Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 38*x^14 + 524*x^12 + 3268*x^10 + 9161*x^8 + 9768*x^6 + 4128*x^4 + 558*x^2 + 9)
gp: K = bnfinit(y^16 + 38*y^14 + 524*y^12 + 3268*y^10 + 9161*y^8 + 9768*y^6 + 4128*y^4 + 558*y^2 + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 38*x^14 + 524*x^12 + 3268*x^10 + 9161*x^8 + 9768*x^6 + 4128*x^4 + 558*x^2 + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 38*x^14 + 524*x^12 + 3268*x^10 + 9161*x^8 + 9768*x^6 + 4128*x^4 + 558*x^2 + 9)
\( x^{16} + 38x^{14} + 524x^{12} + 3268x^{10} + 9161x^{8} + 9768x^{6} + 4128x^{4} + 558x^{2} + 9 \)
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: |
\(954142202181946982400000000\)
\(\medspace = 2^{20}\cdot 3^{6}\cdot 5^{8}\cdot 7^{4}\cdot 191^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(48.55\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\), \(191\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{84}a^{12}-\frac{19}{84}a^{10}+\frac{1}{42}a^{8}+\frac{1}{12}a^{6}-\frac{1}{2}a^{5}-\frac{17}{42}a^{4}-\frac{1}{2}a^{3}+\frac{3}{28}a^{2}-\frac{1}{2}a+\frac{5}{28}$, $\frac{1}{84}a^{13}-\frac{19}{84}a^{11}+\frac{1}{42}a^{9}+\frac{1}{12}a^{7}+\frac{2}{21}a^{5}-\frac{1}{2}a^{4}-\frac{11}{28}a^{3}-\frac{1}{2}a^{2}-\frac{9}{28}a-\frac{1}{2}$, $\frac{1}{12939108}a^{14}+\frac{7222}{3234777}a^{12}-\frac{24961}{4313036}a^{10}-\frac{618505}{12939108}a^{8}+\frac{1326361}{12939108}a^{6}-\frac{809033}{12939108}a^{4}+\frac{309119}{1078259}a^{2}+\frac{290289}{4313036}$, $\frac{1}{12939108}a^{15}+\frac{7222}{3234777}a^{13}-\frac{24961}{4313036}a^{11}-\frac{618505}{12939108}a^{9}+\frac{1326361}{12939108}a^{7}-\frac{809033}{12939108}a^{5}+\frac{309119}{1078259}a^{3}+\frac{290289}{4313036}a$
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
$C_{2}\times C_{42}$, which has order $84$ (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: |
\( -1 \)
(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: |
$\frac{3454}{78897}a^{14}+\frac{42828}{26299}a^{12}+\frac{1706704}{78897}a^{10}+\frac{471967}{3757}a^{8}+\frac{7860532}{26299}a^{6}+\frac{14145368}{78897}a^{4}+\frac{424110}{26299}a^{2}-\frac{1422}{3757}$, $\frac{16442}{190281}a^{14}+\frac{1220987}{380562}a^{12}+\frac{1154704}{27183}a^{10}+\frac{93283201}{380562}a^{8}+\frac{109270300}{190281}a^{6}+\frac{40709911}{126854}a^{4}+\frac{1449635}{63427}a^{2}+\frac{40927}{63427}$, $\frac{41923}{1848444}a^{14}+\frac{393275}{462111}a^{12}+\frac{21211157}{1848444}a^{10}+\frac{126697657}{1848444}a^{8}+\frac{322077125}{1848444}a^{6}+\frac{84620491}{616148}a^{4}+\frac{10446533}{308074}a^{2}+\frac{862747}{616148}$, $\frac{59061}{4313036}a^{14}+\frac{6744659}{12939108}a^{12}+\frac{23332207}{3234777}a^{10}+\frac{83770405}{1848444}a^{8}+\frac{418073801}{3234777}a^{6}+\frac{1856084011}{12939108}a^{4}+\frac{251865539}{4313036}a^{2}+\frac{756870}{154037}$, $\frac{48123}{4313036}a^{14}+\frac{1841743}{4313036}a^{12}+\frac{6417787}{1078259}a^{10}+\frac{162608393}{4313036}a^{8}+\frac{116145067}{1078259}a^{6}+\frac{487437833}{4313036}a^{4}+\frac{106385021}{4313036}a^{2}-\frac{18618433}{2156518}$, $\frac{90631}{2156518}a^{14}+\frac{10418833}{6469554}a^{12}+\frac{72852016}{3234777}a^{10}+\frac{66528847}{462111}a^{8}+\frac{1368967694}{3234777}a^{6}+\frac{1614216673}{3234777}a^{4}+\frac{470918863}{2156518}a^{2}+\frac{6853149}{308074}$, $\frac{97141}{1078259}a^{14}+\frac{43766323}{12939108}a^{12}+\frac{591137315}{12939108}a^{10}+\frac{253425439}{924222}a^{8}+\frac{9148915219}{12939108}a^{6}+\frac{3804057085}{6469554}a^{4}+\frac{656445833}{4313036}a^{2}+\frac{1915547}{616148}$
| 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: | \( 179538.402106 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 179538.402106 \cdot 84}{2\cdot\sqrt{954142202181946982400000000}}\cr\approx \mathstrut & 0.592978627086 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 38*x^14 + 524*x^12 + 3268*x^10 + 9161*x^8 + 9768*x^6 + 4128*x^4 + 558*x^2 + 9)
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 + 38*x^14 + 524*x^12 + 3268*x^10 + 9161*x^8 + 9768*x^6 + 4128*x^4 + 558*x^2 + 9, 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 + 38*x^14 + 524*x^12 + 3268*x^10 + 9161*x^8 + 9768*x^6 + 4128*x^4 + 558*x^2 + 9);
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 + 38*x^14 + 524*x^12 + 3268*x^10 + 9161*x^8 + 9768*x^6 + 4128*x^4 + 558*x^2 + 9);
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
$C_2^7.\PGOPlus(4,3)$ (as 16T1867):
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 73728 |
| The 101 conjugacy class representatives for $C_2^7.\PGOPlus(4,3)$ |
| Character table for $C_2^7.\PGOPlus(4,3)$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 8.8.160881210000.1 |
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 | R | R | R | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.12.16.1 | $x^{12} - 2 x^{11} + 4 x^{10} + 4 x^{9} - 4 x^{7} + 10 x^{6} + 8 x^{5} + 4 x^{4} + 12 x^{3} + 12 x^{2} + 12$ | $6$ | $2$ | $16$ | 12T208 | $[4/3, 4/3, 4/3, 4/3, 2, 2]_{3}^{6}$ | |
|
\(3\)
| 3.8.6.3 | $x^{8} - 6 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
| 3.8.0.1 | $x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(191\)
| $\Q_{191}$ | $x + 172$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{191}$ | $x + 172$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 191.2.0.1 | $x^{2} + 190 x + 19$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 191.2.0.1 | $x^{2} + 190 x + 19$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 191.4.0.1 | $x^{4} + 7 x^{2} + 100 x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 191.6.4.1 | $x^{6} + 570 x^{5} + 108357 x^{4} + 6881042 x^{3} + 2167653 x^{2} + 20869296 x + 1308043810$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |