Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*x^2 - 4*x + 1)
gp: K = bnfinit(y^16 - 5*y^15 + 12*y^14 - 19*y^13 + 21*y^12 - 16*y^11 + 6*y^10 + 12*y^9 - 28*y^8 + 21*y^7 + y^6 - 20*y^5 + 28*y^4 - 20*y^3 + 10*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*x^2 - 4*x + 1)
\( x^{16} - 5 x^{15} + 12 x^{14} - 19 x^{13} + 21 x^{12} - 16 x^{11} + 6 x^{10} + 12 x^{9} - 28 x^{8} + \cdots + 1 \)
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: |
\(3345005787800625\)
\(\medspace = 3^{8}\cdot 5^{4}\cdot 13^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.34\) | 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: | $3^{1/2}5^{1/2}13^{1/2}\approx 13.96424004376894$ | ||
| Ramified primes: |
\(3\), \(5\), \(13\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4201259}a^{15}-\frac{844160}{4201259}a^{14}+\frac{1138268}{4201259}a^{13}-\frac{476410}{4201259}a^{12}-\frac{1634204}{4201259}a^{11}+\frac{273623}{4201259}a^{10}+\frac{795002}{4201259}a^{9}-\frac{1897}{4201259}a^{8}+\frac{682328}{4201259}a^{7}+\frac{2016081}{4201259}a^{6}-\frac{1049503}{4201259}a^{5}-\frac{1487939}{4201259}a^{4}+\frac{743343}{4201259}a^{3}-\frac{867204}{4201259}a^{2}+\frac{2016916}{4201259}a-\frac{107421}{4201259}$
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: |
\( \frac{8947}{8033} a^{15} - \frac{40788}{8033} a^{14} + \frac{87386}{8033} a^{13} - \frac{122437}{8033} a^{12} + \frac{114059}{8033} a^{11} - \frac{64231}{8033} a^{10} - \frac{1220}{8033} a^{9} + \frac{121765}{8033} a^{8} - \frac{194988}{8033} a^{7} + \frac{72428}{8033} a^{6} + \frac{87250}{8033} a^{5} - \frac{166072}{8033} a^{4} + \frac{161088}{8033} a^{3} - \frac{64577}{8033} a^{2} + \frac{24285}{8033} a - \frac{3468}{8033} \)
(order $6$)
| 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: |
$\frac{796875}{4201259}a^{15}-\frac{1213956}{4201259}a^{14}-\frac{2908118}{4201259}a^{13}+\frac{11752044}{4201259}a^{12}-\frac{21469083}{4201259}a^{11}+\frac{23193579}{4201259}a^{10}-\frac{15833414}{4201259}a^{9}+\frac{9183883}{4201259}a^{8}+\frac{15788997}{4201259}a^{7}-\frac{43108574}{4201259}a^{6}+\frac{26127264}{4201259}a^{5}+\frac{7333168}{4201259}a^{4}-\frac{30267134}{4201259}a^{3}+\frac{32911705}{4201259}a^{2}-\frac{12108058}{4201259}a+\frac{3744009}{4201259}$, $\frac{1962229}{4201259}a^{15}-\frac{9047969}{4201259}a^{14}+\frac{18754684}{4201259}a^{13}-\frac{24384095}{4201259}a^{12}+\frac{20878732}{4201259}a^{11}-\frac{9914533}{4201259}a^{10}-\frac{1902350}{4201259}a^{9}+\frac{25174615}{4201259}a^{8}-\frac{40649152}{4201259}a^{7}+\frac{6299933}{4201259}a^{6}+\frac{28920728}{4201259}a^{5}-\frac{28919017}{4201259}a^{4}+\frac{24494445}{4201259}a^{3}-\frac{8502428}{4201259}a^{2}+\frac{2068879}{4201259}a-\frac{3236120}{4201259}$, $\frac{3231593}{4201259}a^{15}-\frac{11650482}{4201259}a^{14}+\frac{19187251}{4201259}a^{13}-\frac{20263098}{4201259}a^{12}+\frac{9588762}{4201259}a^{11}+\frac{3390968}{4201259}a^{10}-\frac{14200458}{4201259}a^{9}+\frac{41317550}{4201259}a^{8}-\frac{32800164}{4201259}a^{7}-\frac{20568620}{4201259}a^{6}+\frac{38421018}{4201259}a^{5}-\frac{34519996}{4201259}a^{4}+\frac{15573192}{4201259}a^{3}+\frac{12043755}{4201259}a^{2}-\frac{2591707}{4201259}a+\frac{4878258}{4201259}$, $\frac{4648396}{4201259}a^{15}-\frac{18262619}{4201259}a^{14}+\frac{33827233}{4201259}a^{13}-\frac{41914424}{4201259}a^{12}+\frac{32910217}{4201259}a^{11}-\frac{14701024}{4201259}a^{10}-\frac{6318493}{4201259}a^{9}+\frac{55051796}{4201259}a^{8}-\frac{64352529}{4201259}a^{7}-\frac{3934792}{4201259}a^{6}+\frac{42617461}{4201259}a^{5}-\frac{47419252}{4201259}a^{4}+\frac{43968314}{4201259}a^{3}-\frac{12198061}{4201259}a^{2}+\frac{8116329}{4201259}a-\frac{3110789}{4201259}$, $\frac{173680}{4201259}a^{15}-\frac{2373477}{4201259}a^{14}+\frac{8345254}{4201259}a^{13}-\frac{15897831}{4201259}a^{12}+\frac{21111097}{4201259}a^{11}-\frac{18604204}{4201259}a^{10}+\frac{9972843}{4201259}a^{9}+\frac{2428501}{4201259}a^{8}-\frac{23393127}{4201259}a^{7}+\frac{28425538}{4201259}a^{6}-\frac{6059325}{4201259}a^{5}-\frac{14206948}{4201259}a^{4}+\frac{28531983}{4201259}a^{3}-\frac{26063124}{4201259}a^{2}+\frac{5397978}{4201259}a-\frac{3289320}{4201259}$, $\frac{1440473}{4201259}a^{15}-\frac{6691533}{4201259}a^{14}+\frac{14769575}{4201259}a^{13}-\frac{22096870}{4201259}a^{12}+\frac{22704388}{4201259}a^{11}-\frac{15575701}{4201259}a^{10}+\frac{3938985}{4201259}a^{9}+\frac{19247364}{4201259}a^{8}-\frac{34689460}{4201259}a^{7}+\frac{19371376}{4201259}a^{6}+\frac{4504900}{4201259}a^{5}-\frac{25864966}{4201259}a^{4}+\frac{32652499}{4201259}a^{3}-\frac{19407763}{4201259}a^{2}+\frac{12203739}{4201259}a-\frac{479904}{4201259}$, $\frac{1130760}{4201259}a^{15}-\frac{3713023}{4201259}a^{14}+\frac{6015181}{4201259}a^{13}-\frac{7338843}{4201259}a^{12}+\frac{6048556}{4201259}a^{11}-\frac{3976834}{4201259}a^{10}+\frac{469513}{4201259}a^{9}+\frac{10194147}{4201259}a^{8}-\frac{7804811}{4201259}a^{7}-\frac{4413315}{4201259}a^{6}+\frac{2019968}{4201259}a^{5}-\frac{6906874}{4201259}a^{4}+\frac{13447586}{4201259}a^{3}-\frac{4738145}{4201259}a^{2}+\frac{7091787}{4201259}a-\frac{569752}{4201259}$
| 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: | \( 22.9306095521 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 22.9306095521 \cdot 1}{6\cdot\sqrt{3345005787800625}}\cr\approx \mathstrut & 0.160511021714 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*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 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*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 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*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 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*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
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
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), 4.0.117.1 x2, 4.2.507.1 x2, \(\Q(\sqrt{-3}, \sqrt{13})\), 8.0.2313441.1 |
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: | 16.4.232292068597265625.1, 16.0.12370583534765625.1 |
| Minimal sibling: | 16.0.12370583534765625.1 |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.8.0.1 | $x^{8} + x^{4} + 3 x^{2} + 4 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| 5.8.4.2 | $x^{8} + 100 x^{4} - 500 x^{2} + 1250$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
|
\(13\)
| 13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |