Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - x^15 - x^14 + x^12 + 13*x^11 + 7*x^10 + 11*x^9 + 4*x^8 + x^7 + 7*x^6 + 23*x^5 + 31*x^4 + 42*x^3 + 24*x^2 + 6*x - 1)
gp: K = bnfinit(y^17 - y^16 - y^15 - y^14 + y^12 + 13*y^11 + 7*y^10 + 11*y^9 + 4*y^8 + y^7 + 7*y^6 + 23*y^5 + 31*y^4 + 42*y^3 + 24*y^2 + 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^16 - x^15 - x^14 + x^12 + 13*x^11 + 7*x^10 + 11*x^9 + 4*x^8 + x^7 + 7*x^6 + 23*x^5 + 31*x^4 + 42*x^3 + 24*x^2 + 6*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^16 - x^15 - x^14 + x^12 + 13*x^11 + 7*x^10 + 11*x^9 + 4*x^8 + x^7 + 7*x^6 + 23*x^5 + 31*x^4 + 42*x^3 + 24*x^2 + 6*x - 1)
\( x^{17} - x^{16} - x^{15} - x^{14} + x^{12} + 13 x^{11} + 7 x^{10} + 11 x^{9} + 4 x^{8} + x^{7} + \cdots - 1 \)
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: |
\(463009808974713123841\)
\(\medspace = 383^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.43\) | 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: | $383^{1/2}\approx 19.570385790780925$ | ||
| Ramified primes: |
\(383\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{65}a^{14}+\frac{11}{65}a^{13}-\frac{3}{13}a^{12}-\frac{31}{65}a^{11}+\frac{9}{65}a^{10}+\frac{1}{13}a^{9}+\frac{28}{65}a^{8}+\frac{12}{65}a^{7}-\frac{28}{65}a^{6}+\frac{24}{65}a^{5}-\frac{31}{65}a^{4}+\frac{3}{65}a^{3}-\frac{6}{65}a^{2}+\frac{2}{13}a+\frac{1}{65}$, $\frac{1}{715}a^{15}-\frac{2}{715}a^{14}-\frac{93}{715}a^{13}-\frac{226}{715}a^{12}-\frac{173}{715}a^{11}-\frac{307}{715}a^{10}+\frac{93}{715}a^{9}-\frac{92}{715}a^{8}+\frac{271}{715}a^{7}-\frac{67}{715}a^{6}-\frac{18}{715}a^{5}-\frac{244}{715}a^{4}-\frac{35}{143}a^{3}+\frac{218}{715}a^{2}-\frac{259}{715}a-\frac{1}{55}$, $\frac{1}{3754465}a^{16}+\frac{382}{750893}a^{15}-\frac{4687}{3754465}a^{14}+\frac{1760433}{3754465}a^{13}-\frac{328105}{750893}a^{12}+\frac{1358022}{3754465}a^{11}+\frac{1837179}{3754465}a^{10}+\frac{466309}{3754465}a^{9}+\frac{1345062}{3754465}a^{8}-\frac{301062}{750893}a^{7}-\frac{4218}{42185}a^{6}-\frac{205108}{750893}a^{5}+\frac{1173782}{3754465}a^{4}-\frac{654867}{3754465}a^{3}-\frac{940898}{3754465}a^{2}+\frac{1634929}{3754465}a+\frac{1364334}{3754465}$
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: | $8$ | 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: |
$\frac{17794}{750893}a^{16}+\frac{199376}{3754465}a^{15}-\frac{119618}{750893}a^{14}-\frac{59041}{3754465}a^{13}-\frac{9616}{3754465}a^{12}-\frac{1695}{750893}a^{11}+\frac{955431}{3754465}a^{10}+\frac{4732503}{3754465}a^{9}+\frac{185949}{3754465}a^{8}+\frac{63138}{68263}a^{7}+\frac{9793}{42185}a^{6}+\frac{168257}{750893}a^{5}+\frac{3140734}{3754465}a^{4}+\frac{9329486}{3754465}a^{3}+\frac{698291}{341315}a^{2}+\frac{11270456}{3754465}a+\frac{3698674}{3754465}$, $\frac{59439}{750893}a^{16}-\frac{573511}{3754465}a^{15}+\frac{168841}{3754465}a^{14}-\frac{100703}{3754465}a^{13}-\frac{183319}{3754465}a^{12}+\frac{13934}{341315}a^{11}+\frac{3391528}{3754465}a^{10}-\frac{675933}{3754465}a^{9}+\frac{4026569}{3754465}a^{8}+\frac{899942}{3754465}a^{7}-\frac{2229}{8437}a^{6}+\frac{2313189}{3754465}a^{5}+\frac{334168}{750893}a^{4}+\frac{5705047}{3754465}a^{3}+\frac{9056613}{3754465}a^{2}+\frac{376484}{341315}a+\frac{1693162}{3754465}$, $\frac{109667}{750893}a^{16}-\frac{1084308}{3754465}a^{15}+\frac{575654}{3754465}a^{14}-\frac{831093}{3754465}a^{13}+\frac{7523}{341315}a^{12}+\frac{387421}{3754465}a^{11}+\frac{620053}{341315}a^{10}-\frac{186458}{288805}a^{9}+\frac{1886628}{750893}a^{8}-\frac{3178152}{3754465}a^{7}+\frac{25748}{42185}a^{6}+\frac{443151}{341315}a^{5}+\frac{770209}{341315}a^{4}+\frac{9093189}{3754465}a^{3}+\frac{18983938}{3754465}a^{2}+\frac{145242}{3754465}a+\frac{2439452}{3754465}$, $\frac{271369}{3754465}a^{16}-\frac{704552}{3754465}a^{15}+\frac{353036}{3754465}a^{14}-\frac{184037}{3754465}a^{13}+\frac{300827}{3754465}a^{12}+\frac{655609}{3754465}a^{11}+\frac{45406}{57761}a^{10}-\frac{824275}{750893}a^{9}+\frac{2424222}{3754465}a^{8}-\frac{3331872}{3754465}a^{7}-\frac{1286}{3835}a^{6}+\frac{2059516}{3754465}a^{5}+\frac{2140761}{3754465}a^{4}+\frac{434102}{3754465}a^{3}+\frac{291822}{3754465}a^{2}-\frac{10532001}{3754465}a-\frac{2936398}{3754465}$, $\frac{1512}{68263}a^{16}+\frac{82117}{3754465}a^{15}-\frac{351509}{3754465}a^{14}+\frac{116854}{3754465}a^{13}-\frac{599897}{3754465}a^{12}+\frac{109819}{3754465}a^{11}+\frac{1414536}{3754465}a^{10}+\frac{270187}{288805}a^{9}+\frac{186981}{3754465}a^{8}+\frac{3189827}{3754465}a^{7}-\frac{12641}{42185}a^{6}+\frac{1780149}{3754465}a^{5}+\frac{1276447}{3754465}a^{4}+\frac{1567078}{750893}a^{3}+\frac{9924196}{3754465}a^{2}+\frac{8036577}{3754465}a+\frac{1254744}{3754465}$, $\frac{27813}{341315}a^{16}-\frac{69417}{3754465}a^{15}-\frac{683889}{3754465}a^{14}-\frac{375197}{3754465}a^{13}-\frac{8923}{3754465}a^{12}+\frac{157059}{3754465}a^{11}+\frac{4389928}{3754465}a^{10}+\frac{4669861}{3754465}a^{9}+\frac{3395244}{3754465}a^{8}+\frac{262353}{288805}a^{7}+\frac{20841}{42185}a^{6}+\frac{931023}{3754465}a^{5}+\frac{9776836}{3754465}a^{4}+\frac{11871193}{3754465}a^{3}+\frac{14342672}{3754465}a^{2}+\frac{2260768}{750893}a+\frac{2083381}{3754465}$, $\frac{665628}{3754465}a^{16}-\frac{647509}{3754465}a^{15}-\frac{181845}{750893}a^{14}+\frac{54429}{3754465}a^{13}-\frac{65216}{341315}a^{12}+\frac{136134}{750893}a^{11}+\frac{800827}{341315}a^{10}+\frac{1044944}{750893}a^{9}+\frac{4911183}{3754465}a^{8}+\frac{6927207}{3754465}a^{7}-\frac{43783}{42185}a^{6}+\frac{543424}{341315}a^{5}+\frac{100498}{26255}a^{4}+\frac{20485768}{3754465}a^{3}+\frac{27202731}{3754465}a^{2}+\frac{16622978}{3754465}a-\frac{586758}{3754465}$, $\frac{762276}{3754465}a^{16}-\frac{401937}{3754465}a^{15}-\frac{1267452}{3754465}a^{14}-\frac{201466}{750893}a^{13}-\frac{22098}{341315}a^{12}+\frac{1053487}{3754465}a^{11}+\frac{959702}{341315}a^{10}+\frac{710981}{288805}a^{9}+\frac{8996164}{3754465}a^{8}+\frac{1131855}{750893}a^{7}+\frac{10146}{42185}a^{6}+\frac{6921}{5251}a^{5}+\frac{1857559}{341315}a^{4}+\frac{28912051}{3754465}a^{3}+\frac{8355836}{750893}a^{2}+\frac{26267647}{3754465}a+\frac{9066576}{3754465}$
| 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: | \( 2610.31631075 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 2610.31631075 \cdot 1}{2\cdot\sqrt{463009808974713123841}}\cr\approx \mathstrut & 0.294670722603 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - x^15 - x^14 + x^12 + 13*x^11 + 7*x^10 + 11*x^9 + 4*x^8 + x^7 + 7*x^6 + 23*x^5 + 31*x^4 + 42*x^3 + 24*x^2 + 6*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 - x^15 - x^14 + x^12 + 13*x^11 + 7*x^10 + 11*x^9 + 4*x^8 + x^7 + 7*x^6 + 23*x^5 + 31*x^4 + 42*x^3 + 24*x^2 + 6*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 - x^15 - x^14 + x^12 + 13*x^11 + 7*x^10 + 11*x^9 + 4*x^8 + x^7 + 7*x^6 + 23*x^5 + 31*x^4 + 42*x^3 + 24*x^2 + 6*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 - x^15 - x^14 + x^12 + 13*x^11 + 7*x^10 + 11*x^9 + 4*x^8 + x^7 + 7*x^6 + 23*x^5 + 31*x^4 + 42*x^3 + 24*x^2 + 6*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 | $17$ | $17$ | ${\href{/padicField/5.2.0.1}{2} }^{8}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $17$ | ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $17$ | $17$ | $17$ | $17$ | $17$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $17$ | ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(383\)
| $\Q_{383}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |