Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 74*x^13 + 88*x^12 + 52*x^11 - 494*x^10 + 1356*x^9 - 2844*x^8 + 5424*x^7 - 7904*x^6 + 3328*x^5 + 22528*x^4 - 75776*x^3 + 131072*x^2 - 131072*x + 65536)
gp: K = bnfinit(y^16 - 8*y^15 + 32*y^14 - 74*y^13 + 88*y^12 + 52*y^11 - 494*y^10 + 1356*y^9 - 2844*y^8 + 5424*y^7 - 7904*y^6 + 3328*y^5 + 22528*y^4 - 75776*y^3 + 131072*y^2 - 131072*y + 65536, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 32*x^14 - 74*x^13 + 88*x^12 + 52*x^11 - 494*x^10 + 1356*x^9 - 2844*x^8 + 5424*x^7 - 7904*x^6 + 3328*x^5 + 22528*x^4 - 75776*x^3 + 131072*x^2 - 131072*x + 65536);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 32*x^14 - 74*x^13 + 88*x^12 + 52*x^11 - 494*x^10 + 1356*x^9 - 2844*x^8 + 5424*x^7 - 7904*x^6 + 3328*x^5 + 22528*x^4 - 75776*x^3 + 131072*x^2 - 131072*x + 65536)
\( x^{16} - 8 x^{15} + 32 x^{14} - 74 x^{13} + 88 x^{12} + 52 x^{11} - 494 x^{10} + 1356 x^{9} + \cdots + 65536 \)
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: |
\(35257923508597298719358976\)
\(\medspace = 2^{20}\cdot 3^{8}\cdot 8461^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(39.51\) | 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: | $2^{4/3}3^{1/2}8461^{1/2}\approx 401.46233149379685$ | ||
| Ramified primes: |
\(2\), \(3\), \(8461\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{10}+\frac{3}{16}a^{7}-\frac{1}{4}a^{6}-\frac{3}{8}a^{5}-\frac{7}{16}a^{4}-\frac{1}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{11}+\frac{3}{32}a^{8}+\frac{1}{8}a^{7}-\frac{3}{16}a^{6}+\frac{9}{32}a^{5}+\frac{7}{16}a^{4}+\frac{1}{16}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{12}-\frac{5}{128}a^{9}+\frac{1}{32}a^{8}+\frac{13}{64}a^{7}+\frac{25}{128}a^{6}+\frac{23}{64}a^{5}-\frac{31}{64}a^{4}+\frac{5}{16}a^{3}+\frac{1}{8}a^{2}$, $\frac{1}{1024}a^{13}-\frac{5}{512}a^{10}+\frac{1}{128}a^{9}-\frac{19}{256}a^{8}+\frac{25}{512}a^{7}+\frac{23}{256}a^{6}-\frac{95}{256}a^{5}-\frac{27}{64}a^{4}-\frac{15}{32}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{3633152}a^{14}+\frac{305}{908288}a^{13}+\frac{35}{28384}a^{12}-\frac{12101}{1816576}a^{11}+\frac{267}{113536}a^{10}+\frac{23477}{908288}a^{9}-\frac{204415}{1816576}a^{8}-\frac{65495}{908288}a^{7}-\frac{156035}{908288}a^{6}+\frac{31875}{113536}a^{5}+\frac{28619}{113536}a^{4}-\frac{8489}{28384}a^{3}-\frac{421}{1774}a^{2}-\frac{221}{1774}a+\frac{4}{887}$, $\frac{1}{1235271680}a^{15}+\frac{3}{154408960}a^{14}-\frac{2217}{15440896}a^{13}-\frac{533957}{617635840}a^{12}-\frac{59545}{30881792}a^{11}-\frac{2658907}{308817920}a^{10}-\frac{6753547}{123527168}a^{9}-\frac{23123021}{308817920}a^{8}+\frac{50090017}{308817920}a^{7}-\frac{1591293}{15440896}a^{6}+\frac{3676723}{38602240}a^{5}-\frac{393543}{965056}a^{4}+\frac{430789}{2412640}a^{3}+\frac{5429}{30158}a^{2}-\frac{58099}{150790}a-\frac{3857}{75395}$
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
$C_{2}\times C_{60}$, which has order $120$ (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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{113853}{617635840} a^{15} + \frac{11319}{19301120} a^{14} - \frac{12439}{15440896} a^{13} - \frac{527919}{308817920} a^{12} + \frac{113519}{15440896} a^{11} - \frac{2164029}{154408960} a^{10} + \frac{630511}{61763584} a^{9} - \frac{1004227}{154408960} a^{8} + \frac{2135959}{154408960} a^{7} + \frac{203777}{7720448} a^{6} - \frac{921543}{2412640} a^{5} + \frac{959917}{965056} a^{4} - \frac{3115679}{2412640} a^{3} - \frac{40607}{60316} a^{2} + \frac{638379}{150790} a - \frac{459713}{75395} \)
(order $4$)
| 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: |
$\frac{183483}{308817920}a^{15}+\frac{1134029}{154408960}a^{14}-\frac{466339}{15440896}a^{13}+\frac{10404671}{154408960}a^{12}-\frac{960235}{15440896}a^{11}-\frac{6925909}{77204480}a^{10}+\frac{15379661}{30881792}a^{9}-\frac{22000493}{19301120}a^{8}+\frac{175411709}{77204480}a^{7}-\frac{34138875}{7720448}a^{6}+\frac{123836257}{19301120}a^{5}-\frac{289787}{965056}a^{4}-\frac{60505673}{2412640}a^{3}+\frac{8654287}{120632}a^{2}-\frac{7611166}{75395}a+\frac{5344869}{75395}$, $\frac{91}{16384}a^{15}+\frac{137}{4096}a^{14}-\frac{103}{1024}a^{13}+\frac{1287}{8192}a^{12}-\frac{9}{256}a^{11}-\frac{2143}{4096}a^{10}+\frac{13189}{8192}a^{9}-\frac{13979}{4096}a^{8}+\frac{27913}{4096}a^{7}-\frac{6233}{512}a^{6}+\frac{5585}{512}a^{5}+\frac{2313}{128}a^{4}-\frac{3061}{32}a^{3}+\frac{379}{2}a^{2}-207a+93$, $\frac{3220307}{1235271680}a^{15}+\frac{5631523}{308817920}a^{14}-\frac{458781}{7720448}a^{13}+\frac{63165919}{617635840}a^{12}-\frac{705661}{15440896}a^{11}-\frac{85407471}{308817920}a^{10}+\frac{118349369}{123527168}a^{9}-\frac{632125503}{308817920}a^{8}+\frac{1263472721}{308817920}a^{7}-\frac{29225301}{3860224}a^{6}+\frac{309311319}{38602240}a^{5}+\frac{15174361}{1930112}a^{4}-\frac{65937419}{1206320}a^{3}+\frac{7197955}{60316}a^{2}-\frac{10554516}{75395}a+\frac{5419929}{75395}$, $\frac{12415361}{1235271680}a^{15}-\frac{9864827}{154408960}a^{14}+\frac{1500433}{7720448}a^{13}-\frac{189611717}{617635840}a^{12}+\frac{2359075}{30881792}a^{11}+\frac{308935853}{308817920}a^{10}-\frac{383993515}{123527168}a^{9}+\frac{2015710379}{308817920}a^{8}-\frac{4050628343}{308817920}a^{7}+\frac{364955595}{15440896}a^{6}-\frac{836905207}{38602240}a^{5}-\frac{16409909}{482528}a^{4}+\frac{13885777}{75395}a^{3}-\frac{22254101}{60316}a^{2}+\frac{60335541}{150790}a-\frac{13452037}{75395}$, $\frac{981511}{617635840}a^{15}+\frac{2995973}{308817920}a^{14}-\frac{224645}{7720448}a^{13}+\frac{14251827}{308817920}a^{12}-\frac{401851}{30881792}a^{11}-\frac{23095423}{154408960}a^{10}+\frac{29217841}{61763584}a^{9}-\frac{38440301}{38602240}a^{8}+\frac{307928983}{154408960}a^{7}-\frac{55214877}{15440896}a^{6}+\frac{15546563}{4825280}a^{5}+\frac{9607867}{1930112}a^{4}-\frac{33250019}{1206320}a^{3}+\frac{6751683}{120632}a^{2}-\frac{4697331}{75395}a+\frac{2081569}{75395}$, $\frac{271239}{154408960}a^{15}+\frac{3269673}{308817920}a^{14}-\frac{509249}{15440896}a^{13}+\frac{4070683}{77204480}a^{12}-\frac{528705}{30881792}a^{11}-\frac{6360377}{38602240}a^{10}+\frac{4067147}{7720448}a^{9}-\frac{173595019}{154408960}a^{8}+\frac{172918599}{77204480}a^{7}-\frac{62918767}{15440896}a^{6}+\frac{18958633}{4825280}a^{5}+\frac{9933927}{1930112}a^{4}-\frac{75012363}{2412640}a^{3}+\frac{955525}{15079}a^{2}-\frac{11019947}{150790}a+\frac{2556549}{75395}$, $\frac{656331}{308817920}a^{15}+\frac{3631881}{308817920}a^{14}-\frac{534127}{15440896}a^{13}+\frac{7613447}{154408960}a^{12}-\frac{74629}{30881792}a^{11}-\frac{14884783}{77204480}a^{10}+\frac{16899099}{30881792}a^{9}-\frac{175603473}{154408960}a^{8}+\frac{87527279}{38602240}a^{7}-\frac{62103659}{15440896}a^{6}+\frac{15272461}{4825280}a^{5}+\frac{13773595}{1930112}a^{4}-\frac{81243721}{2412640}a^{3}+\frac{1870443}{30158}a^{2}-\frac{9743189}{150790}a+\frac{1637123}{75395}$
| 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: | \( 797128.635518811 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 797128.635518811 \cdot 120}{4\cdot\sqrt{35257923508597298719358976}}\cr\approx \mathstrut & 9.78273404040191 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 74*x^13 + 88*x^12 + 52*x^11 - 494*x^10 + 1356*x^9 - 2844*x^8 + 5424*x^7 - 7904*x^6 + 3328*x^5 + 22528*x^4 - 75776*x^3 + 131072*x^2 - 131072*x + 65536)
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 - 8*x^15 + 32*x^14 - 74*x^13 + 88*x^12 + 52*x^11 - 494*x^10 + 1356*x^9 - 2844*x^8 + 5424*x^7 - 7904*x^6 + 3328*x^5 + 22528*x^4 - 75776*x^3 + 131072*x^2 - 131072*x + 65536, 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 - 8*x^15 + 32*x^14 - 74*x^13 + 88*x^12 + 52*x^11 - 494*x^10 + 1356*x^9 - 2844*x^8 + 5424*x^7 - 7904*x^6 + 3328*x^5 + 22528*x^4 - 75776*x^3 + 131072*x^2 - 131072*x + 65536);
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 - 8*x^15 + 32*x^14 - 74*x^13 + 88*x^12 + 52*x^11 - 494*x^10 + 1356*x^9 - 2844*x^8 + 5424*x^7 - 7904*x^6 + 3328*x^5 + 22528*x^4 - 75776*x^3 + 131072*x^2 - 131072*x + 65536);
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
$C_2^4:S_4$ (as 16T747):
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 384 |
| The 26 conjugacy class representatives for $C_2^4:S_4$ |
| Character table for $C_2^4:S_4$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 4.4.304596.1, 8.0.92778723216.1, 8.8.5937838285824.4, 8.0.5937838285824.3 |
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: | 16.0.435283006278978996535296.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.3.0.1}{3} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.6.8.1 | $x^{6} + 2 x^{3} + 2$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ | |
| 2.6.8.1 | $x^{6} + 2 x^{3} + 2$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ | |
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(8461\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |