Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 19*x^14 - 42*x^13 + 83*x^12 - 156*x^11 + 249*x^10 - 300*x^9 + 260*x^8 - 114*x^7 - 9*x^6 + 78*x^5 - 37*x^4 + 12*x^3 + 13*x^2 + 1)
gp: K = bnfinit(y^16 - 6*y^15 + 19*y^14 - 42*y^13 + 83*y^12 - 156*y^11 + 249*y^10 - 300*y^9 + 260*y^8 - 114*y^7 - 9*y^6 + 78*y^5 - 37*y^4 + 12*y^3 + 13*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 19*x^14 - 42*x^13 + 83*x^12 - 156*x^11 + 249*x^10 - 300*x^9 + 260*x^8 - 114*x^7 - 9*x^6 + 78*x^5 - 37*x^4 + 12*x^3 + 13*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 19*x^14 - 42*x^13 + 83*x^12 - 156*x^11 + 249*x^10 - 300*x^9 + 260*x^8 - 114*x^7 - 9*x^6 + 78*x^5 - 37*x^4 + 12*x^3 + 13*x^2 + 1)
\( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 83 x^{12} - 156 x^{11} + 249 x^{10} - 300 x^{9} + 260 x^{8} + \cdots + 1 \)
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: |
\(3125945300120764416\)
\(\medspace = 2^{24}\cdot 3^{8}\cdot 73^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.32\) | 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: | $2^{2}3^{1/2}73^{1/2}\approx 59.19459434779497$ | ||
| Ramified primes: |
\(2\), \(3\), \(73\)
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{18}a^{12}+\frac{1}{3}a^{11}+\frac{5}{18}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{2}{9}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{18}a^{2}-\frac{1}{3}a-\frac{5}{18}$, $\frac{1}{18}a^{13}+\frac{5}{18}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{2}{9}a^{7}+\frac{1}{3}a^{4}-\frac{1}{18}a^{3}-\frac{5}{18}a-\frac{1}{3}$, $\frac{1}{18}a^{14}-\frac{1}{3}a^{11}-\frac{1}{18}a^{10}+\frac{1}{3}a^{9}-\frac{4}{9}a^{8}+\frac{1}{3}a^{7}-\frac{1}{9}a^{6}-\frac{1}{3}a^{5}-\frac{1}{18}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{7}{18}$, $\frac{1}{24700806}a^{15}-\frac{175934}{12350403}a^{14}+\frac{304961}{24700806}a^{13}+\frac{230641}{24700806}a^{12}+\frac{191066}{1372267}a^{11}+\frac{3322979}{8233602}a^{10}+\frac{3426533}{12350403}a^{9}+\frac{4287070}{12350403}a^{8}+\frac{68245}{1372267}a^{7}-\frac{436808}{1372267}a^{6}-\frac{6472159}{24700806}a^{5}+\frac{3883682}{12350403}a^{4}-\frac{870641}{1900062}a^{3}-\frac{7314919}{24700806}a^{2}+\frac{491509}{4116801}a+\frac{2823703}{8233602}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
\( -\frac{282317}{950031} a^{15} + \frac{3303499}{1900062} a^{14} - \frac{4921007}{950031} a^{13} + \frac{20127439}{1900062} a^{12} - \frac{18951347}{950031} a^{11} + \frac{35310566}{950031} a^{10} - \frac{53687543}{950031} a^{9} + \frac{55405682}{950031} a^{8} - \frac{33799666}{950031} a^{7} - \frac{4191059}{950031} a^{6} + \frac{22673984}{950031} a^{5} - \frac{40614541}{1900062} a^{4} + \frac{3390392}{950031} a^{3} + \frac{4974785}{1900062} a^{2} - \frac{1866610}{950031} a - \frac{553565}{950031} \)
(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: |
$\frac{37582}{316677}a^{15}-\frac{842825}{950031}a^{14}+\frac{2883283}{950031}a^{13}-\frac{12724859}{1900062}a^{12}+\frac{11782442}{950031}a^{11}-\frac{14775583}{633354}a^{10}+\frac{12244982}{316677}a^{9}-\frac{42584645}{950031}a^{8}+\frac{29184430}{950031}a^{7}-\frac{556667}{105559}a^{6}-\frac{4464968}{316677}a^{5}+\frac{11307203}{950031}a^{4}-\frac{4155082}{950031}a^{3}-\frac{5664967}{1900062}a^{2}+\frac{398401}{950031}a-\frac{478609}{633354}$, $\frac{14760899}{24700806}a^{15}-\frac{89327455}{24700806}a^{14}+\frac{287109763}{24700806}a^{13}-\frac{323004967}{12350403}a^{12}+\frac{215202371}{4116801}a^{11}-\frac{812938039}{8233602}a^{10}+\frac{1971154921}{12350403}a^{9}-\frac{2450834735}{12350403}a^{8}+\frac{751030598}{4116801}a^{7}-\frac{400506580}{4116801}a^{6}+\frac{454989667}{24700806}a^{5}+\frac{843826885}{24700806}a^{4}-\frac{35684869}{1900062}a^{3}+\frac{104066386}{12350403}a^{2}+\frac{8052715}{1372267}a+\frac{1664001}{2744534}$, $\frac{396466}{950031}a^{15}-\frac{2096479}{950031}a^{14}+\frac{11762209}{1900062}a^{13}-\frac{11730565}{950031}a^{12}+\frac{5076213}{211118}a^{11}-\frac{42897349}{950031}a^{10}+\frac{63409468}{950031}a^{9}-\frac{65252257}{950031}a^{8}+\frac{15891826}{316677}a^{7}-\frac{11397458}{950031}a^{6}+\frac{622865}{950031}a^{5}+\frac{8250364}{950031}a^{4}+\frac{11276057}{1900062}a^{3}+\frac{1367200}{950031}a^{2}+\frac{1777777}{633354}a+\frac{1866610}{950031}$, $\frac{18040751}{24700806}a^{15}+\frac{109700263}{24700806}a^{14}-\frac{173429227}{12350403}a^{13}+\frac{761980315}{24700806}a^{12}-\frac{1496815787}{24700806}a^{11}+\frac{1409508098}{12350403}a^{10}-\frac{2247174988}{12350403}a^{9}+\frac{2676544655}{12350403}a^{8}-\frac{2270750255}{12350403}a^{7}+\frac{956694850}{12350403}a^{6}+\frac{301670621}{24700806}a^{5}-\frac{1373105689}{24700806}a^{4}+\frac{27185332}{950031}a^{3}-\frac{149113093}{24700806}a^{2}-\frac{191265247}{24700806}a+\frac{21398398}{12350403}$, $\frac{2664949}{12350403}a^{15}+\frac{15532826}{12350403}a^{14}-\frac{44877788}{12350403}a^{13}+\frac{174519095}{24700806}a^{12}-\frac{52764248}{4116801}a^{11}+\frac{65480911}{2744534}a^{10}-\frac{435781111}{12350403}a^{9}+\frac{391911842}{12350403}a^{8}-\frac{44839310}{4116801}a^{7}-\frac{21405810}{1372267}a^{6}+\frac{261236500}{12350403}a^{5}-\frac{128444648}{12350403}a^{4}-\frac{1564180}{950031}a^{3}+\frac{109155217}{24700806}a^{2}-\frac{1647736}{1372267}a-\frac{198131}{8233602}$, $\frac{11362933}{24700806}a^{15}+\frac{66151523}{24700806}a^{14}-\frac{33169399}{4116801}a^{13}+\frac{69762086}{4116801}a^{12}-\frac{816129707}{24700806}a^{11}+\frac{1546826449}{24700806}a^{10}-\frac{1202306282}{12350403}a^{9}+\frac{1350198250}{12350403}a^{8}-\frac{1088275973}{12350403}a^{7}+\frac{440313502}{12350403}a^{6}+\frac{11346307}{24700806}a^{5}-\frac{443701601}{24700806}a^{4}+\frac{794884}{105559}a^{3}-\frac{11581772}{4116801}a^{2}-\frac{41710267}{24700806}a+\frac{20212859}{24700806}$, $\frac{1555481}{24700806}a^{15}-\frac{1503038}{4116801}a^{14}+\frac{13673411}{12350403}a^{13}-\frac{19064257}{8233602}a^{12}+\frac{36405467}{8233602}a^{11}-\frac{68199263}{8233602}a^{10}+\frac{160548340}{12350403}a^{9}-\frac{59949374}{4116801}a^{8}+\frac{44684656}{4116801}a^{7}-\frac{14027887}{4116801}a^{6}+\frac{28315951}{24700806}a^{5}-\frac{6021943}{4116801}a^{4}+\frac{3886990}{950031}a^{3}-\frac{17322623}{8233602}a^{2}-\frac{3267097}{8233602}a+\frac{2289437}{2744534}$
| 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: | \( 1742.7885280839505 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1742.7885280839505 \cdot 2}{12\cdot\sqrt{3125945300120764416}}\cr\approx \mathstrut & 0.399063219459381 \end{aligned}\]
# 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 + 83*x^12 - 156*x^11 + 249*x^10 - 300*x^9 + 260*x^8 - 114*x^7 - 9*x^6 + 78*x^5 - 37*x^4 + 12*x^3 + 13*x^2 + 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 - 6*x^15 + 19*x^14 - 42*x^13 + 83*x^12 - 156*x^11 + 249*x^10 - 300*x^9 + 260*x^8 - 114*x^7 - 9*x^6 + 78*x^5 - 37*x^4 + 12*x^3 + 13*x^2 + 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 - 6*x^15 + 19*x^14 - 42*x^13 + 83*x^12 - 156*x^11 + 249*x^10 - 300*x^9 + 260*x^8 - 114*x^7 - 9*x^6 + 78*x^5 - 37*x^4 + 12*x^3 + 13*x^2 + 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 - 6*x^15 + 19*x^14 - 42*x^13 + 83*x^12 - 156*x^11 + 249*x^10 - 300*x^9 + 260*x^8 - 114*x^7 - 9*x^6 + 78*x^5 - 37*x^4 + 12*x^3 + 13*x^2 + 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
$C_2^3.C_2^4$ (as 16T223):
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 128 |
| The 32 conjugacy class representatives for $C_2^3.C_2^4$ |
| Character table for $C_2^3.C_2^4$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.0.1168.1, 4.0.10512.3, \(\Q(\zeta_{12})\), 8.0.110502144.3 |
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: | 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 | R | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.16.2 | $x^{8} + 4 x^{6} + 8 x^{5} + 16 x^{4} + 16 x^{3} + 40 x^{2} + 48 x + 84$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(73\)
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |