Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 17*x^13 + 20*x^12 - 44*x^11 - 25*x^10 + 22*x^9 + 24*x^8 + 68*x^7 + 29*x^6 - 133*x^5 - 130*x^4 + 48*x^3 + 105*x^2 + 38*x + 2)
gp: K = bnfinit(y^16 - 2*y^15 - 8*y^14 + 17*y^13 + 20*y^12 - 44*y^11 - 25*y^10 + 22*y^9 + 24*y^8 + 68*y^7 + 29*y^6 - 133*y^5 - 130*y^4 + 48*y^3 + 105*y^2 + 38*y + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 8*x^14 + 17*x^13 + 20*x^12 - 44*x^11 - 25*x^10 + 22*x^9 + 24*x^8 + 68*x^7 + 29*x^6 - 133*x^5 - 130*x^4 + 48*x^3 + 105*x^2 + 38*x + 2);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 8*x^14 + 17*x^13 + 20*x^12 - 44*x^11 - 25*x^10 + 22*x^9 + 24*x^8 + 68*x^7 + 29*x^6 - 133*x^5 - 130*x^4 + 48*x^3 + 105*x^2 + 38*x + 2)
\( x^{16} - 2 x^{15} - 8 x^{14} + 17 x^{13} + 20 x^{12} - 44 x^{11} - 25 x^{10} + 22 x^{9} + 24 x^{8} + \cdots + 2 \)
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: | $[6, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-10612039879318707392816\)
\(\medspace = -\,2^{4}\cdot 11^{3}\cdot 163^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.80\) | 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: | $2\cdot 11^{3/4}163^{2/3}\approx 360.4676089578411$ | ||
| Ramified primes: |
\(2\), \(11\), \(163\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2035}a^{14}+\frac{192}{2035}a^{13}-\frac{571}{2035}a^{12}+\frac{301}{2035}a^{11}+\frac{3}{37}a^{10}+\frac{10}{407}a^{9}-\frac{1}{407}a^{8}-\frac{983}{2035}a^{7}+\frac{412}{2035}a^{6}-\frac{431}{2035}a^{5}-\frac{357}{2035}a^{4}+\frac{12}{407}a^{3}-\frac{328}{2035}a^{2}-\frac{134}{2035}a-\frac{753}{2035}$, $\frac{1}{20504660}a^{15}-\frac{3821}{20504660}a^{14}+\frac{641413}{1864060}a^{13}+\frac{1690146}{5126165}a^{12}-\frac{1435432}{5126165}a^{11}-\frac{245050}{1025233}a^{10}+\frac{673747}{4100932}a^{9}+\frac{3969017}{20504660}a^{8}-\frac{8057239}{20504660}a^{7}-\frac{359527}{20504660}a^{6}-\frac{4878147}{10252330}a^{5}-\frac{9540019}{20504660}a^{4}+\frac{1095887}{20504660}a^{3}-\frac{670025}{4100932}a^{2}-\frac{2439519}{5126165}a-\frac{3085153}{10252330}$
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: | $10$ | 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: |
$\frac{32851329}{20504660}a^{15}-\frac{87594989}{20504660}a^{14}-\frac{18695643}{1864060}a^{13}+\frac{175040439}{5126165}a^{12}+\frac{48513167}{5126165}a^{11}-\frac{80342750}{1025233}a^{10}+\frac{53852091}{4100932}a^{9}+\frac{595216013}{20504660}a^{8}+\frac{342950109}{20504660}a^{7}+\frac{2010541397}{20504660}a^{6}-\frac{211608953}{10252330}a^{5}-\frac{4133993691}{20504660}a^{4}-\frac{1450773077}{20504660}a^{3}+\frac{535839383}{4100932}a^{2}+\frac{404667854}{5126165}a+\frac{28379733}{10252330}$, $\frac{6414223}{5126165}a^{15}-\frac{17131276}{5126165}a^{14}-\frac{3607052}{466015}a^{13}+\frac{27048085}{1025233}a^{12}+\frac{36642798}{5126165}a^{11}-\frac{60672999}{1025233}a^{10}+\frac{8825778}{1025233}a^{9}+\frac{103149436}{5126165}a^{8}+\frac{85714502}{5126165}a^{7}+\frac{382521628}{5126165}a^{6}-\frac{68576669}{5126165}a^{5}-\frac{794703601}{5126165}a^{4}-\frac{303178984}{5126165}a^{3}+\frac{485511734}{5126165}a^{2}+\frac{339560269}{5126165}a+\frac{30062441}{5126165}$, $\frac{3641181}{10252330}a^{15}-\frac{9760331}{10252330}a^{14}-\frac{1995657}{932030}a^{13}+\frac{37978307}{5126165}a^{12}+\frac{229938}{138545}a^{11}-\frac{16484828}{1025233}a^{10}+\frac{6066727}{2050466}a^{9}+\frac{45467517}{10252330}a^{8}+\frac{47931891}{10252330}a^{7}+\frac{206842203}{10252330}a^{6}-\frac{15578437}{5126165}a^{5}-\frac{421636449}{10252330}a^{4}-\frac{170525273}{10252330}a^{3}+\frac{52282133}{2050466}a^{2}+\frac{81973237}{5126165}a+\frac{1266372}{5126165}$, $\frac{10723551}{20504660}a^{15}-\frac{27154403}{20504660}a^{14}-\frac{583091}{169460}a^{13}+\frac{54750794}{5126165}a^{12}+\frac{4511803}{1025233}a^{11}-\frac{25512815}{1025233}a^{10}+\frac{5313321}{4100932}a^{9}+\frac{188761707}{20504660}a^{8}+\frac{139296907}{20504660}a^{7}+\frac{670172439}{20504660}a^{6}-\frac{913233}{277090}a^{5}-\frac{264616289}{4100932}a^{4}-\frac{646609323}{20504660}a^{3}+\frac{801164601}{20504660}a^{2}+\frac{152817243}{5126165}a+\frac{8535623}{2050466}$, $\frac{501625}{2050466}a^{15}-\frac{5960003}{10252330}a^{14}-\frac{139881}{84730}a^{13}+\frac{23692054}{5126165}a^{12}+\frac{12954001}{5126165}a^{11}-\frac{10780046}{1025233}a^{10}-\frac{2058981}{2050466}a^{9}+\frac{7069339}{2050466}a^{8}+\frac{40133939}{10252330}a^{7}+\frac{145805109}{10252330}a^{6}+\frac{8892899}{5126165}a^{5}-\frac{287814569}{10252330}a^{4}-\frac{36017517}{2050466}a^{3}+\frac{143230359}{10252330}a^{2}+\frac{77408731}{5126165}a+\frac{15977282}{5126165}$, $\frac{1410479}{1864060}a^{15}-\frac{796811}{372812}a^{14}-\frac{1605531}{372812}a^{13}+\frac{7673198}{466015}a^{12}+\frac{760093}{466015}a^{11}-\frac{3264152}{93203}a^{10}+\frac{3737649}{372812}a^{9}+\frac{17009143}{1864060}a^{8}+\frac{19224187}{1864060}a^{7}+\frac{431931}{10076}a^{6}-\frac{2601153}{186406}a^{5}-\frac{15191799}{169460}a^{4}-\frac{44263587}{1864060}a^{3}+\frac{9615503}{169460}a^{2}+\frac{3115973}{93203}a+\frac{1576367}{932030}$, $\frac{368491}{466015}a^{15}-\frac{1109177}{466015}a^{14}-\frac{1881019}{466015}a^{13}+\frac{1660526}{93203}a^{12}-\frac{729924}{466015}a^{11}-\frac{3292214}{93203}a^{10}+\frac{134169}{8473}a^{9}+\frac{2326997}{466015}a^{8}+\frac{5862974}{466015}a^{7}+\frac{19652641}{466015}a^{6}-\frac{10036868}{466015}a^{5}-\frac{41280032}{466015}a^{4}-\frac{5722943}{466015}a^{3}+\frac{27124133}{466015}a^{2}+\frac{11935808}{466015}a+\frac{349447}{466015}$, $\frac{22863091}{20504660}a^{15}-\frac{63623431}{20504660}a^{14}-\frac{12189717}{1864060}a^{13}+\frac{123875716}{5126165}a^{12}+\frac{19245728}{5126165}a^{11}-\frac{54119686}{1025233}a^{10}+\frac{52179925}{4100932}a^{9}+\frac{338417327}{20504660}a^{8}+\frac{288502671}{20504660}a^{7}+\frac{1320420063}{20504660}a^{6}-\frac{185156787}{10252330}a^{5}-\frac{2803441389}{20504660}a^{4}-\frac{814724363}{20504660}a^{3}+\frac{363122537}{4100932}a^{2}+\frac{264219251}{5126165}a+\frac{14828747}{10252330}$, $\frac{1809326}{5126165}a^{15}-\frac{3898254}{5126165}a^{14}-\frac{1258078}{466015}a^{13}+\frac{32591142}{5126165}a^{12}+\frac{31528524}{5126165}a^{11}-\frac{16587375}{1025233}a^{10}-\frac{7094049}{1025233}a^{9}+\frac{44783502}{5126165}a^{8}+\frac{7688462}{1025233}a^{7}+\frac{116844722}{5126165}a^{6}+\frac{42361499}{5126165}a^{5}-\frac{252415613}{5126165}a^{4}-\frac{200363458}{5126165}a^{3}+\frac{109942694}{5126165}a^{2}+\frac{170959276}{5126165}a+\frac{53807833}{5126165}$, $\frac{1446263}{5126165}a^{15}-\frac{101953}{138545}a^{14}-\frac{881637}{466015}a^{13}+\frac{6354233}{1025233}a^{12}+\frac{11440348}{5126165}a^{11}-\frac{15904443}{1025233}a^{10}+\frac{2968346}{1025233}a^{9}+\frac{37290326}{5126165}a^{8}+\frac{2480397}{5126165}a^{7}+\frac{99764463}{5126165}a^{6}-\frac{27304289}{5126165}a^{5}-\frac{189520026}{5126165}a^{4}-\frac{65405104}{5126165}a^{3}+\frac{139606884}{5126165}a^{2}+\frac{1928132}{138545}a-\frac{6692219}{5126165}$
| 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: | \( 248731.376036 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{5}\cdot 248731.376036 \cdot 1}{2\cdot\sqrt{10612039879318707392816}}\cr\approx \mathstrut & 0.756624597539 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 17*x^13 + 20*x^12 - 44*x^11 - 25*x^10 + 22*x^9 + 24*x^8 + 68*x^7 + 29*x^6 - 133*x^5 - 130*x^4 + 48*x^3 + 105*x^2 + 38*x + 2)
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 - 2*x^15 - 8*x^14 + 17*x^13 + 20*x^12 - 44*x^11 - 25*x^10 + 22*x^9 + 24*x^8 + 68*x^7 + 29*x^6 - 133*x^5 - 130*x^4 + 48*x^3 + 105*x^2 + 38*x + 2, 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 - 2*x^15 - 8*x^14 + 17*x^13 + 20*x^12 - 44*x^11 - 25*x^10 + 22*x^9 + 24*x^8 + 68*x^7 + 29*x^6 - 133*x^5 - 130*x^4 + 48*x^3 + 105*x^2 + 38*x + 2);
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 - 2*x^15 - 8*x^14 + 17*x^13 + 20*x^12 - 44*x^11 - 25*x^10 + 22*x^9 + 24*x^8 + 68*x^7 + 29*x^6 - 133*x^5 - 130*x^4 + 48*x^3 + 105*x^2 + 38*x + 2);
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
$C_4^4:\SL(2,3)$ (as 16T1672):
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 6144 |
| The 56 conjugacy class representatives for $C_4^4:\SL(2,3)$ |
| Character table for $C_4^4:\SL(2,3)$ |
Intermediate fields
| 4.4.26569.1, 8.6.7765029371.1 |
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
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.2.10612039879318707392816.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.12.0.1 | $x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(11\)
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.4.3.1 | $x^{4} + 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(163\)
| 163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 163.6.4.1 | $x^{6} + 477 x^{5} + 75849 x^{4} + 4021913 x^{3} + 229449 x^{2} + 12362361 x + 655078752$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 163.6.4.1 | $x^{6} + 477 x^{5} + 75849 x^{4} + 4021913 x^{3} + 229449 x^{2} + 12362361 x + 655078752$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |