Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64)
gp: K = bnfinit(y^17 - 2*y^16 + 4*y^15 - 18*y^14 + 10*y^13 - 16*y^12 + 44*y^11 + 10*y^10 + 72*y^9 + 18*y^8 + 134*y^7 + 88*y^6 + 275*y^5 + 200*y^4 + 220*y^3 + 240*y^2 + 80*y + 64, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64)
\( x^{17} - 2 x^{16} + 4 x^{15} - 18 x^{14} + 10 x^{13} - 16 x^{12} + 44 x^{11} + 10 x^{10} + 72 x^{9} + \cdots + 64 \)
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: |
\(2007233999721653909999521\)
\(\medspace = 1091^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.89\) | 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: | $1091^{1/2}\approx 33.03028912982749$ | ||
| Ramified primes: |
\(1091\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{16}a^{11}+\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{3}{16}a^{5}+\frac{1}{16}a^{4}+\frac{7}{16}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{12}-\frac{1}{8}a^{10}+\frac{1}{16}a^{9}-\frac{1}{8}a^{8}-\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}-\frac{1}{16}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{10}-\frac{1}{16}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{16}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{12}-\frac{1}{32}a^{10}+\frac{1}{16}a^{9}+\frac{1}{32}a^{8}+\frac{1}{16}a^{7}+\frac{3}{32}a^{6}+\frac{1}{16}a^{5}-\frac{5}{32}a^{4}+\frac{1}{4}a$, $\frac{1}{32}a^{15}-\frac{1}{32}a^{13}-\frac{1}{32}a^{11}+\frac{1}{16}a^{10}+\frac{1}{32}a^{9}+\frac{1}{16}a^{8}+\frac{3}{32}a^{7}+\frac{1}{16}a^{6}-\frac{5}{32}a^{5}+\frac{1}{4}a^{2}$, $\frac{1}{8286671488}a^{16}-\frac{68849395}{8286671488}a^{15}+\frac{120169979}{8286671488}a^{14}-\frac{31185489}{8286671488}a^{13}-\frac{231372529}{8286671488}a^{12}-\frac{155375939}{8286671488}a^{11}+\frac{626061315}{8286671488}a^{10}+\frac{5191329}{123681664}a^{9}-\frac{370481855}{8286671488}a^{8}+\frac{21699013}{8286671488}a^{7}+\frac{508174493}{8286671488}a^{6}+\frac{100883231}{487451264}a^{5}-\frac{61534825}{258958484}a^{4}+\frac{612237787}{2071667872}a^{3}-\frac{15242284}{64739621}a^{2}+\frac{225047811}{517916968}a-\frac{42857061}{129479242}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: |
$\frac{27638595}{2071667872}a^{16}-\frac{59758569}{2071667872}a^{15}+\frac{114372541}{2071667872}a^{14}-\frac{503032675}{2071667872}a^{13}+\frac{334125613}{2071667872}a^{12}-\frac{406806641}{2071667872}a^{11}+\frac{1238202533}{2071667872}a^{10}+\frac{1409071}{30920416}a^{9}+\frac{1958013207}{2071667872}a^{8}-\frac{7560353}{2071667872}a^{7}+\frac{3443761159}{2071667872}a^{6}+\frac{87982853}{121862816}a^{5}+\frac{1789572533}{517916968}a^{4}+\frac{120955080}{64739621}a^{3}+\frac{122601464}{64739621}a^{2}+\frac{114222595}{64739621}a+\frac{18608725}{64739621}$, $\frac{74996645}{8286671488}a^{16}-\frac{205870467}{8286671488}a^{15}+\frac{363052387}{8286671488}a^{14}-\frac{1487967265}{8286671488}a^{13}+\frac{1598759023}{8286671488}a^{12}-\frac{974199819}{8286671488}a^{11}+\frac{3897530875}{8286671488}a^{10}-\frac{22599039}{123681664}a^{9}+\frac{2849665209}{8286671488}a^{8}-\frac{3311805323}{8286671488}a^{7}+\frac{8005073533}{8286671488}a^{6}+\frac{98372871}{487451264}a^{5}+\frac{2892843841}{2071667872}a^{4}-\frac{706562537}{2071667872}a^{3}-\frac{35088549}{129479242}a^{2}+\frac{155291909}{517916968}a-\frac{57018825}{129479242}$, $\frac{2943}{16474496}a^{16}+\frac{175443}{16474496}a^{15}-\frac{331575}{16474496}a^{14}+\frac{523089}{16474496}a^{13}-\frac{2910035}{16474496}a^{12}+\frac{900435}{16474496}a^{11}-\frac{430519}{16474496}a^{10}+\frac{91623}{245888}a^{9}+\frac{5013987}{16474496}a^{8}+\frac{8137731}{16474496}a^{7}+\frac{3300847}{16474496}a^{6}+\frac{1130201}{969088}a^{5}+\frac{5287503}{4118624}a^{4}+\frac{7902041}{4118624}a^{3}+\frac{1221713}{514828}a^{2}+\frac{1069403}{1029656}a+\frac{410853}{257414}$, $\frac{51006195}{8286671488}a^{16}-\frac{84014117}{8286671488}a^{15}+\frac{206234617}{8286671488}a^{14}-\frac{904092839}{8286671488}a^{13}+\frac{310716933}{8286671488}a^{12}-\frac{1332582669}{8286671488}a^{11}+\frac{2042321881}{8286671488}a^{10}+\frac{7832479}{123681664}a^{9}+\frac{6201114547}{8286671488}a^{8}+\frac{4591316475}{8286671488}a^{7}+\frac{13197412471}{8286671488}a^{6}+\frac{511090041}{487451264}a^{5}+\frac{2063680363}{1035833936}a^{4}+\frac{610088117}{2071667872}a^{3}+\frac{244385619}{258958484}a^{2}+\frac{10682897}{517916968}a+\frac{32284239}{129479242}$, $\frac{63150545}{4143335744}a^{16}+\frac{4481207}{4143335744}a^{15}-\frac{71206375}{4143335744}a^{14}-\frac{708225295}{4143335744}a^{13}-\frac{1434825827}{4143335744}a^{12}+\frac{662249911}{4143335744}a^{11}+\frac{3615040137}{4143335744}a^{10}+\frac{66601915}{61840832}a^{9}+\frac{2621739563}{4143335744}a^{8}+\frac{2761000251}{4143335744}a^{7}+\frac{10252795383}{4143335744}a^{6}+\frac{955589709}{243725632}a^{5}+\frac{8820575161}{2071667872}a^{4}+\frac{5781445111}{1035833936}a^{3}+\frac{356048950}{64739621}a^{2}+\frac{306534399}{129479242}a+\frac{88535823}{64739621}$, $\frac{3099}{15232852}a^{16}-\frac{778193}{121862816}a^{15}+\frac{1500569}{121862816}a^{14}-\frac{3431815}{121862816}a^{13}+\frac{13064755}{121862816}a^{12}-\frac{8090567}{121862816}a^{11}+\frac{14644221}{121862816}a^{10}-\frac{472633}{1818848}a^{9}+\frac{8511623}{121862816}a^{8}-\frac{50977705}{121862816}a^{7}+\frac{469325}{121862816}a^{6}-\frac{97581585}{121862816}a^{5}-\frac{74389117}{121862816}a^{4}-\frac{42872641}{30465704}a^{3}-\frac{13169777}{15232852}a^{2}-\frac{13181633}{15232852}a-\frac{3576494}{3808213}$, $\frac{67814601}{8286671488}a^{16}-\frac{234095699}{8286671488}a^{15}+\frac{444375935}{8286671488}a^{14}-\frac{1457245705}{8286671488}a^{13}+\frac{2013105723}{8286671488}a^{12}-\frac{731979019}{8286671488}a^{11}+\frac{1138003775}{8286671488}a^{10}+\frac{22149825}{123681664}a^{9}-\frac{5003388539}{8286671488}a^{8}+\frac{6808792053}{8286671488}a^{7}-\frac{2569440695}{8286671488}a^{6}+\frac{264565311}{487451264}a^{5}+\frac{1248537237}{2071667872}a^{4}-\frac{2948667503}{2071667872}a^{3}+\frac{477137409}{517916968}a^{2}-\frac{411165849}{517916968}a-\frac{50269591}{129479242}$, $\frac{97026975}{8286671488}a^{16}-\frac{358034209}{8286671488}a^{15}+\frac{541845141}{8286671488}a^{14}-\frac{1921291211}{8286671488}a^{13}+\frac{3189896617}{8286671488}a^{12}+\frac{11706927}{8286671488}a^{11}+\frac{3429878237}{8286671488}a^{10}-\frac{88858701}{123681664}a^{9}-\frac{2669839113}{8286671488}a^{8}-\frac{5809955937}{8286671488}a^{7}+\frac{9331308083}{8286671488}a^{6}-\frac{234659075}{487451264}a^{5}-\frac{409691155}{1035833936}a^{4}-\frac{6765012675}{2071667872}a^{3}-\frac{382921001}{129479242}a^{2}+\frac{342505529}{517916968}a-\frac{1331749}{129479242}$
| 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: | \( 1757145.67579 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 1757145.67579 \cdot 1}{2\cdot\sqrt{2007233999721653909999521}}\cr\approx \mathstrut & 3.01264329152 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64)
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^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64, 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^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64);
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^16 + 4*x^15 - 18*x^14 + 10*x^13 - 16*x^12 + 44*x^11 + 10*x^10 + 72*x^9 + 18*x^8 + 134*x^7 + 88*x^6 + 275*x^5 + 200*x^4 + 220*x^3 + 240*x^2 + 80*x + 64);
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 | ${\href{/padicField/2.2.0.1}{2} }^{8}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $17$ | $17$ | $17$ | $17$ | $17$ | ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $17$ | $17$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $17$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $17$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $17$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(1091\)
| $\Q_{1091}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |