Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 3*x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 35*x^10 - 52*x^9 - 12*x^8 - 9*x^7 + 119*x^6 + 301*x^5 + 364*x^4 + 315*x^3 + 171*x^2 + 54*x + 9)
gp: K = bnfinit(y^16 - 3*y^15 + 3*y^14 - 7*y^13 + 14*y^12 - 7*y^11 + 35*y^10 - 52*y^9 - 12*y^8 - 9*y^7 + 119*y^6 + 301*y^5 + 364*y^4 + 315*y^3 + 171*y^2 + 54*y + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 3*x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 35*x^10 - 52*x^9 - 12*x^8 - 9*x^7 + 119*x^6 + 301*x^5 + 364*x^4 + 315*x^3 + 171*x^2 + 54*x + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 3*x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 35*x^10 - 52*x^9 - 12*x^8 - 9*x^7 + 119*x^6 + 301*x^5 + 364*x^4 + 315*x^3 + 171*x^2 + 54*x + 9)
\( x^{16} - 3 x^{15} + 3 x^{14} - 7 x^{13} + 14 x^{12} - 7 x^{11} + 35 x^{10} - 52 x^{9} - 12 x^{8} + \cdots + 9 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(18226469195621535744\)
\(\medspace = 2^{12}\cdot 3^{8}\cdot 7^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.99\) | 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: | $2\cdot 3^{1/2}7^{7/8}\approx 19.013033264982823$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{6}a^{11}-\frac{1}{2}a^{9}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{3}a^{4}-\frac{1}{6}a^{3}+\frac{1}{3}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{12}a^{12}-\frac{1}{12}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{12}a^{8}-\frac{1}{2}a^{6}-\frac{1}{12}a^{5}+\frac{1}{12}a^{4}-\frac{1}{4}a^{3}+\frac{1}{12}a^{2}+\frac{1}{4}$, $\frac{1}{12}a^{13}+\frac{1}{6}a^{9}-\frac{1}{12}a^{8}+\frac{1}{6}a^{7}-\frac{5}{12}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{216}a^{14}+\frac{5}{216}a^{13}-\frac{1}{36}a^{11}-\frac{17}{108}a^{10}-\frac{7}{72}a^{9}+\frac{11}{72}a^{8}+\frac{35}{216}a^{7}-\frac{71}{216}a^{6}+\frac{13}{36}a^{5}-\frac{23}{54}a^{4}+\frac{11}{24}a^{3}-\frac{1}{3}a^{2}+\frac{1}{12}a-\frac{5}{24}$, $\frac{1}{41252735592}a^{15}+\frac{31150517}{20626367796}a^{14}-\frac{447468851}{41252735592}a^{13}+\frac{113481049}{3437727966}a^{12}-\frac{1108913333}{20626367796}a^{11}+\frac{8354902291}{41252735592}a^{10}-\frac{115394299}{2291818644}a^{9}-\frac{10110970187}{20626367796}a^{8}-\frac{1514980751}{5156591949}a^{7}+\frac{12170476667}{41252735592}a^{6}-\frac{1918420237}{5156591949}a^{5}-\frac{16366308175}{41252735592}a^{4}+\frac{2026709297}{4583637288}a^{3}+\frac{57285023}{572954661}a^{2}-\frac{436717279}{4583637288}a-\frac{1263785047}{4583637288}$
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{1039523}{7916472} a^{15} - \frac{11897951}{23749416} a^{14} + \frac{4653719}{5937354} a^{13} - \frac{164850}{109951} a^{12} + \frac{2961091}{989559} a^{11} - \frac{75869143}{23749416} a^{10} + \frac{54194945}{7916472} a^{9} - \frac{31995623}{2638824} a^{8} + \frac{177338813}{23749416} a^{7} - \frac{69913669}{11874708} a^{6} + \frac{79590245}{3958236} a^{5} + \frac{569641345}{23749416} a^{4} + \frac{32545709}{1319412} a^{3} + \frac{5444831}{329853} a^{2} + \frac{9002555}{2638824} a + \frac{1484351}{1319412} \)
(order $6$)
| 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: |
$\frac{222047468}{5156591949}a^{15}-\frac{3014607037}{20626367796}a^{14}+\frac{2140450339}{10313183898}a^{13}-\frac{1637586851}{3437727966}a^{12}+\frac{4935940525}{5156591949}a^{11}-\frac{9725818109}{10313183898}a^{10}+\frac{5481712775}{2291818644}a^{9}-\frac{39220093903}{10313183898}a^{8}+\frac{40927643323}{20626367796}a^{7}-\frac{33091254157}{10313183898}a^{6}+\frac{41463763631}{5156591949}a^{5}+\frac{50145704929}{5156591949}a^{4}+\frac{31687158865}{2291818644}a^{3}+\frac{24502766177}{2291818644}a^{2}+\frac{6877447123}{2291818644}a+\frac{182810149}{572954661}$, $\frac{8055385645}{41252735592}a^{15}-\frac{29651735633}{41252735592}a^{14}+\frac{5338495025}{5156591949}a^{13}-\frac{6563224109}{3437727966}a^{12}+\frac{19458726745}{5156591949}a^{11}-\frac{145076169131}{41252735592}a^{10}+\frac{38619953629}{4583637288}a^{9}-\frac{620013831391}{41252735592}a^{8}+\frac{255020987027}{41252735592}a^{7}-\frac{55854913865}{20626367796}a^{6}+\frac{477682357691}{20626367796}a^{5}+\frac{1762293316505}{41252735592}a^{4}+\frac{44294937983}{1145909322}a^{3}+\frac{65169206603}{2291818644}a^{2}+\frac{46292274401}{4583637288}a+\frac{2664790897}{2291818644}$, $\frac{1276768271}{41252735592}a^{15}-\frac{4036933007}{20626367796}a^{14}+\frac{20328942533}{41252735592}a^{13}-\frac{1425706052}{1718863983}a^{12}+\frac{31139172503}{20626367796}a^{11}-\frac{97306084795}{41252735592}a^{10}+\frac{7522602811}{2291818644}a^{9}-\frac{32411695576}{5156591949}a^{8}+\frac{43144027046}{5156591949}a^{7}-\frac{206340767393}{41252735592}a^{6}+\frac{53729465405}{10313183898}a^{5}-\frac{132796339769}{41252735592}a^{4}-\frac{34960655045}{4583637288}a^{3}-\frac{12184155239}{2291818644}a^{2}-\frac{15226414967}{4583637288}a-\frac{3901003511}{4583637288}$, $\frac{574950473}{4583637288}a^{15}-\frac{840733021}{2291818644}a^{14}+\frac{461828507}{1527879096}a^{13}-\frac{520216913}{763939548}a^{12}+\frac{825017483}{572954661}a^{11}-\frac{45116489}{169764344}a^{10}+\frac{644010350}{190984887}a^{9}-\frac{3005658694}{572954661}a^{8}-\frac{4797468667}{1145909322}a^{7}+\frac{3892830677}{1527879096}a^{6}+\frac{29592744409}{2291818644}a^{5}+\frac{61804878433}{1527879096}a^{4}+\frac{21515880007}{509293032}a^{3}+\frac{8527826327}{254646516}a^{2}+\frac{7570101113}{509293032}a+\frac{271283441}{169764344}$, $\frac{2169854017}{41252735592}a^{15}-\frac{1378535551}{5156591949}a^{14}+\frac{22174282489}{41252735592}a^{13}-\frac{2871661463}{3437727966}a^{12}+\frac{32533355755}{20626367796}a^{11}-\frac{86435753189}{41252735592}a^{10}+\frac{1764902633}{572954661}a^{9}-\frac{133435937999}{20626367796}a^{8}+\frac{127804607725}{20626367796}a^{7}-\frac{36687643933}{41252735592}a^{6}+\frac{24117914621}{5156591949}a^{5}+\frac{177498528365}{41252735592}a^{4}-\frac{35813689333}{4583637288}a^{3}-\frac{15885800293}{2291818644}a^{2}-\frac{30986743165}{4583637288}a-\frac{12271701631}{4583637288}$, $\frac{1465916425}{20626367796}a^{15}-\frac{5833219259}{20626367796}a^{14}+\frac{2282983759}{5156591949}a^{13}-\frac{1298920283}{1718863983}a^{12}+\frac{7691701442}{5156591949}a^{11}-\frac{31878516515}{20626367796}a^{10}+\frac{7307905963}{2291818644}a^{9}-\frac{126346357867}{20626367796}a^{8}+\frac{69942141989}{20626367796}a^{7}-\frac{6661129385}{10313183898}a^{6}+\frac{83771495243}{10313183898}a^{5}+\frac{267608154881}{20626367796}a^{4}+\frac{10030670281}{1145909322}a^{3}+\frac{1846402666}{572954661}a^{2}-\frac{188271421}{2291818644}a-\frac{595329239}{1145909322}$, $\frac{191390615}{1718863983}a^{15}-\frac{785442649}{1718863983}a^{14}+\frac{2175303907}{3437727966}a^{13}-\frac{1823254399}{2291818644}a^{12}+\frac{11989407019}{6875455932}a^{11}-\frac{10579498693}{6875455932}a^{10}+\frac{743242943}{254646516}a^{9}-\frac{52526308481}{6875455932}a^{8}+\frac{5205279227}{3437727966}a^{7}+\frac{28673168723}{3437727966}a^{6}+\frac{49854006131}{6875455932}a^{5}+\frac{123008710879}{6875455932}a^{4}-\frac{1124574911}{763939548}a^{3}-\frac{10337969563}{763939548}a^{2}-\frac{1944613346}{190984887}a-\frac{1980209279}{763939548}$
| 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: | \( 4928.02836346 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4928.02836346 \cdot 1}{6\cdot\sqrt{18226469195621535744}}\cr\approx \mathstrut & 0.467314893402 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 3*x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 35*x^10 - 52*x^9 - 12*x^8 - 9*x^7 + 119*x^6 + 301*x^5 + 364*x^4 + 315*x^3 + 171*x^2 + 54*x + 9)
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^16 - 3*x^15 + 3*x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 35*x^10 - 52*x^9 - 12*x^8 - 9*x^7 + 119*x^6 + 301*x^5 + 364*x^4 + 315*x^3 + 171*x^2 + 54*x + 9, 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^16 - 3*x^15 + 3*x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 35*x^10 - 52*x^9 - 12*x^8 - 9*x^7 + 119*x^6 + 301*x^5 + 364*x^4 + 315*x^3 + 171*x^2 + 54*x + 9);
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^16 - 3*x^15 + 3*x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 35*x^10 - 52*x^9 - 12*x^8 - 9*x^7 + 119*x^6 + 301*x^5 + 364*x^4 + 315*x^3 + 171*x^2 + 54*x + 9);
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 32 |
| The 11 conjugacy class representatives for $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}) \), 4.0.1372.1, 4.0.12348.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.152473104.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(7\)
| 7.8.7.1 | $x^{8} + 7$ | $8$ | $1$ | $7$ | $D_{8}$ | $[\ ]_{8}^{2}$ |
| 7.8.7.1 | $x^{8} + 7$ | $8$ | $1$ | $7$ | $D_{8}$ | $[\ ]_{8}^{2}$ |