Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 5*x^15 - 19*x^14 + 26*x^13 - 30*x^12 + 44*x^11 - 48*x^10 + 39*x^9 - 62*x^8 + 30*x^7 + 112*x^6 - 4*x^5 + 16*x^4 - 62*x^3 + 180*x^2 + 35*x + 49)
gp: K = bnfinit(y^17 - y^16 + 5*y^15 - 19*y^14 + 26*y^13 - 30*y^12 + 44*y^11 - 48*y^10 + 39*y^9 - 62*y^8 + 30*y^7 + 112*y^6 - 4*y^5 + 16*y^4 - 62*y^3 + 180*y^2 + 35*y + 49, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^16 + 5*x^15 - 19*x^14 + 26*x^13 - 30*x^12 + 44*x^11 - 48*x^10 + 39*x^9 - 62*x^8 + 30*x^7 + 112*x^6 - 4*x^5 + 16*x^4 - 62*x^3 + 180*x^2 + 35*x + 49);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^16 + 5*x^15 - 19*x^14 + 26*x^13 - 30*x^12 + 44*x^11 - 48*x^10 + 39*x^9 - 62*x^8 + 30*x^7 + 112*x^6 - 4*x^5 + 16*x^4 - 62*x^3 + 180*x^2 + 35*x + 49)
\( x^{17} - x^{16} + 5 x^{15} - 19 x^{14} + 26 x^{13} - 30 x^{12} + 44 x^{11} - 48 x^{10} + 39 x^{9} + \cdots + 49 \)
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: |
\(1837263966425479287178561\)
\(\medspace = 13^{8}\cdot 83^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.75\) | 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: | $13^{1/2}83^{1/2}\approx 32.848135411313685$ | ||
| Ramified primes: |
\(13\), \(83\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13}a^{11}-\frac{4}{13}a^{10}+\frac{3}{13}a^{9}+\frac{1}{13}a^{8}-\frac{6}{13}a^{7}-\frac{2}{13}a^{6}-\frac{5}{13}a^{5}-\frac{6}{13}a^{4}-\frac{4}{13}a^{3}+\frac{3}{13}a^{2}+\frac{1}{13}a-\frac{4}{13}$, $\frac{1}{143}a^{12}+\frac{1}{143}a^{11}+\frac{9}{143}a^{10}+\frac{29}{143}a^{9}-\frac{27}{143}a^{8}-\frac{6}{143}a^{7}+\frac{50}{143}a^{6}-\frac{70}{143}a^{5}-\frac{8}{143}a^{4}+\frac{35}{143}a^{3}-\frac{36}{143}a^{2}+\frac{53}{143}a+\frac{58}{143}$, $\frac{1}{143}a^{13}-\frac{3}{143}a^{11}+\frac{64}{143}a^{10}+\frac{54}{143}a^{9}+\frac{10}{143}a^{8}-\frac{21}{143}a^{7}+\frac{45}{143}a^{6}-\frac{2}{11}a^{5}-\frac{34}{143}a^{4}-\frac{27}{143}a^{3}+\frac{56}{143}a^{2}-\frac{6}{143}a-\frac{14}{143}$, $\frac{1}{143}a^{14}+\frac{1}{143}a^{11}+\frac{59}{143}a^{10}+\frac{42}{143}a^{9}-\frac{25}{143}a^{8}-\frac{6}{143}a^{7}-\frac{30}{143}a^{6}-\frac{57}{143}a^{5}+\frac{59}{143}a^{4}-\frac{4}{143}a^{3}-\frac{2}{11}a^{2}-\frac{64}{143}a+\frac{9}{143}$, $\frac{1}{1001}a^{15}+\frac{2}{1001}a^{14}-\frac{3}{1001}a^{13}-\frac{30}{1001}a^{11}+\frac{69}{1001}a^{10}+\frac{2}{7}a^{9}+\frac{271}{1001}a^{8}-\frac{380}{1001}a^{7}-\frac{30}{77}a^{6}+\frac{302}{1001}a^{5}-\frac{326}{1001}a^{4}-\frac{450}{1001}a^{3}-\frac{116}{1001}a^{2}-\frac{36}{91}a+\frac{16}{143}$, $\frac{1}{1221517297}a^{16}-\frac{36983}{1221517297}a^{15}-\frac{3908517}{1221517297}a^{14}+\frac{3036619}{1221517297}a^{13}+\frac{28187}{1221517297}a^{12}-\frac{4920906}{174502471}a^{11}+\frac{553749892}{1221517297}a^{10}-\frac{206982179}{1221517297}a^{9}-\frac{3703743}{28407379}a^{8}-\frac{30236515}{111047027}a^{7}-\frac{10006336}{174502471}a^{6}+\frac{147306151}{1221517297}a^{5}+\frac{33562096}{174502471}a^{4}+\frac{505944163}{1221517297}a^{3}+\frac{171129719}{1221517297}a^{2}-\frac{52795685}{1221517297}a-\frac{65347462}{174502471}$
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: |
$\frac{286461}{174502471}a^{16}-\frac{15284718}{1221517297}a^{15}+\frac{36438890}{1221517297}a^{14}-\frac{124030904}{1221517297}a^{13}+\frac{52243729}{174502471}a^{12}-\frac{644671298}{1221517297}a^{11}+\frac{844616781}{1221517297}a^{10}-\frac{856580871}{1221517297}a^{9}+\frac{13330214}{28407379}a^{8}-\frac{157626929}{1221517297}a^{7}+\frac{184946382}{1221517297}a^{6}-\frac{22783931}{111047027}a^{5}-\frac{533920887}{1221517297}a^{4}+\frac{306656289}{1221517297}a^{3}-\frac{99744767}{1221517297}a^{2}-\frac{397253566}{1221517297}a+\frac{18064715}{174502471}$, $\frac{10969272}{1221517297}a^{16}-\frac{10814072}{1221517297}a^{15}+\frac{44379175}{1221517297}a^{14}-\frac{207937650}{1221517297}a^{13}+\frac{232414513}{1221517297}a^{12}-\frac{174242786}{1221517297}a^{11}+\frac{381430037}{1221517297}a^{10}-\frac{26056185}{111047027}a^{9}+\frac{3416067}{28407379}a^{8}-\frac{64629225}{174502471}a^{7}-\frac{276081340}{1221517297}a^{6}+\frac{1647722359}{1221517297}a^{5}+\frac{70175972}{1221517297}a^{4}-\frac{509754771}{1221517297}a^{3}-\frac{1415875551}{1221517297}a^{2}+\frac{1404759199}{1221517297}a+\frac{13387099}{15863861}$, $\frac{13994556}{1221517297}a^{16}-\frac{2406141}{174502471}a^{15}+\frac{10542646}{174502471}a^{14}-\frac{270292213}{1221517297}a^{13}+\frac{401361544}{1221517297}a^{12}-\frac{464271151}{1221517297}a^{11}+\frac{486921801}{1221517297}a^{10}-\frac{60410722}{174502471}a^{9}+\frac{10683215}{28407379}a^{8}-\frac{714268228}{1221517297}a^{7}+\frac{221047940}{1221517297}a^{6}+\frac{1528652925}{1221517297}a^{5}+\frac{11651099}{1221517297}a^{4}+\frac{35623590}{174502471}a^{3}+\frac{154875372}{1221517297}a^{2}+\frac{1843831299}{1221517297}a+\frac{30486646}{174502471}$, $\frac{30607249}{1221517297}a^{16}-\frac{39442071}{1221517297}a^{15}+\frac{133550878}{1221517297}a^{14}-\frac{622772699}{1221517297}a^{13}+\frac{822214384}{1221517297}a^{12}-\frac{696022043}{1221517297}a^{11}+\frac{1249111505}{1221517297}a^{10}-\frac{1101608700}{1221517297}a^{9}+\frac{8138532}{28407379}a^{8}-\frac{16864398}{15863861}a^{7}+\frac{237126347}{1221517297}a^{6}+\frac{5053790571}{1221517297}a^{5}-\frac{1046352691}{1221517297}a^{4}-\frac{166756952}{111047027}a^{3}-\frac{5601596281}{1221517297}a^{2}+\frac{2768432012}{1221517297}a+\frac{57576708}{174502471}$, $\frac{756572}{111047027}a^{16}-\frac{873883}{174502471}a^{15}+\frac{582863}{15863861}a^{14}-\frac{146099365}{1221517297}a^{13}+\frac{211423081}{1221517297}a^{12}-\frac{254913886}{1221517297}a^{11}+\frac{414014642}{1221517297}a^{10}-\frac{63495258}{174502471}a^{9}+\frac{10106444}{28407379}a^{8}-\frac{560005977}{1221517297}a^{7}+\frac{17803008}{111047027}a^{6}+\frac{908825847}{1221517297}a^{5}-\frac{272637945}{1221517297}a^{4}+\frac{100410949}{174502471}a^{3}-\frac{1090398502}{1221517297}a^{2}+\frac{1826610407}{1221517297}a+\frac{46722762}{174502471}$, $\frac{4524560}{1221517297}a^{16}-\frac{8338731}{1221517297}a^{15}+\frac{15775693}{1221517297}a^{14}-\frac{7815568}{111047027}a^{13}+\frac{103038198}{1221517297}a^{12}+\frac{18588939}{1221517297}a^{11}-\frac{31964139}{174502471}a^{10}+\frac{494913288}{1221517297}a^{9}-\frac{2073144}{4058197}a^{8}+\frac{359198041}{1221517297}a^{7}-\frac{69734341}{1221517297}a^{6}+\frac{608769283}{1221517297}a^{5}-\frac{37789634}{111047027}a^{4}-\frac{306116632}{1221517297}a^{3}+\frac{158163409}{174502471}a^{2}+\frac{33694121}{1221517297}a-\frac{23952628}{174502471}$, $\frac{22760503}{1221517297}a^{16}-\frac{18878680}{1221517297}a^{15}+\frac{115403368}{1221517297}a^{14}-\frac{443584558}{1221517297}a^{13}+\frac{527063664}{1221517297}a^{12}-\frac{775322969}{1221517297}a^{11}+\frac{1283380299}{1221517297}a^{10}-\frac{926386928}{1221517297}a^{9}+\frac{16120849}{28407379}a^{8}-\frac{1270893098}{1221517297}a^{7}-\frac{89582219}{1221517297}a^{6}+\frac{259496519}{111047027}a^{5}+\frac{658268210}{1221517297}a^{4}+\frac{2371217384}{1221517297}a^{3}-\frac{4248974269}{1221517297}a^{2}-\frac{667256999}{1221517297}a-\frac{13281267}{15863861}$, $\frac{292239}{7227913}a^{16}-\frac{307747}{8542079}a^{15}+\frac{1619781}{8542079}a^{14}-\frac{72085344}{93962869}a^{13}+\frac{7522534}{8542079}a^{12}-\frac{96423204}{93962869}a^{11}+\frac{22035605}{13423267}a^{10}-\frac{83871743}{93962869}a^{9}+\frac{110797}{312169}a^{8}-\frac{152350126}{93962869}a^{7}-\frac{14895108}{93962869}a^{6}+\frac{452528826}{93962869}a^{5}+\frac{176964278}{93962869}a^{4}+\frac{100831842}{93962869}a^{3}-\frac{68136217}{13423267}a^{2}-\frac{125584716}{93962869}a-\frac{1403797}{1220297}$
| 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: | \( 257874.899874 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 257874.899874 \cdot 1}{2\cdot\sqrt{1837263966425479287178561}}\cr\approx \mathstrut & 0.462127954868 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 5*x^15 - 19*x^14 + 26*x^13 - 30*x^12 + 44*x^11 - 48*x^10 + 39*x^9 - 62*x^8 + 30*x^7 + 112*x^6 - 4*x^5 + 16*x^4 - 62*x^3 + 180*x^2 + 35*x + 49)
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 + 5*x^15 - 19*x^14 + 26*x^13 - 30*x^12 + 44*x^11 - 48*x^10 + 39*x^9 - 62*x^8 + 30*x^7 + 112*x^6 - 4*x^5 + 16*x^4 - 62*x^3 + 180*x^2 + 35*x + 49, 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 + 5*x^15 - 19*x^14 + 26*x^13 - 30*x^12 + 44*x^11 - 48*x^10 + 39*x^9 - 62*x^8 + 30*x^7 + 112*x^6 - 4*x^5 + 16*x^4 - 62*x^3 + 180*x^2 + 35*x + 49);
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 + 5*x^15 - 19*x^14 + 26*x^13 - 30*x^12 + 44*x^11 - 48*x^10 + 39*x^9 - 62*x^8 + 30*x^7 + 112*x^6 - 4*x^5 + 16*x^4 - 62*x^3 + 180*x^2 + 35*x + 49);
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 | $17$ | $17$ | $17$ | ${\href{/padicField/7.2.0.1}{2} }^{8}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | $17$ | $17$ | $17$ | $17$ | ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $17$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. 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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(83\)
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |