Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 - 10*x^13 + 28*x^12 - 25*x^11 + 36*x^10 - 50*x^9 + 35*x^8 - 50*x^7 + 36*x^6 - 25*x^5 + 28*x^4 - 10*x^3 + 8*x^2 - 5*x + 1)
gp: K = bnfinit(y^16 - 5*y^15 + 8*y^14 - 10*y^13 + 28*y^12 - 25*y^11 + 36*y^10 - 50*y^9 + 35*y^8 - 50*y^7 + 36*y^6 - 25*y^5 + 28*y^4 - 10*y^3 + 8*y^2 - 5*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 8*x^14 - 10*x^13 + 28*x^12 - 25*x^11 + 36*x^10 - 50*x^9 + 35*x^8 - 50*x^7 + 36*x^6 - 25*x^5 + 28*x^4 - 10*x^3 + 8*x^2 - 5*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 8*x^14 - 10*x^13 + 28*x^12 - 25*x^11 + 36*x^10 - 50*x^9 + 35*x^8 - 50*x^7 + 36*x^6 - 25*x^5 + 28*x^4 - 10*x^3 + 8*x^2 - 5*x + 1)
\( x^{16} - 5 x^{15} + 8 x^{14} - 10 x^{13} + 28 x^{12} - 25 x^{11} + 36 x^{10} - 50 x^{9} + 35 x^{8} + \cdots + 1 \)
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: |
\(1308342779541015625\)
\(\medspace = 5^{14}\cdot 11^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.56\) | 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: | $5^{7/8}11^{1/2}\approx 13.561105554803268$ | ||
| Ramified primes: |
\(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{239}a^{14}+\frac{109}{239}a^{13}+\frac{5}{239}a^{12}-\frac{27}{239}a^{11}+\frac{52}{239}a^{10}-\frac{45}{239}a^{9}+\frac{112}{239}a^{8}+\frac{96}{239}a^{7}+\frac{112}{239}a^{6}-\frac{45}{239}a^{5}+\frac{52}{239}a^{4}-\frac{27}{239}a^{3}+\frac{5}{239}a^{2}+\frac{109}{239}a+\frac{1}{239}$, $\frac{1}{4541}a^{15}-\frac{6}{4541}a^{14}-\frac{1297}{4541}a^{13}-\frac{841}{4541}a^{12}-\frac{1145}{4541}a^{11}+\frac{189}{4541}a^{10}-\frac{210}{4541}a^{9}+\frac{122}{4541}a^{8}-\frac{1607}{4541}a^{7}-\frac{1692}{4541}a^{6}+\frac{208}{4541}a^{5}-\frac{271}{4541}a^{4}+\frac{242}{4541}a^{3}-\frac{1183}{4541}a^{2}-\frac{823}{4541}a-\frac{1310}{4541}$
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
$C_{2}$, 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: |
\( \frac{791}{4541} a^{15} - \frac{2409}{4541} a^{14} + \frac{776}{4541} a^{13} - \frac{8724}{4541} a^{12} + \frac{34767}{4541} a^{11} - \frac{10519}{4541} a^{10} + \frac{55677}{4541} a^{9} - \frac{73149}{4541} a^{8} + \frac{43055}{4541} a^{7} - \frac{109395}{4541} a^{6} + \frac{54822}{4541} a^{5} - \frac{61050}{4541} a^{4} + \frac{73831}{4541} a^{3} - \frac{11327}{4541} a^{2} + \frac{30594}{4541} a - \frac{12148}{4541} \)
(order $10$)
| 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: |
$\frac{103}{239}a^{15}-\frac{596}{239}a^{14}+\frac{1213}{239}a^{13}-\frac{1429}{239}a^{12}+\frac{2644}{239}a^{11}-\frac{3050}{239}a^{10}+\frac{3670}{239}a^{9}-\frac{3851}{239}a^{8}+\frac{2935}{239}a^{7}-\frac{2839}{239}a^{6}+\frac{1553}{239}a^{5}-\frac{957}{239}a^{4}+\frac{193}{239}a^{3}-\frac{88}{239}a^{2}+\frac{84}{239}a+\frac{127}{239}$, $\frac{3880}{4541}a^{15}-\frac{20031}{4541}a^{14}+\frac{35339}{4541}a^{13}-\frac{49971}{4541}a^{12}+\frac{124202}{4541}a^{11}-\frac{119457}{4541}a^{10}+\frac{174245}{4541}a^{9}-\frac{234428}{4541}a^{8}+\frac{166225}{4541}a^{7}-\frac{238739}{4541}a^{6}+\frac{656}{19}a^{5}-\frac{115103}{4541}a^{4}+\frac{120136}{4541}a^{3}-\frac{28254}{4541}a^{2}+\frac{39895}{4541}a-\frac{11795}{4541}$, $\frac{3089}{4541}a^{15}-\frac{13214}{4541}a^{14}+\frac{15525}{4541}a^{13}-\frac{23748}{4541}a^{12}+\frac{83944}{4541}a^{11}-\frac{38657}{4541}a^{10}+\frac{110930}{4541}a^{9}-\frac{135306}{4541}a^{8}+\frac{69533}{4541}a^{7}-\frac{166945}{4541}a^{6}+\frac{71619}{4541}a^{5}-\frac{74592}{4541}a^{4}+\frac{90765}{4541}a^{3}-\frac{3969}{4541}a^{2}+\frac{31132}{4541}a-\frac{13403}{4541}$, $\frac{1061}{4541}a^{15}-\frac{5910}{4541}a^{14}+\frac{8641}{4541}a^{13}+\frac{15}{4541}a^{12}+\frac{7995}{4541}a^{11}-\frac{11891}{4541}a^{10}-\frac{20821}{4541}a^{9}+\frac{3415}{4541}a^{8}-\frac{26491}{4541}a^{7}+\frac{40473}{4541}a^{6}-\frac{22341}{4541}a^{5}+\frac{49510}{4541}a^{4}-\frac{23469}{4541}a^{3}+\frac{14056}{4541}a^{2}-\frac{19742}{4541}a+\frac{4633}{4541}$, $\frac{1918}{4541}a^{15}-\frac{8392}{4541}a^{14}+\frac{8973}{4541}a^{13}-\frac{8108}{4541}a^{12}+\frac{40209}{4541}a^{11}-\frac{11304}{4541}a^{10}+\frac{29166}{4541}a^{9}-\frac{52752}{4541}a^{8}-\frac{3998}{4541}a^{7}-\frac{49057}{4541}a^{6}+\frac{13510}{4541}a^{5}+\frac{5534}{4541}a^{4}+\frac{34908}{4541}a^{3}+\frac{8004}{4541}a^{2}+\frac{5364}{4541}a-\frac{7373}{4541}$, $\frac{2650}{4541}a^{15}-\frac{8756}{4541}a^{14}-\frac{1869}{4541}a^{13}+\frac{9455}{4541}a^{12}+\frac{42382}{4541}a^{11}+\frac{27712}{4541}a^{10}+\frac{12056}{4541}a^{9}-\frac{43609}{4541}a^{8}-\frac{35287}{4541}a^{7}-\frac{73577}{4541}a^{6}+\frac{2670}{4541}a^{5}-\frac{10628}{4541}a^{4}+\frac{62427}{4541}a^{3}+\frac{11355}{4541}a^{2}+\frac{19079}{4541}a-\frac{8655}{4541}$, $\frac{889}{4541}a^{15}-\frac{7253}{4541}a^{14}+\frac{22801}{4541}a^{13}-\frac{35225}{4541}a^{12}+\frac{42010}{4541}a^{11}-\frac{72538}{4541}a^{10}+\frac{99469}{4541}a^{9}-\frac{79224}{4541}a^{8}+\frac{108192}{4541}a^{7}-\frac{79815}{4541}a^{6}+\frac{62381}{4541}a^{5}-\frac{68247}{4541}a^{4}+\frac{12655}{4541}a^{3}-\frac{25934}{4541}a^{2}+\frac{12792}{4541}a+\frac{528}{4541}$
| 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: | \( 826.91591824 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 826.91591824 \cdot 2}{10\cdot\sqrt{1308342779541015625}}\cr\approx \mathstrut & 0.35121214361 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 - 10*x^13 + 28*x^12 - 25*x^11 + 36*x^10 - 50*x^9 + 35*x^8 - 50*x^7 + 36*x^6 - 25*x^5 + 28*x^4 - 10*x^3 + 8*x^2 - 5*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 - 5*x^15 + 8*x^14 - 10*x^13 + 28*x^12 - 25*x^11 + 36*x^10 - 50*x^9 + 35*x^8 - 50*x^7 + 36*x^6 - 25*x^5 + 28*x^4 - 10*x^3 + 8*x^2 - 5*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 - 5*x^15 + 8*x^14 - 10*x^13 + 28*x^12 - 25*x^11 + 36*x^10 - 50*x^9 + 35*x^8 - 50*x^7 + 36*x^6 - 25*x^5 + 28*x^4 - 10*x^3 + 8*x^2 - 5*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 - 5*x^15 + 8*x^14 - 10*x^13 + 28*x^12 - 25*x^11 + 36*x^10 - 50*x^9 + 35*x^8 - 50*x^7 + 36*x^6 - 25*x^5 + 28*x^4 - 10*x^3 + 8*x^2 - 5*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
$\OD_{16}:C_2$ (as 16T36):
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 11 conjugacy class representatives for $\OD_{16}:C_2$ |
| Character table for $\OD_{16}:C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{5})\), 4.4.15125.1, \(\Q(\sqrt{5}, \sqrt{-11})\), 8.0.9453125.1 x2, 8.4.1143828125.1 x2, 8.0.228765625.1 |
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 8 siblings: | 8.0.9453125.1, 8.4.1143828125.1 |
| Degree 16 siblings: | 16.0.10812750244140625.1, 16.4.1308342779541015625.1 |
| Minimal sibling: | 8.0.9453125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
|
\(11\)
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |