Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 19*x^14 - 42*x^13 + 72*x^12 - 78*x^11 + 25*x^10 + 42*x^9 - 44*x^8 + 30*x^7 - 63*x^6 + 18*x^5 + 75*x^4 + 54*x^2 + 36)
gp: K = bnfinit(y^16 - 6*y^15 + 19*y^14 - 42*y^13 + 72*y^12 - 78*y^11 + 25*y^10 + 42*y^9 - 44*y^8 + 30*y^7 - 63*y^6 + 18*y^5 + 75*y^4 + 54*y^2 + 36, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 19*x^14 - 42*x^13 + 72*x^12 - 78*x^11 + 25*x^10 + 42*x^9 - 44*x^8 + 30*x^7 - 63*x^6 + 18*x^5 + 75*x^4 + 54*x^2 + 36);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 19*x^14 - 42*x^13 + 72*x^12 - 78*x^11 + 25*x^10 + 42*x^9 - 44*x^8 + 30*x^7 - 63*x^6 + 18*x^5 + 75*x^4 + 54*x^2 + 36)
\( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 72 x^{12} - 78 x^{11} + 25 x^{10} + 42 x^{9} - 44 x^{8} + \cdots + 36 \)
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: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{60}a^{12}-\frac{1}{5}a^{11}-\frac{7}{30}a^{10}-\frac{1}{10}a^{9}-\frac{1}{4}a^{8}+\frac{1}{5}a^{7}-\frac{2}{15}a^{6}+\frac{2}{5}a^{5}-\frac{29}{60}a^{4}-\frac{1}{2}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{60}a^{13}-\frac{2}{15}a^{11}+\frac{1}{10}a^{10}+\frac{1}{20}a^{9}+\frac{1}{5}a^{8}-\frac{7}{30}a^{7}-\frac{1}{5}a^{6}+\frac{19}{60}a^{5}-\frac{3}{10}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{60}a^{14}+\frac{11}{60}a^{10}-\frac{1}{10}a^{9}-\frac{7}{30}a^{8}-\frac{1}{10}a^{7}-\frac{1}{4}a^{6}-\frac{1}{10}a^{5}-\frac{1}{6}a^{4}-\frac{2}{5}a^{3}-\frac{1}{10}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{80336828340}a^{15}-\frac{572925}{77620124}a^{14}-\frac{100294997}{20084207085}a^{13}-\frac{17589571}{26778942780}a^{12}+\frac{472473209}{8926314260}a^{11}+\frac{1311620507}{8926314260}a^{10}-\frac{560544520}{4016841417}a^{9}+\frac{2196852241}{26778942780}a^{8}-\frac{4002857579}{80336828340}a^{7}-\frac{4854117907}{26778942780}a^{6}-\frac{959820444}{2231578565}a^{5}-\frac{3267404971}{26778942780}a^{4}-\frac{2294915356}{6694735695}a^{3}-\frac{105944353}{892631426}a^{2}+\frac{865563837}{2231578565}a+\frac{1603234}{2231578565}$
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
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: |
\( -\frac{3185850}{446315713} a^{15} + \frac{3243449}{116430186} a^{14} - \frac{49757887}{1338947139} a^{13} - \frac{87617969}{2677894278} a^{12} + \frac{345103691}{1338947139} a^{11} - \frac{363363785}{446315713} a^{10} + \frac{643770137}{446315713} a^{9} - \frac{2742687563}{2677894278} a^{8} - \frac{651671911}{1338947139} a^{7} + \frac{1397083097}{1338947139} a^{6} - \frac{311184223}{1338947139} a^{5} + \frac{380673089}{892631426} a^{4} - \frac{151506467}{446315713} a^{3} - \frac{1223953495}{892631426} a^{2} - \frac{13536530}{446315713} a - \frac{44064382}{446315713} \)
(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: |
$\frac{33797}{2237670}a^{15}-\frac{383}{4324}a^{14}+\frac{584459}{2237670}a^{13}-\frac{777311}{1491780}a^{12}+\frac{580291}{745890}a^{11}-\frac{829543}{1491780}a^{10}-\frac{1272427}{2237670}a^{9}+\frac{2184529}{1491780}a^{8}-\frac{1659529}{2237670}a^{7}-\frac{92507}{1491780}a^{6}-\frac{91693}{149178}a^{5}-\frac{256931}{1491780}a^{4}+\frac{890021}{372945}a^{3}-\frac{38982}{124315}a^{2}-\frac{1654}{124315}a+\frac{58087}{124315}$, $\frac{16399889}{8033682834}a^{15}-\frac{976405}{77620124}a^{14}+\frac{3339250421}{80336828340}a^{13}-\frac{111745021}{1338947139}a^{12}+\frac{1246318637}{13389471390}a^{11}+\frac{434569999}{8926314260}a^{10}-\frac{35462985337}{80336828340}a^{9}+\frac{12528776797}{13389471390}a^{8}-\frac{38170653887}{40168414170}a^{7}+\frac{318668431}{26778942780}a^{6}+\frac{30165767923}{26778942780}a^{5}-\frac{12397641313}{13389471390}a^{4}-\frac{4530413443}{13389471390}a^{3}+\frac{588983677}{4463157130}a^{2}+\frac{1976837503}{2231578565}a+\frac{2224619108}{2231578565}$, $\frac{213395183}{80336828340}a^{15}-\frac{5969561}{388100620}a^{14}+\frac{2380241969}{80336828340}a^{13}+\frac{38078009}{6694735695}a^{12}-\frac{874910717}{5355788556}a^{11}+\frac{14531133199}{26778942780}a^{10}-\frac{91435577491}{80336828340}a^{9}+\frac{8343708794}{6694735695}a^{8}-\frac{5057535583}{80336828340}a^{7}-\frac{6458611331}{5355788556}a^{6}+\frac{4301514403}{5355788556}a^{5}-\frac{1938785471}{13389471390}a^{4}+\frac{8747721113}{13389471390}a^{3}+\frac{1864698553}{4463157130}a^{2}-\frac{4135618844}{2231578565}a-\frac{90132523}{446315713}$, $\frac{49118263}{26778942780}a^{15}-\frac{3900113}{582150930}a^{14}-\frac{18619271}{5355788556}a^{13}+\frac{900503233}{13389471390}a^{12}-\frac{5452411379}{26778942780}a^{11}+\frac{1059176131}{2677894278}a^{10}-\frac{2486227945}{5355788556}a^{9}-\frac{2311274713}{13389471390}a^{8}+\frac{7726484389}{5355788556}a^{7}-\frac{10532521309}{6694735695}a^{6}-\frac{8794657043}{26778942780}a^{5}+\frac{10293881131}{6694735695}a^{4}-\frac{806754528}{2231578565}a^{3}+\frac{1626662007}{4463157130}a^{2}-\frac{1452405042}{2231578565}a+\frac{820898179}{2231578565}$, $\frac{237119059}{26778942780}a^{15}-\frac{14254993}{232860372}a^{14}+\frac{2718656221}{13389471390}a^{13}-\frac{3089837584}{6694735695}a^{12}+\frac{4378440275}{5355788556}a^{11}-\frac{9038777731}{8926314260}a^{10}+\frac{3601263814}{6694735695}a^{9}+\frac{165533098}{1338947139}a^{8}-\frac{137658479}{5355788556}a^{7}-\frac{132763331}{5355788556}a^{6}-\frac{9360163124}{6694735695}a^{5}+\frac{5715501577}{4463157130}a^{4}+\frac{378832445}{446315713}a^{3}-\frac{562355127}{2231578565}a^{2}-\frac{1954213874}{2231578565}a-\frac{2545959899}{2231578565}$, $\frac{1457854423}{80336828340}a^{15}-\frac{10388089}{97025155}a^{14}+\frac{5343406319}{16067365668}a^{13}-\frac{4871860058}{6694735695}a^{12}+\frac{6711331057}{5355788556}a^{11}-\frac{18446269001}{13389471390}a^{10}+\frac{44932062661}{80336828340}a^{9}+\frac{363643184}{1338947139}a^{8}+\frac{935013485}{16067365668}a^{7}-\frac{639959633}{2677894278}a^{6}-\frac{15746014069}{26778942780}a^{5}-\frac{1227925349}{6694735695}a^{4}+\frac{1890686080}{1338947139}a^{3}+\frac{2826088779}{4463157130}a^{2}+\frac{2646629996}{2231578565}a+\frac{31115627}{2231578565}$, $\frac{107321201}{20084207085}a^{15}-\frac{2599889}{97025155}a^{14}+\frac{5872635821}{80336828340}a^{13}-\frac{320643796}{2231578565}a^{12}+\frac{1509060319}{6694735695}a^{11}-\frac{1199883118}{6694735695}a^{10}-\frac{155765123}{16067365668}a^{9}-\frac{60515309}{13389471390}a^{8}+\frac{10792843993}{40168414170}a^{7}-\frac{1228646861}{4463157130}a^{6}-\frac{273004991}{26778942780}a^{5}-\frac{141862213}{892631426}a^{4}-\frac{2957337739}{13389471390}a^{3}+\frac{840420927}{2231578565}a^{2}-\frac{465377909}{2231578565}a-\frac{55197101}{446315713}$
| 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: | \( 2681.6222586 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2681.6222586 \cdot 1}{6\cdot\sqrt{3482851737600000000}}\cr\approx \mathstrut & 0.58172510229 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 19*x^14 - 42*x^13 + 72*x^12 - 78*x^11 + 25*x^10 + 42*x^9 - 44*x^8 + 30*x^7 - 63*x^6 + 18*x^5 + 75*x^4 + 54*x^2 + 36)
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 - 6*x^15 + 19*x^14 - 42*x^13 + 72*x^12 - 78*x^11 + 25*x^10 + 42*x^9 - 44*x^8 + 30*x^7 - 63*x^6 + 18*x^5 + 75*x^4 + 54*x^2 + 36, 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 - 6*x^15 + 19*x^14 - 42*x^13 + 72*x^12 - 78*x^11 + 25*x^10 + 42*x^9 - 44*x^8 + 30*x^7 - 63*x^6 + 18*x^5 + 75*x^4 + 54*x^2 + 36);
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 - 6*x^15 + 19*x^14 - 42*x^13 + 72*x^12 - 78*x^11 + 25*x^10 + 42*x^9 - 44*x^8 + 30*x^7 - 63*x^6 + 18*x^5 + 75*x^4 + 54*x^2 + 36);
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
$C_2\times D_4$ (as 16T9):
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 16 |
| The 10 conjugacy class representatives for $D_4\times C_2$ |
| Character table for $D_4\times C_2$ |
Intermediate fields
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 8 siblings: | 8.4.1866240000.2, 8.0.29160000.1, 8.0.74649600.1, 8.0.1866240000.7 |
| Minimal sibling: | 8.0.29160000.1 |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
|
\(3\)
| 3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |
| 3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |