Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 8*x^13 + 14*x^12 - 10*x^11 + 20*x^10 - 34*x^9 + 20*x^8 - 12*x^7 + 20*x^6 - 10*x^5 - x^4 - 4*x^3 + 8*x^2 - 4*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + 2*y^14 - 8*y^13 + 14*y^12 - 10*y^11 + 20*y^10 - 34*y^9 + 20*y^8 - 12*y^7 + 20*y^6 - 10*y^5 - y^4 - 4*y^3 + 8*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 2*x^14 - 8*x^13 + 14*x^12 - 10*x^11 + 20*x^10 - 34*x^9 + 20*x^8 - 12*x^7 + 20*x^6 - 10*x^5 - x^4 - 4*x^3 + 8*x^2 - 4*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 2*x^14 - 8*x^13 + 14*x^12 - 10*x^11 + 20*x^10 - 34*x^9 + 20*x^8 - 12*x^7 + 20*x^6 - 10*x^5 - x^4 - 4*x^3 + 8*x^2 - 4*x + 1)
\( x^{16} - 2 x^{15} + 2 x^{14} - 8 x^{13} + 14 x^{12} - 10 x^{11} + 20 x^{10} - 34 x^{9} + 20 x^{8} + \cdots + 1 \)
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: |
\(31673320900000000\)
\(\medspace = 2^{8}\cdot 5^{8}\cdot 13^{2}\cdot 37^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.75\) | 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: | $2^{2/3}5^{1/2}13^{1/2}37^{1/2}\approx 77.84741646319638$ | ||
| Ramified primes: |
\(2\), \(5\), \(13\), \(37\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{764738}a^{15}+\frac{13434}{382369}a^{14}+\frac{15245}{382369}a^{13}-\frac{11575}{58826}a^{12}+\frac{288939}{764738}a^{11}+\frac{85372}{382369}a^{10}-\frac{154331}{764738}a^{9}+\frac{11545}{29413}a^{8}+\frac{258603}{764738}a^{7}-\frac{129239}{764738}a^{6}-\frac{27617}{58826}a^{5}-\frac{106779}{764738}a^{4}+\frac{145245}{764738}a^{3}+\frac{10582}{29413}a^{2}+\frac{37301}{382369}a+\frac{88719}{382369}$
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: |
$\frac{77073}{764738}a^{15}-\frac{56570}{382369}a^{14}+\frac{298265}{764738}a^{13}-\frac{26549}{29413}a^{12}+\frac{989725}{764738}a^{11}-\frac{1771037}{764738}a^{10}+\frac{2275903}{764738}a^{9}-\frac{194473}{58826}a^{8}+\frac{2074292}{382369}a^{7}-\frac{1400790}{382369}a^{6}+\frac{61591}{29413}a^{5}-\frac{2726463}{764738}a^{4}+\frac{997779}{764738}a^{3}+\frac{22822}{29413}a^{2}+\frac{632200}{382369}a-\frac{513049}{764738}$, $\frac{40499}{58826}a^{15}-\frac{65629}{58826}a^{14}+\frac{28385}{29413}a^{13}-\frac{301685}{58826}a^{12}+\frac{227505}{29413}a^{11}-\frac{240935}{58826}a^{10}+\frac{717243}{58826}a^{9}-\frac{1131655}{58826}a^{8}+\frac{428943}{58826}a^{7}-\frac{165227}{29413}a^{6}+\frac{383133}{29413}a^{5}-\frac{115850}{29413}a^{4}-\frac{58427}{29413}a^{3}-\frac{279465}{58826}a^{2}+\frac{148584}{29413}a-\frac{31886}{29413}$, $a$, $\frac{227244}{382369}a^{15}-\frac{535169}{764738}a^{14}+\frac{143280}{382369}a^{13}-\frac{121188}{29413}a^{12}+\frac{3469669}{764738}a^{11}-\frac{307109}{764738}a^{10}+\frac{3532237}{382369}a^{9}-\frac{636371}{58826}a^{8}-\frac{693847}{382369}a^{7}-\frac{2254911}{764738}a^{6}+\frac{373241}{58826}a^{5}+\frac{2411173}{764738}a^{4}-\frac{1221707}{764738}a^{3}-\frac{178771}{58826}a^{2}+\frac{527273}{382369}a+\frac{145084}{382369}$, $\frac{61489}{382369}a^{15}+\frac{122375}{764738}a^{14}+\frac{44403}{382369}a^{13}-\frac{28814}{29413}a^{12}-\frac{793197}{764738}a^{11}-\frac{38003}{764738}a^{10}+\frac{1122090}{382369}a^{9}+\frac{178065}{58826}a^{8}-\frac{339736}{382369}a^{7}-\frac{2680471}{764738}a^{6}-\frac{170207}{58826}a^{5}+\frac{1760181}{764738}a^{4}+\frac{1865989}{764738}a^{3}+\frac{38261}{58826}a^{2}-\frac{460884}{382369}a-\frac{31864}{382369}$, $\frac{2543}{29413}a^{15}-\frac{31563}{58826}a^{14}+\frac{36217}{58826}a^{13}-\frac{24208}{29413}a^{12}+\frac{217339}{58826}a^{11}-\frac{110366}{29413}a^{10}+\frac{76091}{58826}a^{9}-\frac{487753}{58826}a^{8}+\frac{523171}{58826}a^{7}+\frac{100409}{58826}a^{6}+\frac{77356}{29413}a^{5}-\frac{174659}{29413}a^{4}-\frac{69245}{29413}a^{3}+\frac{43058}{29413}a^{2}+\frac{203963}{58826}a-\frac{58825}{29413}$, $\frac{222881}{764738}a^{15}+\frac{85799}{764738}a^{14}-\frac{202547}{764738}a^{13}-\frac{92171}{58826}a^{12}-\frac{360414}{382369}a^{11}+\frac{1115292}{382369}a^{10}+\frac{1189737}{382369}a^{9}+\frac{135571}{58826}a^{8}-\frac{3171676}{382369}a^{7}+\frac{43159}{382369}a^{6}-\frac{46067}{58826}a^{5}+\frac{4224689}{764738}a^{4}-\frac{538171}{764738}a^{3}-\frac{42902}{29413}a^{2}+\frac{52397}{764738}a+\frac{331342}{382369}$
| 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: | \( 27.3219740008 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 27.3219740008 \cdot 1}{2\cdot\sqrt{31673320900000000}}\cr\approx \mathstrut & 0.186455081806 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 8*x^13 + 14*x^12 - 10*x^11 + 20*x^10 - 34*x^9 + 20*x^8 - 12*x^7 + 20*x^6 - 10*x^5 - x^4 - 4*x^3 + 8*x^2 - 4*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^15 + 2*x^14 - 8*x^13 + 14*x^12 - 10*x^11 + 20*x^10 - 34*x^9 + 20*x^8 - 12*x^7 + 20*x^6 - 10*x^5 - x^4 - 4*x^3 + 8*x^2 - 4*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^15 + 2*x^14 - 8*x^13 + 14*x^12 - 10*x^11 + 20*x^10 - 34*x^9 + 20*x^8 - 12*x^7 + 20*x^6 - 10*x^5 - x^4 - 4*x^3 + 8*x^2 - 4*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^15 + 2*x^14 - 8*x^13 + 14*x^12 - 10*x^11 + 20*x^10 - 34*x^9 + 20*x^8 - 12*x^7 + 20*x^6 - 10*x^5 - x^4 - 4*x^3 + 8*x^2 - 4*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
$(C_2^3\times C_4):S_4$ (as 16T1046):
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 768 |
| The 40 conjugacy class representatives for $(C_2^3\times C_4):S_4$ |
| Character table for $(C_2^3\times C_4):S_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.3700.1, 8.0.13690000.1 |
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 | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(13\)
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
|
\(37\)
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.8.4.1 | $x^{8} + 3700 x^{7} + 5133910 x^{6} + 3166256548 x^{5} + 732510094073 x^{4} + 136269235536 x^{3} + 4476368972260 x^{2} + 17928293629116 x + 2173698901413$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |