Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 3*x^13 + 3*x^12 - x^11 + 6*x^10 + 13*x^9 + 8*x^8 + 19*x^7 + 4*x^5 + 12*x^4 - 17*x^3 + 12*x^2 - x + 1)
gp: K = bnfinit(y^16 + 2*y^14 - 3*y^13 + 3*y^12 - y^11 + 6*y^10 + 13*y^9 + 8*y^8 + 19*y^7 + 4*y^5 + 12*y^4 - 17*y^3 + 12*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 2*x^14 - 3*x^13 + 3*x^12 - x^11 + 6*x^10 + 13*x^9 + 8*x^8 + 19*x^7 + 4*x^5 + 12*x^4 - 17*x^3 + 12*x^2 - x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 2*x^14 - 3*x^13 + 3*x^12 - x^11 + 6*x^10 + 13*x^9 + 8*x^8 + 19*x^7 + 4*x^5 + 12*x^4 - 17*x^3 + 12*x^2 - x + 1)
\( x^{16} + 2 x^{14} - 3 x^{13} + 3 x^{12} - x^{11} + 6 x^{10} + 13 x^{9} + 8 x^{8} + 19 x^{7} + 4 x^{5} + \cdots + 1 \)
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: |
\(311829406284765625\)
\(\medspace = 5^{8}\cdot 61^{2}\cdot 97^{2}\cdot 151^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.40\) | 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: | $5^{1/2}61^{1/2}97^{1/2}151^{1/2}\approx 2113.6071063468726$ | ||
| Ramified primes: |
\(5\), \(61\), \(97\), \(151\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13204342879}a^{15}+\frac{426218564}{13204342879}a^{14}+\frac{4785173784}{13204342879}a^{13}-\frac{1740421044}{13204342879}a^{12}-\frac{46594851}{145102669}a^{11}+\frac{6384881411}{13204342879}a^{10}-\frac{1295086740}{13204342879}a^{9}-\frac{727618934}{1886334697}a^{8}-\frac{1860190520}{13204342879}a^{7}-\frac{278001824}{1886334697}a^{6}+\frac{445311679}{1886334697}a^{5}+\frac{5467659082}{13204342879}a^{4}+\frac{4741944437}{13204342879}a^{3}+\frac{3251137528}{13204342879}a^{2}+\frac{492357784}{13204342879}a+\frac{1406077067}{13204342879}$
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: | $7$ | 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: |
$\frac{2046541278}{13204342879}a^{15}-\frac{1464904067}{13204342879}a^{14}+\frac{3534843860}{13204342879}a^{13}-\frac{9242507712}{13204342879}a^{12}+\frac{102299823}{145102669}a^{11}-\frac{5282146287}{13204342879}a^{10}+\frac{13010375328}{13204342879}a^{9}+\frac{2568156611}{1886334697}a^{8}-\frac{4941030873}{13204342879}a^{7}+\frac{2395017490}{1886334697}a^{6}-\frac{5180744906}{1886334697}a^{5}-\frac{4498096170}{13204342879}a^{4}+\frac{15880118116}{13204342879}a^{3}-\frac{44921199124}{13204342879}a^{2}+\frac{45876334808}{13204342879}a-\frac{9653123990}{13204342879}$, $\frac{105526350}{1886334697}a^{15}+\frac{202237044}{1886334697}a^{14}+\frac{116447195}{1886334697}a^{13}+\frac{78278890}{1886334697}a^{12}-\frac{30756953}{145102669}a^{11}+\frac{712064957}{1886334697}a^{10}+\frac{292982983}{1886334697}a^{9}+\frac{2280552153}{1886334697}a^{8}+\frac{3034772660}{1886334697}a^{7}+\frac{2376990318}{1886334697}a^{6}+\frac{3578997518}{1886334697}a^{5}-\frac{735022625}{1886334697}a^{4}+\frac{1292490665}{1886334697}a^{3}-\frac{193581348}{1886334697}a^{2}-\frac{3925546562}{1886334697}a+\frac{2646519404}{1886334697}$, $\frac{195079170}{1015718683}a^{15}+\frac{3495126}{1015718683}a^{14}+\frac{419954382}{1015718683}a^{13}-\frac{515228772}{1015718683}a^{12}+\frac{91566464}{145102669}a^{11}-\frac{163839703}{1015718683}a^{10}+\frac{1048759078}{1015718683}a^{9}+\frac{371129408}{145102669}a^{8}+\frac{1688783256}{1015718683}a^{7}+\frac{646417164}{145102669}a^{6}+\frac{179725552}{145102669}a^{5}+\frac{1865179704}{1015718683}a^{4}+\frac{2869494195}{1015718683}a^{3}-\frac{4059747575}{1015718683}a^{2}+\frac{2242374230}{1015718683}a-\frac{641468012}{1015718683}$, $\frac{420209068}{13204342879}a^{15}-\frac{1947327300}{13204342879}a^{14}-\frac{511073310}{13204342879}a^{13}-\frac{5297610488}{13204342879}a^{12}+\frac{55571216}{145102669}a^{11}-\frac{1613966178}{13204342879}a^{10}+\frac{2141010462}{13204342879}a^{9}-\frac{995729181}{1886334697}a^{8}-\frac{30751224904}{13204342879}a^{7}-\frac{3665782031}{1886334697}a^{6}-\frac{5869012549}{1886334697}a^{5}-\frac{12164137220}{13204342879}a^{4}+\frac{8826562942}{13204342879}a^{3}-\frac{23580124556}{13204342879}a^{2}+\frac{21629229403}{13204342879}a+\frac{1015701600}{13204342879}$, $\frac{970264962}{13204342879}a^{15}-\frac{807557614}{13204342879}a^{14}+\frac{3068616906}{13204342879}a^{13}-\frac{3260492084}{13204342879}a^{12}+\frac{87237387}{145102669}a^{11}-\frac{4490375010}{13204342879}a^{10}+\frac{6903779380}{13204342879}a^{9}+\frac{1342451670}{1886334697}a^{8}+\frac{4456310133}{13204342879}a^{7}+\frac{4813504296}{1886334697}a^{6}+\frac{1651036415}{1886334697}a^{5}+\frac{38274820228}{13204342879}a^{4}+\frac{34016778325}{13204342879}a^{3}-\frac{11607661733}{13204342879}a^{2}+\frac{41571598702}{13204342879}a-\frac{6904558914}{13204342879}$, $a$, $\frac{94876911}{13204342879}a^{15}+\frac{696238304}{13204342879}a^{14}+\frac{2410191081}{13204342879}a^{13}+\frac{1805731822}{13204342879}a^{12}+\frac{12296654}{145102669}a^{11}-\frac{4803799493}{13204342879}a^{10}+\frac{1961532858}{13204342879}a^{9}+\frac{1221443551}{1886334697}a^{8}+\frac{24081230127}{13204342879}a^{7}+\frac{5423856083}{1886334697}a^{6}+\frac{3987244241}{1886334697}a^{5}+\frac{22073190046}{13204342879}a^{4}-\frac{10738698330}{13204342879}a^{3}-\frac{419831336}{13204342879}a^{2}+\frac{12128086312}{13204342879}a-\frac{6417866824}{13204342879}$
| 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: | \( 90.9680140734 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 90.9680140734 \cdot 1}{2\cdot\sqrt{311829406284765625}}\cr\approx \mathstrut & 0.197851401508 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 3*x^13 + 3*x^12 - x^11 + 6*x^10 + 13*x^9 + 8*x^8 + 19*x^7 + 4*x^5 + 12*x^4 - 17*x^3 + 12*x^2 - 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 + 2*x^14 - 3*x^13 + 3*x^12 - x^11 + 6*x^10 + 13*x^9 + 8*x^8 + 19*x^7 + 4*x^5 + 12*x^4 - 17*x^3 + 12*x^2 - 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 + 2*x^14 - 3*x^13 + 3*x^12 - x^11 + 6*x^10 + 13*x^9 + 8*x^8 + 19*x^7 + 4*x^5 + 12*x^4 - 17*x^3 + 12*x^2 - 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 + 2*x^14 - 3*x^13 + 3*x^12 - x^11 + 6*x^10 + 13*x^9 + 8*x^8 + 19*x^7 + 4*x^5 + 12*x^4 - 17*x^3 + 12*x^2 - 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
$C_2^6.S_4^2:C_2^2$ (as 16T1884):
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 147456 |
| The 136 conjugacy class representatives for $C_2^6.S_4^2:C_2^2$ |
| Character table for $C_2^6.S_4^2:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 8.2.5756875.1 |
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: | 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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(61\)
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(97\)
| 97.4.2.1 | $x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 97.6.0.1 | $x^{6} + 92 x^{3} + 58 x^{2} + 88 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 97.6.0.1 | $x^{6} + 92 x^{3} + 58 x^{2} + 88 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(151\)
| 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.4.2.1 | $x^{4} + 298 x^{3} + 22515 x^{2} + 46786 x + 3373376$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |