Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 10*x^13 + 7*x^12 + 97*x^10 + 388*x^8 - 280*x^7 + 881*x^6 - 1030*x^5 + 1488*x^4 - 1080*x^3 + 624*x^2 - 320*x + 256)
gp: K = bnfinit(y^16 + 2*y^14 - 10*y^13 + 7*y^12 + 97*y^10 + 388*y^8 - 280*y^7 + 881*y^6 - 1030*y^5 + 1488*y^4 - 1080*y^3 + 624*y^2 - 320*y + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 2*x^14 - 10*x^13 + 7*x^12 + 97*x^10 + 388*x^8 - 280*x^7 + 881*x^6 - 1030*x^5 + 1488*x^4 - 1080*x^3 + 624*x^2 - 320*x + 256);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 2*x^14 - 10*x^13 + 7*x^12 + 97*x^10 + 388*x^8 - 280*x^7 + 881*x^6 - 1030*x^5 + 1488*x^4 - 1080*x^3 + 624*x^2 - 320*x + 256)
\( x^{16} + 2 x^{14} - 10 x^{13} + 7 x^{12} + 97 x^{10} + 388 x^{8} - 280 x^{7} + 881 x^{6} - 1030 x^{5} + \cdots + 256 \)
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: |
\(592620628869750390625\)
\(\medspace = 5^{8}\cdot 79^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.87\) | 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: | $5^{1/2}79^{1/2}\approx 19.87460691435179$ | ||
| Ramified primes: |
\(5\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}+\frac{1}{16}a^{5}-\frac{3}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{12}+\frac{1}{32}a^{10}-\frac{1}{16}a^{8}-\frac{3}{16}a^{7}-\frac{5}{32}a^{6}+\frac{1}{16}a^{5}-\frac{3}{32}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{64}a^{13}+\frac{1}{64}a^{11}-\frac{1}{32}a^{9}-\frac{3}{32}a^{8}-\frac{5}{64}a^{7}+\frac{1}{32}a^{6}-\frac{3}{64}a^{5}+\frac{1}{16}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{14}-\frac{1}{64}a^{12}-\frac{1}{16}a^{10}-\frac{3}{32}a^{9}-\frac{1}{64}a^{8}+\frac{7}{32}a^{7}+\frac{7}{64}a^{6}-\frac{5}{32}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{52\!\cdots\!36}a^{15}-\frac{35086997524683}{52\!\cdots\!36}a^{14}+\frac{2542751559787}{52\!\cdots\!36}a^{13}-\frac{54918452568947}{52\!\cdots\!36}a^{12}-\frac{4359126684583}{659031307428992}a^{11}+\frac{40744589346203}{659031307428992}a^{10}+\frac{445096294637401}{52\!\cdots\!36}a^{9}-\frac{137015898364947}{52\!\cdots\!36}a^{8}-\frac{611943946095227}{52\!\cdots\!36}a^{7}+\frac{751707198247265}{52\!\cdots\!36}a^{6}-\frac{625129700738757}{26\!\cdots\!68}a^{5}-\frac{39846022485565}{659031307428992}a^{4}-\frac{239903240447867}{659031307428992}a^{3}-\frac{131148514860975}{329515653714496}a^{2}-\frac{30694114410031}{82378913428624}a+\frac{7283193052735}{20594728357156}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{5}$, which has order $5$
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: |
$\frac{10106461}{32049959936}a^{15}+\frac{85551809}{32049959936}a^{14}+\frac{69485871}{32049959936}a^{13}+\frac{19696441}{32049959936}a^{12}-\frac{89828273}{4006244992}a^{11}-\frac{4036257}{4006244992}a^{10}+\frac{1683608117}{32049959936}a^{9}+\frac{8311895481}{32049959936}a^{8}+\frac{8573441729}{32049959936}a^{7}+\frac{25886590685}{32049959936}a^{6}+\frac{29273271}{16024979968}a^{5}+\frac{3629872079}{4006244992}a^{4}-\frac{4367946175}{4006244992}a^{3}+\frac{1158073989}{2003122496}a^{2}-\frac{59906859}{500780624}a-\frac{26847105}{125195156}$, $\frac{2875190234447}{52\!\cdots\!36}a^{15}-\frac{212468533205}{52\!\cdots\!36}a^{14}-\frac{8058591785371}{52\!\cdots\!36}a^{13}-\frac{43761689492205}{52\!\cdots\!36}a^{12}+\frac{2496974694411}{659031307428992}a^{11}+\frac{13100262275377}{659031307428992}a^{10}+\frac{336553258216663}{52\!\cdots\!36}a^{9}-\frac{264715320267309}{52\!\cdots\!36}a^{8}-\frac{128020934145205}{52\!\cdots\!36}a^{7}-\frac{19\!\cdots\!81}{52\!\cdots\!36}a^{6}-\frac{647047926809563}{26\!\cdots\!68}a^{5}-\frac{563475593330147}{659031307428992}a^{4}+\frac{556293826737451}{659031307428992}a^{3}-\frac{224506883237201}{329515653714496}a^{2}+\frac{27043356139503}{82378913428624}a+\frac{3933944287129}{20594728357156}$, $\frac{10835746248211}{26\!\cdots\!68}a^{15}+\frac{3192448628055}{26\!\cdots\!68}a^{14}+\frac{25086289924105}{26\!\cdots\!68}a^{13}-\frac{98011593267761}{26\!\cdots\!68}a^{12}+\frac{3197200337945}{164757826857248}a^{11}+\frac{45291990465}{329515653714496}a^{10}+\frac{10\!\cdots\!43}{26\!\cdots\!68}a^{9}+\frac{320630246242015}{26\!\cdots\!68}a^{8}+\frac{44\!\cdots\!07}{26\!\cdots\!68}a^{7}-\frac{13\!\cdots\!81}{26\!\cdots\!68}a^{6}+\frac{50\!\cdots\!21}{13\!\cdots\!84}a^{5}-\frac{927292694929073}{329515653714496}a^{4}+\frac{17\!\cdots\!07}{329515653714496}a^{3}-\frac{459663300951001}{164757826857248}a^{2}+\frac{21731236990053}{41189456714312}a-\frac{14580886312157}{10297364178578}$, $\frac{11360250584039}{52\!\cdots\!36}a^{15}+\frac{5661990144051}{52\!\cdots\!36}a^{14}+\frac{21118695038269}{52\!\cdots\!36}a^{13}-\frac{97419150513445}{52\!\cdots\!36}a^{12}+\frac{2537743872069}{659031307428992}a^{11}+\frac{5623662220093}{659031307428992}a^{10}+\frac{10\!\cdots\!79}{52\!\cdots\!36}a^{9}+\frac{541662881041755}{52\!\cdots\!36}a^{8}+\frac{44\!\cdots\!55}{52\!\cdots\!36}a^{7}-\frac{747546100226425}{52\!\cdots\!36}a^{6}+\frac{39\!\cdots\!01}{26\!\cdots\!68}a^{5}-\frac{754774697301659}{659031307428992}a^{4}+\frac{10\!\cdots\!35}{659031307428992}a^{3}-\frac{350438197214449}{329515653714496}a^{2}-\frac{3242576870641}{82378913428624}a-\frac{19804003588239}{20594728357156}$, $\frac{4357061849239}{52\!\cdots\!36}a^{15}+\frac{11443281143619}{52\!\cdots\!36}a^{14}+\frac{21805437395885}{52\!\cdots\!36}a^{13}-\frac{14838434494293}{52\!\cdots\!36}a^{12}-\frac{7492005532951}{659031307428992}a^{11}-\frac{6540948143323}{659031307428992}a^{10}+\frac{429766856754015}{52\!\cdots\!36}a^{9}+\frac{11\!\cdots\!83}{52\!\cdots\!36}a^{8}+\frac{31\!\cdots\!51}{52\!\cdots\!36}a^{7}+\frac{39\!\cdots\!83}{52\!\cdots\!36}a^{6}+\frac{25\!\cdots\!89}{26\!\cdots\!68}a^{5}+\frac{305872501481877}{659031307428992}a^{4}+\frac{286292393468099}{659031307428992}a^{3}+\frac{63868686941919}{329515653714496}a^{2}+\frac{11375259200887}{82378913428624}a+\frac{3886163271753}{20594728357156}$, $\frac{1259816771867}{52\!\cdots\!36}a^{15}+\frac{1211083142471}{52\!\cdots\!36}a^{14}+\frac{5467411630473}{52\!\cdots\!36}a^{13}-\frac{7011052261969}{52\!\cdots\!36}a^{12}-\frac{988594764581}{659031307428992}a^{11}-\frac{732065703951}{659031307428992}a^{10}+\frac{128613160363843}{52\!\cdots\!36}a^{9}+\frac{198244004137167}{52\!\cdots\!36}a^{8}+\frac{618321472748391}{52\!\cdots\!36}a^{7}+\frac{233596003243019}{52\!\cdots\!36}a^{6}+\frac{782153570278553}{26\!\cdots\!68}a^{5}+\frac{157377655948965}{659031307428992}a^{4}-\frac{62444811211337}{659031307428992}a^{3}+\frac{186736863901771}{329515653714496}a^{2}-\frac{21512706760065}{82378913428624}a+\frac{11348103819709}{20594728357156}$, $\frac{9076765331201}{52\!\cdots\!36}a^{15}-\frac{5440586733547}{52\!\cdots\!36}a^{14}+\frac{8958368263179}{52\!\cdots\!36}a^{13}-\frac{98159529378835}{52\!\cdots\!36}a^{12}+\frac{11457242706293}{659031307428992}a^{11}+\frac{5045380518811}{659031307428992}a^{10}+\frac{772708849357145}{52\!\cdots\!36}a^{9}-\frac{466073053661363}{52\!\cdots\!36}a^{8}+\frac{27\!\cdots\!41}{52\!\cdots\!36}a^{7}-\frac{44\!\cdots\!31}{52\!\cdots\!36}a^{6}+\frac{25\!\cdots\!95}{26\!\cdots\!68}a^{5}-\frac{15\!\cdots\!97}{659031307428992}a^{4}+\frac{903929408716325}{659031307428992}a^{3}-\frac{566406810724447}{329515653714496}a^{2}-\frac{46208083230807}{82378913428624}a-\frac{20023423848373}{20594728357156}$
| 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: | \( 5928.11603186 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5928.11603186 \cdot 5}{2\cdot\sqrt{592620628869750390625}}\cr\approx \mathstrut & 1.47879259977 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 10*x^13 + 7*x^12 + 97*x^10 + 388*x^8 - 280*x^7 + 881*x^6 - 1030*x^5 + 1488*x^4 - 1080*x^3 + 624*x^2 - 320*x + 256)
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^14 - 10*x^13 + 7*x^12 + 97*x^10 + 388*x^8 - 280*x^7 + 881*x^6 - 1030*x^5 + 1488*x^4 - 1080*x^3 + 624*x^2 - 320*x + 256, 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^14 - 10*x^13 + 7*x^12 + 97*x^10 + 388*x^8 - 280*x^7 + 881*x^6 - 1030*x^5 + 1488*x^4 - 1080*x^3 + 624*x^2 - 320*x + 256);
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^14 - 10*x^13 + 7*x^12 + 97*x^10 + 388*x^8 - 280*x^7 + 881*x^6 - 1030*x^5 + 1488*x^4 - 1080*x^3 + 624*x^2 - 320*x + 256);
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 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{-395}) \), \(\Q(\sqrt{-79}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-79})\), 4.0.31205.1 x2, 4.2.1975.1 x2, 8.0.24343800625.1, 8.0.4868760125.1 x4, 8.2.308149375.1 x4 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(79\)
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |