Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 24*x^14 - 50*x^13 + 65*x^12 - 40*x^11 - 28*x^10 + 94*x^9 - 94*x^8 + 16*x^7 + 82*x^6 - 128*x^5 + 108*x^4 - 59*x^3 + 22*x^2 - 6*x + 1)
gp: K = bnfinit(y^16 - 7*y^15 + 24*y^14 - 50*y^13 + 65*y^12 - 40*y^11 - 28*y^10 + 94*y^9 - 94*y^8 + 16*y^7 + 82*y^6 - 128*y^5 + 108*y^4 - 59*y^3 + 22*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 24*x^14 - 50*x^13 + 65*x^12 - 40*x^11 - 28*x^10 + 94*x^9 - 94*x^8 + 16*x^7 + 82*x^6 - 128*x^5 + 108*x^4 - 59*x^3 + 22*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 + 24*x^14 - 50*x^13 + 65*x^12 - 40*x^11 - 28*x^10 + 94*x^9 - 94*x^8 + 16*x^7 + 82*x^6 - 128*x^5 + 108*x^4 - 59*x^3 + 22*x^2 - 6*x + 1)
\( x^{16} - 7 x^{15} + 24 x^{14} - 50 x^{13} + 65 x^{12} - 40 x^{11} - 28 x^{10} + 94 x^{9} - 94 x^{8} + \cdots + 1 \)
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: |
\(12370583534765625\)
\(\medspace = 3^{8}\cdot 5^{8}\cdot 13^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.13\) | 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2641}a^{15}-\frac{27}{2641}a^{14}+\frac{564}{2641}a^{13}-\frac{766}{2641}a^{12}-\frac{461}{2641}a^{11}+\frac{1257}{2641}a^{10}+\frac{1242}{2641}a^{9}-\frac{977}{2641}a^{8}+\frac{959}{2641}a^{7}-\frac{677}{2641}a^{6}+\frac{3}{19}a^{5}-\frac{545}{2641}a^{4}+\frac{444}{2641}a^{3}-\frac{1016}{2641}a^{2}-\frac{786}{2641}a-\frac{132}{2641}$
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: |
\( \frac{413}{139} a^{15} - \frac{2672}{139} a^{14} + \frac{8447}{139} a^{13} - \frac{15840}{139} a^{12} + \frac{17412}{139} a^{11} - \frac{5445}{139} a^{10} - \frac{16227}{139} a^{9} + \frac{30318}{139} a^{8} - \frac{20377}{139} a^{7} - \frac{7578}{139} a^{6} + 226 a^{5} - \frac{34516}{139} a^{4} + \frac{22688}{139} a^{3} - \frac{9002}{139} a^{2} + \frac{2449}{139} a - \frac{445}{139} \)
(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: |
$\frac{6700}{2641}a^{15}-\frac{43568}{2641}a^{14}+\frac{139502}{2641}a^{13}-\frac{267478}{2641}a^{12}+\frac{307626}{2641}a^{11}-\frac{124376}{2641}a^{10}-\frac{238081}{2641}a^{9}+\frac{505570}{2641}a^{8}-\frac{385839}{2641}a^{7}-\frac{64687}{2641}a^{6}+\frac{3589}{19}a^{5}-\frac{609068}{2641}a^{4}+\frac{442081}{2641}a^{3}-\frac{202059}{2641}a^{2}+\frac{63338}{2641}a-\frac{12870}{2641}$, $\frac{4059}{2641}a^{15}-\frac{25081}{2641}a^{14}+\frac{76118}{2641}a^{13}-\frac{135428}{2641}a^{12}+\frac{135961}{2641}a^{11}-\frac{18736}{2641}a^{10}-\frac{164133}{2641}a^{9}+\frac{257316}{2641}a^{8}-\frac{137585}{2641}a^{7}-\frac{106943}{2641}a^{6}+\frac{2031}{19}a^{5}-\frac{271020}{2641}a^{4}+\frac{156853}{2641}a^{3}-\frac{46240}{2641}a^{2}+\frac{5236}{2641}a+\frac{335}{2641}$, $\frac{194}{139}a^{15}-\frac{1207}{139}a^{14}+\frac{3637}{139}a^{13}-\frac{6407}{139}a^{12}+\frac{6337}{139}a^{11}-\frac{782}{139}a^{10}-\frac{7723}{139}a^{9}+\frac{11873}{139}a^{8}-\frac{6191}{139}a^{7}-\frac{5126}{139}a^{6}+94a^{5}-\frac{12600}{139}a^{4}+\frac{7323}{139}a^{3}-\frac{2365}{139}a^{2}+\frac{555}{139}a-\frac{171}{139}$, $\frac{4855}{2641}a^{15}-\frac{30727}{2641}a^{14}+\frac{97220}{2641}a^{13}-\frac{185272}{2641}a^{12}+\frac{212693}{2641}a^{11}-\frac{85128}{2641}a^{10}-\frac{165876}{2641}a^{9}+\frac{351154}{2641}a^{8}-\frac{266879}{2641}a^{7}-\frac{46328}{2641}a^{6}+\frac{2519}{19}a^{5}-\frac{422253}{2641}a^{4}+\frac{304279}{2641}a^{3}-\frac{139265}{2641}a^{2}+\frac{45112}{2641}a-\frac{9661}{2641}$, $\frac{899}{2641}a^{15}-\frac{3145}{2641}a^{14}+\frac{2605}{2641}a^{13}+\frac{11231}{2641}a^{12}-\frac{39417}{2641}a^{11}+\frac{60438}{2641}a^{10}-\frac{37559}{2641}a^{9}-\frac{30562}{2641}a^{8}+\frac{93610}{2641}a^{7}-\frac{80423}{2641}a^{6}+\frac{18}{19}a^{5}+\frac{77860}{2641}a^{4}-\frac{94711}{2641}a^{3}+\frac{66427}{2641}a^{2}-\frac{22595}{2641}a+\frac{5459}{2641}$, $\frac{219}{139}a^{15}-\frac{1465}{139}a^{14}+\frac{4810}{139}a^{13}-\frac{9433}{139}a^{12}+\frac{11075}{139}a^{11}-\frac{4663}{139}a^{10}-\frac{8504}{139}a^{9}+\frac{18445}{139}a^{8}-\frac{14186}{139}a^{7}-\frac{2452}{139}a^{6}+132a^{5}-\frac{21916}{139}a^{4}+\frac{15365}{139}a^{3}-\frac{6637}{139}a^{2}+\frac{2033}{139}a-\frac{413}{139}$, $\frac{7017}{2641}a^{15}-\frac{44204}{2641}a^{14}+\frac{136061}{2641}a^{13}-\frac{246200}{2641}a^{12}+\frac{253924}{2641}a^{11}-\frac{48109}{2641}a^{10}-\frac{285414}{2641}a^{9}+\frac{465243}{2641}a^{8}-\frac{261424}{2641}a^{7}-\frac{178938}{2641}a^{6}+\frac{3609}{19}a^{5}-\frac{491323}{2641}a^{4}+\frac{294960}{2641}a^{3}-\frac{106853}{2641}a^{2}+\frac{30738}{2641}a-\frac{7176}{2641}$
| 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: | \( 50.440336623 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 50.440336623 \cdot 1}{6\cdot\sqrt{12370583534765625}}\cr\approx \mathstrut & 0.18359911337 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 24*x^14 - 50*x^13 + 65*x^12 - 40*x^11 - 28*x^10 + 94*x^9 - 94*x^8 + 16*x^7 + 82*x^6 - 128*x^5 + 108*x^4 - 59*x^3 + 22*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 - 7*x^15 + 24*x^14 - 50*x^13 + 65*x^12 - 40*x^11 - 28*x^10 + 94*x^9 - 94*x^8 + 16*x^7 + 82*x^6 - 128*x^5 + 108*x^4 - 59*x^3 + 22*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 - 7*x^15 + 24*x^14 - 50*x^13 + 65*x^12 - 40*x^11 - 28*x^10 + 94*x^9 - 94*x^8 + 16*x^7 + 82*x^6 - 128*x^5 + 108*x^4 - 59*x^3 + 22*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 - 7*x^15 + 24*x^14 - 50*x^13 + 65*x^12 - 40*x^11 - 28*x^10 + 94*x^9 - 94*x^8 + 16*x^7 + 82*x^6 - 128*x^5 + 108*x^4 - 59*x^3 + 22*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
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
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), 4.0.117.1, 4.0.2925.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.8555625.1 |
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, 16.0.3345005787800625.1 |
| 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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(13\)
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |