{929061441}a^{14}-\frac{4358033}{929061441}a^{13}+\frac{13622674}{309687147}a^{12}-\frac{13836964}{929061441}a^{11}-\frac{2470456}{132723063}a^{10}-\frac{10593104}{103229049}a^{9}+\frac{18158150}{929061441}a^{8}+\frac{23823571}{929061441}a^{7}+\frac{28153693}{929061441}a^{6}-\frac{26253538}{103229049}a^{5}-\frac{80177674}{929061441}a^{4}+\frac{523508}{3428271}a^{3}+\frac{42304928}{309687147}a^{2}-\frac{252479222}{929061441}a-\frac{25772827}{929061441}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: | $8$ | 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: |
$\frac{5058764}{309687147}a^{16}-\frac{5063015}{309687147}a^{15}+\frac{29333086}{309687147}a^{14}-\frac{94529168}{309687147}a^{13}+\frac{32909749}{103229049}a^{12}-\frac{72119429}{309687147}a^{11}+\frac{150513775}{309687147}a^{10}-\frac{304282733}{309687147}a^{9}+\frac{43037576}{44241021}a^{8}-\frac{12053269}{309687147}a^{7}+\frac{365183384}{309687147}a^{6}-\frac{351135787}{309687147}a^{5}+\frac{46394085}{34409683}a^{4}-\frac{44446}{1142757}a^{3}+\frac{339896507}{309687147}a^{2}-\frac{82353247}{309687147}a+\frac{190036100}{309687147}$, $\frac{29624603}{929061441}a^{16}-\frac{9577340}{309687147}a^{15}+\frac{133593995}{929061441}a^{14}-\frac{540316552}{929061441}a^{13}+\frac{39523760}{103229049}a^{12}+\frac{157692991}{929061441}a^{11}+\frac{645882172}{929061441}a^{10}-\frac{446925742}{309687147}a^{9}+\frac{776504149}{929061441}a^{8}+\frac{819813956}{929061441}a^{7}+\frac{151833074}{132723063}a^{6}-\frac{140345330}{44241021}a^{5}+\frac{318146140}{929061441}a^{4}+\frac{581197}{3428271}a^{3}+\frac{609613525}{309687147}a^{2}+\frac{245899793}{929061441}a+\frac{1108740004}{929061441}$, $\frac{12783010}{929061441}a^{16}-\frac{3803824}{309687147}a^{15}+\frac{6308431}{132723063}a^{14}-\frac{220330004}{929061441}a^{13}+\frac{33714518}{309687147}a^{12}+\frac{280972655}{929061441}a^{11}+\frac{269820878}{929061441}a^{10}-\frac{336249202}{309687147}a^{9}+\frac{61167515}{929061441}a^{8}+\frac{1061596873}{929061441}a^{7}+\frac{1431892195}{929061441}a^{6}-\frac{770622560}{309687147}a^{5}-\frac{1407472771}{929061441}a^{4}+\frac{223826}{489753}a^{3}+\frac{540222044}{309687147}a^{2}+\frac{604388467}{929061441}a+\frac{482010887}{929061441}$, $\frac{1203424}{132723063}a^{16}+\frac{4437260}{309687147}a^{15}-\frac{9756479}{929061441}a^{14}-\frac{38199815}{929061441}a^{13}-\frac{43041428}{103229049}a^{12}+\frac{737860052}{929061441}a^{11}+\frac{231032273}{929061441}a^{10}-\frac{185272165}{309687147}a^{9}-\frac{909063895}{929061441}a^{8}+\frac{1022166646}{929061441}a^{7}+\frac{1876360606}{929061441}a^{6}-\frac{90908099}{44241021}a^{5}-\frac{1209996418}{929061441}a^{4}+\frac{320543}{489753}a^{3}+\frac{217807222}{103229049}a^{2}-\frac{1068931244}{929061441}a+\frac{119092682}{132723063}$, $\frac{332137}{9577953}a^{16}-\frac{20051}{456093}a^{15}+\frac{194083}{1368279}a^{14}-\frac{6197504}{9577953}a^{13}+\frac{1611772}{3192651}a^{12}+\frac{5208605}{9577953}a^{11}+\frac{2960759}{9577953}a^{10}-\frac{734684}{354739}a^{9}+\frac{6553445}{9577953}a^{8}+\frac{24835168}{9577953}a^{7}+\frac{1049014}{9577953}a^{6}-\frac{1520390}{354739}a^{5}+\frac{1428857}{1368279}a^{4}+\frac{9155}{5049}a^{3}+\frac{845995}{3192651}a^{2}+\frac{4711711}{9577953}a+\frac{6047747}{9577953}$, $\frac{213931}{54650673}a^{16}+\frac{79819}{18216891}a^{15}+\frac{211303}{54650673}a^{14}-\frac{2165531}{54650673}a^{13}-\frac{2528969}{18216891}a^{12}+\frac{11169626}{54650673}a^{11}+\frac{1319156}{7807239}a^{10}+\frac{559793}{2602413}a^{9}-\frac{6120292}{7807239}a^{8}+\frac{917032}{54650673}a^{7}+\frac{34953742}{54650673}a^{6}+\frac{2923507}{2602413}a^{5}-\frac{142905745}{54650673}a^{4}-\frac{154276}{201663}a^{3}+\frac{7777292}{6072297}a^{2}+\frac{108120511}{54650673}a-\frac{55531276}{54650673}$, $\frac{3007804}{309687147}a^{16}-\frac{1275764}{44241021}a^{15}+\frac{19223453}{309687147}a^{14}-\frac{84447512}{309687147}a^{13}+\frac{16558620}{34409683}a^{12}-\frac{64222747}{309687147}a^{11}+\frac{10544839}{44241021}a^{10}-\frac{317701282}{309687147}a^{9}+\frac{28002066}{34409683}a^{8}+\frac{40314664}{309687147}a^{7}+\frac{8548031}{44241021}a^{6}-\frac{523474739}{309687147}a^{5}+\frac{266495002}{309687147}a^{4}+\frac{27879}{126973}a^{3}+\frac{67155796}{103229049}a^{2}-\frac{96733708}{103229049}a-\frac{20855612}{103229049}$, $a$
| 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: | \( 323467.453718 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 323467.453718 \cdot 1}{2\cdot\sqrt{930227631978098127294721}}\cr\approx \mathstrut & 0.814657234861 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 4*x^15 - 18*x^14 + 11*x^13 + 14*x^12 + 21*x^11 - 62*x^10 + 14*x^9 + 50*x^8 + 54*x^7 - 121*x^6 - 7*x^5 + 36*x^4 + 64*x^3 - 23*x^2 + 43*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 + 4*x^15 - 18*x^14 + 11*x^13 + 14*x^12 + 21*x^11 - 62*x^10 + 14*x^9 + 50*x^8 + 54*x^7 - 121*x^6 - 7*x^5 + 36*x^4 + 64*x^3 - 23*x^2 + 43*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 + 4*x^15 - 18*x^14 + 11*x^13 + 14*x^12 + 21*x^11 - 62*x^10 + 14*x^9 + 50*x^8 + 54*x^7 - 121*x^6 - 7*x^5 + 36*x^4 + 64*x^3 - 23*x^2 + 43*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 + 4*x^15 - 18*x^14 + 11*x^13 + 14*x^12 + 21*x^11 - 62*x^10 + 14*x^9 + 50*x^8 + 54*x^7 - 121*x^6 - 7*x^5 + 36*x^4 + 64*x^3 - 23*x^2 + 43*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 solvable group of order 34 |
| The 10 conjugacy class representatives for $D_{17}$ |
| Character table for $D_{17}$ |
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]
Sibling fields
| Galois closure: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $17$ | ${\href{/padicField/3.2.0.1}{2} }^{8}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $17$ | ${\href{/padicField/7.2.0.1}{2} }^{8}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $17$ | ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $17$ | ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $17$ | $17$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17$ | ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $17$ | $17$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(991\)
| $\Q_{991}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |