Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 7*x^13 + 12*x^12 - 11*x^11 + 14*x^10 - 9*x^9 + 5*x^8 - 3*x^7 + x^6 - 8*x^5 + 27*x^4 - 14*x^3 + 2*x^2 - x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + 3*y^14 - 7*y^13 + 12*y^12 - 11*y^11 + 14*y^10 - 9*y^9 + 5*y^8 - 3*y^7 + y^6 - 8*y^5 + 27*y^4 - 14*y^3 + 2*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 3*x^14 - 7*x^13 + 12*x^12 - 11*x^11 + 14*x^10 - 9*x^9 + 5*x^8 - 3*x^7 + x^6 - 8*x^5 + 27*x^4 - 14*x^3 + 2*x^2 - x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 3*x^14 - 7*x^13 + 12*x^12 - 11*x^11 + 14*x^10 - 9*x^9 + 5*x^8 - 3*x^7 + x^6 - 8*x^5 + 27*x^4 - 14*x^3 + 2*x^2 - x + 1)
\( x^{16} - 2 x^{15} + 3 x^{14} - 7 x^{13} + 12 x^{12} - 11 x^{11} + 14 x^{10} - 9 x^{9} + 5 x^{8} + \cdots + 1 \)
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: |
\(590315622400000000\)
\(\medspace = 2^{18}\cdot 5^{8}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.90\) | 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: | $2^{3/2}5^{1/2}7^{1/2}\approx 16.73320053068151$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{22}a^{14}-\frac{5}{22}a^{13}+\frac{2}{11}a^{12}+\frac{7}{22}a^{11}-\frac{5}{11}a^{10}+\frac{9}{22}a^{9}+\frac{3}{11}a^{8}+\frac{1}{22}a^{7}-\frac{5}{22}a^{6}-\frac{1}{11}a^{5}-\frac{1}{11}a^{3}-\frac{1}{2}a^{2}+\frac{3}{22}a-\frac{7}{22}$, $\frac{1}{66933218}a^{15}-\frac{25660}{3042419}a^{14}+\frac{480789}{66933218}a^{13}-\frac{5930557}{66933218}a^{12}+\frac{25266519}{66933218}a^{11}-\frac{21204785}{66933218}a^{10}-\frac{33391197}{66933218}a^{9}+\frac{22434575}{66933218}a^{8}+\frac{358299}{3042419}a^{7}-\frac{15138007}{66933218}a^{6}+\frac{107056}{1154021}a^{5}+\frac{8885161}{33466609}a^{4}+\frac{16346199}{66933218}a^{3}-\frac{9030696}{33466609}a^{2}-\frac{11571050}{33466609}a+\frac{3076313}{66933218}$
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: |
$\frac{1181281}{66933218}a^{15}-\frac{4141605}{66933218}a^{14}+\frac{301465}{66933218}a^{13}-\frac{405698}{33466609}a^{12}+\frac{10429955}{66933218}a^{11}+\frac{322702}{3042419}a^{10}-\frac{32732157}{66933218}a^{9}+\frac{7033875}{33466609}a^{8}-\frac{22687558}{33466609}a^{7}+\frac{21818707}{66933218}a^{6}-\frac{67638}{1154021}a^{5}+\frac{7023443}{33466609}a^{4}+\frac{30191373}{66933218}a^{3}+\frac{19899919}{66933218}a^{2}-\frac{83031158}{33466609}a+\frac{39655621}{66933218}$, $a$, $\frac{52870}{3042419}a^{15}-\frac{42010}{3042419}a^{14}-\frac{96315}{3042419}a^{13}+\frac{111131}{3042419}a^{12}-\frac{135638}{3042419}a^{11}+\frac{951941}{3042419}a^{10}-\frac{1578869}{3042419}a^{9}+\frac{1551329}{3042419}a^{8}-\frac{1698179}{3042419}a^{7}+\frac{1439307}{3042419}a^{6}-\frac{5282}{104911}a^{5}-\frac{317574}{3042419}a^{4}+\frac{84828}{3042419}a^{3}+\frac{1976}{3042419}a^{2}-\frac{4898893}{3042419}a-\frac{9011}{3042419}$, $\frac{19373935}{66933218}a^{15}-\frac{15600729}{33466609}a^{14}+\frac{20466758}{33466609}a^{13}-\frac{110092737}{66933218}a^{12}+\frac{86405265}{33466609}a^{11}-\frac{110336971}{66933218}a^{10}+\frac{83554752}{33466609}a^{9}-\frac{53205699}{66933218}a^{8}+\frac{4648895}{66933218}a^{7}-\frac{15890987}{66933218}a^{6}-\frac{149834}{1154021}a^{5}-\frac{92104142}{33466609}a^{4}+\frac{452553895}{66933218}a^{3}-\frac{32117660}{33466609}a^{2}-\frac{10154707}{6084838}a+\frac{23987377}{66933218}$, $\frac{1492671}{33466609}a^{15}+\frac{945415}{33466609}a^{14}-\frac{4250815}{33466609}a^{13}+\frac{953184}{33466609}a^{12}-\frac{10685853}{33466609}a^{11}+\frac{35393346}{33466609}a^{10}-\frac{25494481}{33466609}a^{9}+\frac{41650798}{33466609}a^{8}-\frac{40042840}{33466609}a^{7}+\frac{15332912}{33466609}a^{6}-\frac{733405}{1154021}a^{5}-\frac{21850639}{33466609}a^{4}+\frac{20923251}{33466609}a^{3}+\frac{80553925}{33466609}a^{2}-\frac{47047117}{33466609}a-\frac{11964353}{33466609}$, $\frac{36057}{33466609}a^{15}-\frac{11356317}{66933218}a^{14}+\frac{9232768}{33466609}a^{13}-\frac{13844010}{33466609}a^{12}+\frac{34227156}{33466609}a^{11}-\frac{4678385}{3042419}a^{10}+\frac{38759345}{33466609}a^{9}-\frac{55741325}{33466609}a^{8}+\frac{8459223}{33466609}a^{7}+\frac{3336687}{66933218}a^{6}-\frac{269017}{1154021}a^{5}+\frac{26271049}{33466609}a^{4}+\frac{44870434}{33466609}a^{3}-\frac{259999889}{66933218}a^{2}+\frac{43026039}{33466609}a+\frac{7254297}{66933218}$, $\frac{1174533}{33466609}a^{15}-\frac{628332}{3042419}a^{14}+\frac{7439151}{66933218}a^{13}-\frac{24081505}{66933218}a^{12}+\frac{57792453}{66933218}a^{11}-\frac{39844691}{66933218}a^{10}+\frac{9003755}{66933218}a^{9}-\frac{73834485}{66933218}a^{8}-\frac{7053461}{6084838}a^{7}-\frac{64825467}{66933218}a^{6}-\frac{338582}{1154021}a^{5}-\frac{22692532}{33466609}a^{4}+\frac{92832165}{33466609}a^{3}-\frac{54150061}{33466609}a^{2}-\frac{131150661}{66933218}a-\frac{16074321}{66933218}$
| 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: | \( 200.438586587 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 200.438586587 \cdot 1}{2\cdot\sqrt{590315622400000000}}\cr\approx \mathstrut & 0.316845938467 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 7*x^13 + 12*x^12 - 11*x^11 + 14*x^10 - 9*x^9 + 5*x^8 - 3*x^7 + x^6 - 8*x^5 + 27*x^4 - 14*x^3 + 2*x^2 - 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 + 3*x^14 - 7*x^13 + 12*x^12 - 11*x^11 + 14*x^10 - 9*x^9 + 5*x^8 - 3*x^7 + x^6 - 8*x^5 + 27*x^4 - 14*x^3 + 2*x^2 - 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 + 3*x^14 - 7*x^13 + 12*x^12 - 11*x^11 + 14*x^10 - 9*x^9 + 5*x^8 - 3*x^7 + x^6 - 8*x^5 + 27*x^4 - 14*x^3 + 2*x^2 - 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 + 3*x^14 - 7*x^13 + 12*x^12 - 11*x^11 + 14*x^10 - 9*x^9 + 5*x^8 - 3*x^7 + x^6 - 8*x^5 + 27*x^4 - 14*x^3 + 2*x^2 - 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
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 32 |
| The 14 conjugacy class representatives for $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), 4.0.392.1, 4.0.9800.2, \(\Q(\sqrt{5}, \sqrt{-7})\), 8.0.96040000.1 |
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
| Galois closure: | deg 32 |
| Degree 16 siblings: | 16.0.60448319733760000.1, 16.4.771024486400000000.1 |
| 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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.4.6.3 | $x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(7\)
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |