Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 42*x^13 + 30*x^12 + 2*x^11 - 50*x^10 + 184*x^9 - 140*x^8 - 522*x^7 + 1272*x^6 - 1220*x^5 + 621*x^4 - 186*x^3 + 40*x^2 - 8*x + 1)
gp: K = bnfinit(y^16 - 8*y^15 + 26*y^14 - 42*y^13 + 30*y^12 + 2*y^11 - 50*y^10 + 184*y^9 - 140*y^8 - 522*y^7 + 1272*y^6 - 1220*y^5 + 621*y^4 - 186*y^3 + 40*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 26*x^14 - 42*x^13 + 30*x^12 + 2*x^11 - 50*x^10 + 184*x^9 - 140*x^8 - 522*x^7 + 1272*x^6 - 1220*x^5 + 621*x^4 - 186*x^3 + 40*x^2 - 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 26*x^14 - 42*x^13 + 30*x^12 + 2*x^11 - 50*x^10 + 184*x^9 - 140*x^8 - 522*x^7 + 1272*x^6 - 1220*x^5 + 621*x^4 - 186*x^3 + 40*x^2 - 8*x + 1)
\( x^{16} - 8 x^{15} + 26 x^{14} - 42 x^{13} + 30 x^{12} + 2 x^{11} - 50 x^{10} + 184 x^{9} - 140 x^{8} + \cdots + 1 \)
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: |
\(3482851737600000000\)
\(\medspace = 2^{24}\cdot 3^{12}\cdot 5^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.42\) | 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: | $2^{3/2}3^{3/4}5^{1/2}\approx 14.416868484808525$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13}a^{11}+\frac{1}{13}a^{10}-\frac{2}{13}a^{9}-\frac{5}{13}a^{8}+\frac{2}{13}a^{7}+\frac{6}{13}a^{6}+\frac{6}{13}a^{5}+\frac{1}{13}a^{4}-\frac{2}{13}a^{3}-\frac{1}{13}a^{2}+\frac{3}{13}$, $\frac{1}{13}a^{12}-\frac{3}{13}a^{10}-\frac{3}{13}a^{9}-\frac{6}{13}a^{8}+\frac{4}{13}a^{7}-\frac{5}{13}a^{5}-\frac{3}{13}a^{4}+\frac{1}{13}a^{3}+\frac{1}{13}a^{2}+\frac{3}{13}a-\frac{3}{13}$, $\frac{1}{13}a^{13}+\frac{1}{13}a^{9}+\frac{2}{13}a^{8}+\frac{6}{13}a^{7}+\frac{2}{13}a^{5}+\frac{4}{13}a^{4}-\frac{5}{13}a^{3}-\frac{3}{13}a-\frac{4}{13}$, $\frac{1}{46787}a^{14}-\frac{7}{46787}a^{13}+\frac{406}{46787}a^{12}+\frac{1254}{46787}a^{11}+\frac{50}{46787}a^{10}-\frac{1147}{3599}a^{9}-\frac{20804}{46787}a^{8}-\frac{7584}{46787}a^{7}-\frac{15107}{46787}a^{6}+\frac{22137}{46787}a^{5}+\frac{18177}{46787}a^{4}+\frac{1412}{3599}a^{3}-\frac{22158}{46787}a^{2}-\frac{108}{3599}a-\frac{93}{46787}$, $\frac{1}{46787}a^{15}+\frac{357}{46787}a^{13}+\frac{497}{46787}a^{12}+\frac{1630}{46787}a^{11}-\frac{10962}{46787}a^{10}-\frac{6414}{46787}a^{9}-\frac{158}{3599}a^{8}-\frac{3413}{46787}a^{7}+\frac{13561}{46787}a^{6}+\frac{7582}{46787}a^{5}+\frac{8833}{46787}a^{4}-\frac{23230}{46787}a^{3}-\frac{12550}{46787}a^{2}-\frac{20718}{46787}a-\frac{11448}{46787}$
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
$C_{2}$, which has order $2$
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: |
\( \frac{97369}{46787} a^{14} - \frac{681583}{46787} a^{13} + \frac{1828690}{46787} a^{12} - \frac{2111561}{46787} a^{11} + \frac{409289}{46787} a^{10} + \frac{1065136}{46787} a^{9} - \frac{3890436}{46787} a^{8} + \frac{13791291}{46787} a^{7} + \frac{1008725}{46787} a^{6} - \frac{52834862}{46787} a^{5} + \frac{70792012}{46787} a^{4} - \frac{36412307}{46787} a^{3} + \frac{8595437}{46787} a^{2} - \frac{1657200}{46787} a + \frac{395657}{46787} \)
(order $12$)
| 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: |
$\frac{1369179}{46787}a^{15}-\frac{10155687}{46787}a^{14}+\frac{29674834}{46787}a^{13}-\frac{40171176}{46787}a^{12}+\frac{17558952}{46787}a^{11}+\frac{13079219}{46787}a^{10}-\frac{4679605}{3599}a^{9}+\frac{216414834}{46787}a^{8}-\frac{65345389}{46787}a^{7}-\frac{753649359}{46787}a^{6}+\frac{1302138334}{46787}a^{5}-\frac{908242970}{46787}a^{4}+\frac{317329716}{46787}a^{3}-\frac{1120212}{767}a^{2}+\frac{14755530}{46787}a-\frac{2364015}{46787}$, $\frac{302450}{46787}a^{15}-\frac{2256717}{46787}a^{14}+\frac{510109}{3599}a^{13}-\frac{9010960}{46787}a^{12}+\frac{3888774}{46787}a^{11}+\frac{3101820}{46787}a^{10}-\frac{13561766}{46787}a^{9}+\frac{48168712}{46787}a^{8}-\frac{15708305}{46787}a^{7}-\frac{168817002}{46787}a^{6}+\frac{293899995}{46787}a^{5}-\frac{3302372}{767}a^{4}+\frac{66000778}{46787}a^{3}-\frac{13124482}{46787}a^{2}+\frac{2831662}{46787}a-\frac{394153}{46787}$, $\frac{785166}{46787}a^{15}-\frac{5801996}{46787}a^{14}+\frac{16874714}{46787}a^{13}-\frac{1745620}{3599}a^{12}+\frac{9750901}{46787}a^{11}+\frac{7466333}{46787}a^{10}-\frac{34699397}{46787}a^{9}+\frac{123334652}{46787}a^{8}-\frac{34730564}{46787}a^{7}-\frac{430699135}{46787}a^{6}+\frac{735767876}{46787}a^{5}-\frac{510274589}{46787}a^{4}+\frac{178152681}{46787}a^{3}-\frac{37917325}{46787}a^{2}+\frac{7958506}{46787}a-\frac{1259985}{46787}$, $\frac{1136367}{46787}a^{15}-\frac{8398977}{46787}a^{14}+\frac{24404534}{46787}a^{13}-\frac{2513703}{3599}a^{12}+\frac{13680259}{46787}a^{11}+\frac{11200700}{46787}a^{10}-\frac{50110995}{46787}a^{9}+\frac{178259860}{46787}a^{8}-\frac{49468935}{46787}a^{7}-\frac{627141532}{46787}a^{6}+\frac{1063484571}{46787}a^{5}-\frac{723933510}{46787}a^{4}+\frac{244645891}{46787}a^{3}-\frac{52372667}{46787}a^{2}+\frac{11592283}{46787}a-\frac{1631319}{46787}$, $\frac{76129}{46787}a^{14}-\frac{532903}{46787}a^{13}+\frac{1425366}{46787}a^{12}-\frac{1624457}{46787}a^{11}+\frac{265034}{46787}a^{10}+\frac{864831}{46787}a^{9}-\frac{3009743}{46787}a^{8}+\frac{10753853}{46787}a^{7}+\frac{929783}{46787}a^{6}-\frac{41788056}{46787}a^{5}+\frac{54779707}{46787}a^{4}-\frac{26484436}{46787}a^{3}+\frac{5653343}{46787}a^{2}-\frac{1308451}{46787}a+\frac{359136}{46787}$, $\frac{155652}{46787}a^{15}-\frac{1197713}{46787}a^{14}+\frac{3682901}{46787}a^{13}-\frac{5424945}{46787}a^{12}+\frac{3048587}{46787}a^{11}+\frac{1196462}{46787}a^{10}-\frac{7406485}{46787}a^{9}+\frac{2031419}{3599}a^{8}-\frac{1061598}{3599}a^{7}-\frac{85212099}{46787}a^{6}+\frac{13221135}{3599}a^{5}-\frac{10647857}{3599}a^{4}+\frac{55985287}{46787}a^{3}-\frac{12586048}{46787}a^{2}+\frac{2410258}{46787}a-\frac{426030}{46787}$, $\frac{594590}{46787}a^{15}-\frac{4459425}{46787}a^{14}+\frac{13220585}{46787}a^{13}-\frac{1407630}{3599}a^{12}+\frac{8518677}{46787}a^{11}+\frac{5642219}{46787}a^{10}-\frac{26941232}{46787}a^{9}+\frac{95831154}{46787}a^{8}-\frac{34966683}{46787}a^{7}-\frac{329139391}{46787}a^{6}+\frac{591642218}{46787}a^{5}-\frac{424625709}{46787}a^{4}+\frac{150396158}{46787}a^{3}-\frac{32158410}{46787}a^{2}+\frac{7017173}{46787}a-\frac{1136367}{46787}$
| 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: | \( 1368.08464147 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1368.08464147 \cdot 2}{12\cdot\sqrt{3482851737600000000}}\cr\approx \mathstrut & 0.296779002131 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 42*x^13 + 30*x^12 + 2*x^11 - 50*x^10 + 184*x^9 - 140*x^8 - 522*x^7 + 1272*x^6 - 1220*x^5 + 621*x^4 - 186*x^3 + 40*x^2 - 8*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^16 - 8*x^15 + 26*x^14 - 42*x^13 + 30*x^12 + 2*x^11 - 50*x^10 + 184*x^9 - 140*x^8 - 522*x^7 + 1272*x^6 - 1220*x^5 + 621*x^4 - 186*x^3 + 40*x^2 - 8*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^16 - 8*x^15 + 26*x^14 - 42*x^13 + 30*x^12 + 2*x^11 - 50*x^10 + 184*x^9 - 140*x^8 - 522*x^7 + 1272*x^6 - 1220*x^5 + 621*x^4 - 186*x^3 + 40*x^2 - 8*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^16 - 8*x^15 + 26*x^14 - 42*x^13 + 30*x^12 + 2*x^11 - 50*x^10 + 184*x^9 - 140*x^8 - 522*x^7 + 1272*x^6 - 1220*x^5 + 621*x^4 - 186*x^3 + 40*x^2 - 8*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 32 |
| The 14 conjugacy class representatives for $C_4:D_4$ |
| Character table for $C_4:D_4$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.0.2880.1, 4.0.320.1, 4.0.10800.2 x2, 4.2.43200.3 x2, \(\Q(\zeta_{12})\), 8.0.1866240000.14, 8.0.8294400.2, 8.0.4665600.1 |
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
| Galois closure: | deg 32 |
| Degree 16 siblings: | 16.0.217678233600000000.2, 16.0.5572562780160000.1, 16.4.3482851737600000000.1 |
| Minimal sibling: | 16.0.217678233600000000.2 |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $24$ | |||
|
\(3\)
| 3.16.12.1 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 36 x^{12} + 120 x^{11} + 312 x^{10} + 352 x^{9} - 522 x^{8} - 2664 x^{7} - 3672 x^{6} - 1440 x^{5} + 4292 x^{4} + 7720 x^{3} + 6408 x^{2} + 2592 x + 433$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $[\ ]_{4}^{4}$ |
|
\(5\)
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |