Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 - 39*x^12 + 234*x^11 - 381*x^10 + 46*x^9 + 654*x^8 - 890*x^7 + 425*x^6 + 78*x^5 - 176*x^4 + 64*x^3 + 2*x^2 - 6*x + 1)
gp: K = bnfinit(y^16 - 8*y^15 + 24*y^14 - 28*y^13 - 39*y^12 + 234*y^11 - 381*y^10 + 46*y^9 + 654*y^8 - 890*y^7 + 425*y^6 + 78*y^5 - 176*y^4 + 64*y^3 + 2*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 - 39*x^12 + 234*x^11 - 381*x^10 + 46*x^9 + 654*x^8 - 890*x^7 + 425*x^6 + 78*x^5 - 176*x^4 + 64*x^3 + 2*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 - 39*x^12 + 234*x^11 - 381*x^10 + 46*x^9 + 654*x^8 - 890*x^7 + 425*x^6 + 78*x^5 - 176*x^4 + 64*x^3 + 2*x^2 - 6*x + 1)
\( x^{16} - 8 x^{15} + 24 x^{14} - 28 x^{13} - 39 x^{12} + 234 x^{11} - 381 x^{10} + 46 x^{9} + 654 x^{8} + \cdots + 1 \)
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: | $[12, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(905475719089201229649\)
\(\medspace = 3^{8}\cdot 19^{4}\cdot 97^{2}\cdot 103^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.41\) | 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: | $3^{1/2}19^{1/2}97^{1/2}103^{1/2}\approx 754.6436245009959$ | ||
| Ramified primes: |
\(3\), \(19\), \(97\), \(103\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$
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 $1$ (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: | $13$ | 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: |
$\frac{4169}{2}a^{15}-\frac{31315}{2}a^{14}+42378a^{13}-\frac{75325}{2}a^{12}-\frac{199381}{2}a^{11}+\frac{878115}{2}a^{10}-\frac{1159371}{2}a^{9}-187277a^{8}+1271663a^{7}-1233900a^{6}+\frac{566469}{2}a^{5}+\frac{601575}{2}a^{4}-\frac{439733}{2}a^{3}+26025a^{2}+16856a-4268$, $2327a^{15}-17384a^{14}+46644a^{13}-40459a^{12}-\frac{224357}{2}a^{11}+\frac{970261}{2}a^{10}-629728a^{9}-226412a^{8}+1402051a^{7}-1328701a^{6}+\frac{570727}{2}a^{5}+\frac{665437}{2}a^{4}-\frac{466845}{2}a^{3}+\frac{50625}{2}a^{2}+\frac{36149}{2}a-4393$, $\frac{4169}{2}a^{15}-15747a^{14}+\frac{86009}{2}a^{13}-\frac{78323}{2}a^{12}-98841a^{11}+\frac{887549}{2}a^{10}-596126a^{9}-170802a^{8}+\frac{2576307}{2}a^{7}-1280091a^{6}+312573a^{5}+303688a^{4}-231347a^{3}+29607a^{2}+17577a-4628$, $\frac{4169}{2}a^{15}-\frac{31041}{2}a^{14}+41419a^{13}-\frac{70735}{2}a^{12}-\frac{201987}{2}a^{11}+431842a^{10}-554520a^{9}-212517a^{8}+1246479a^{7}-1163224a^{6}+\frac{476475}{2}a^{5}+296470a^{4}-202337a^{3}+\frac{41049}{2}a^{2}+\frac{31539}{2}a-\frac{7437}{2}$, $\frac{297}{2}a^{15}-\frac{2221}{2}a^{14}+2986a^{13}-\frac{5221}{2}a^{12}-7128a^{11}+\frac{62029}{2}a^{10}-\frac{80949}{2}a^{9}-\frac{28015}{2}a^{8}+\frac{179051}{2}a^{7}-85731a^{6}+\frac{38343}{2}a^{5}+21083a^{4}-15191a^{3}+1778a^{2}+\frac{2329}{2}a-297$, $a^{15}-8a^{14}+24a^{13}-28a^{12}-39a^{11}+234a^{10}-381a^{9}+46a^{8}+654a^{7}-890a^{6}+425a^{5}+78a^{4}-176a^{3}+64a^{2}+2a-6$, $a^{15}-8a^{14}+24a^{13}-28a^{12}-39a^{11}+234a^{10}-381a^{9}+46a^{8}+654a^{7}-890a^{6}+425a^{5}+78a^{4}-176a^{3}+64a^{2}+2a-5$, $\frac{4167}{2}a^{15}-15603a^{14}+\frac{84057}{2}a^{13}-\frac{73711}{2}a^{12}-100094a^{11}+436375a^{10}-\frac{1141527}{2}a^{9}-195891a^{8}+\frac{2524933}{2}a^{7}-1209028a^{6}+267538a^{5}+\frac{598509}{2}a^{4}-\frac{427515}{2}a^{3}+\frac{48205}{2}a^{2}+\frac{32969}{2}a-\frac{8157}{2}$, $\frac{3835}{2}a^{15}-\frac{28893}{2}a^{14}+39286a^{13}-\frac{70735}{2}a^{12}-\frac{182603}{2}a^{11}+406172a^{10}-\frac{1082357}{2}a^{9}-\frac{328791}{2}a^{8}+\frac{2355717}{2}a^{7}-1157278a^{6}+274484a^{5}+\frac{556381}{2}a^{4}-\frac{415503}{2}a^{3}+\frac{51237}{2}a^{2}+15857a-4096$, $\frac{9889}{2}a^{15}-\frac{73879}{2}a^{14}+\frac{198241}{2}a^{13}-\frac{171987}{2}a^{12}-238342a^{11}+\frac{2061777}{2}a^{10}-\frac{2676661}{2}a^{9}-\frac{961549}{2}a^{8}+\frac{5958589}{2}a^{7}-\frac{5648031}{2}a^{6}+606988a^{5}+\frac{1413741}{2}a^{4}-496158a^{3}+\frac{107777}{2}a^{2}+38401a-\frac{18687}{2}$, $\frac{757}{2}a^{15}-\frac{5679}{2}a^{14}+\frac{15339}{2}a^{13}-\frac{13551}{2}a^{12}-18144a^{11}+79540a^{10}-\frac{209153}{2}a^{9}-\frac{69509}{2}a^{8}+230342a^{7}-222103a^{6}+49995a^{5}+\frac{109271}{2}a^{4}-\frac{78867}{2}a^{3}+4528a^{2}+3040a-757$, $303a^{15}-\frac{4741}{2}a^{14}+\frac{13651}{2}a^{13}-7082a^{12}-\frac{27087}{2}a^{11}+68793a^{10}-\frac{203657}{2}a^{9}-9240a^{8}+\frac{403741}{2}a^{7}-229024a^{6}+\frac{147287}{2}a^{5}+\frac{92605}{2}a^{4}-\frac{88909}{2}a^{3}+7868a^{2}+3213a-1035$, $\frac{8489}{2}a^{15}-\frac{63661}{2}a^{14}+85930a^{13}-\frac{151653}{2}a^{12}-203480a^{11}+\frac{1782605}{2}a^{10}-\frac{2341813}{2}a^{9}-\frac{781615}{2}a^{8}+\frac{5159819}{2}a^{7}-2485979a^{6}+\frac{1119695}{2}a^{5}+610909a^{4}-441174a^{3}+50931a^{2}+\frac{67863}{2}a-8489$
| 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: | \( 61811.3534598 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 61811.3534598 \cdot 1}{2\cdot\sqrt{905475719089201229649}}\cr\approx \mathstrut & 0.166080841950 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 - 39*x^12 + 234*x^11 - 381*x^10 + 46*x^9 + 654*x^8 - 890*x^7 + 425*x^6 + 78*x^5 - 176*x^4 + 64*x^3 + 2*x^2 - 6*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 + 24*x^14 - 28*x^13 - 39*x^12 + 234*x^11 - 381*x^10 + 46*x^9 + 654*x^8 - 890*x^7 + 425*x^6 + 78*x^5 - 176*x^4 + 64*x^3 + 2*x^2 - 6*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 + 24*x^14 - 28*x^13 - 39*x^12 + 234*x^11 - 381*x^10 + 46*x^9 + 654*x^8 - 890*x^7 + 425*x^6 + 78*x^5 - 176*x^4 + 64*x^3 + 2*x^2 - 6*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 + 24*x^14 - 28*x^13 - 39*x^12 + 234*x^11 - 381*x^10 + 46*x^9 + 654*x^8 - 890*x^7 + 425*x^6 + 78*x^5 - 176*x^4 + 64*x^3 + 2*x^2 - 6*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
$C_2^8.S_4$ (as 16T1664):
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 6144 |
| The 105 conjugacy class representatives for $C_2^8.S_4$ |
| Character table for $C_2^8.S_4$ |
Intermediate fields
| 4.4.1957.1, 8.8.30091123593.1, 8.6.103405923.1, 8.6.1114486059.1 |
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: | 16.0.1242079175705351481.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(19\)
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(97\)
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 97.4.2.1 | $x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 97.4.0.1 | $x^{4} + 6 x^{2} + 80 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 97.4.0.1 | $x^{4} + 6 x^{2} + 80 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(103\)
| 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |