Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 + 608*x^12 - 4520*x^10 + 17782*x^8 - 36792*x^6 + 36288*x^4 - 12312*x^2 + 81)
gp: K = bnfinit(y^16 - 40*y^14 + 608*y^12 - 4520*y^10 + 17782*y^8 - 36792*y^6 + 36288*y^4 - 12312*y^2 + 81, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 40*x^14 + 608*x^12 - 4520*x^10 + 17782*x^8 - 36792*x^6 + 36288*x^4 - 12312*x^2 + 81);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 40*x^14 + 608*x^12 - 4520*x^10 + 17782*x^8 - 36792*x^6 + 36288*x^4 - 12312*x^2 + 81)
\( x^{16} - 40x^{14} + 608x^{12} - 4520x^{10} + 17782x^{8} - 36792x^{6} + 36288x^{4} - 12312x^{2} + 81 \)
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: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(134588851614250885095358464\)
\(\medspace = 2^{58}\cdot 3^{4}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.96\) | 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: | $2^{29/8}3^{1/2}7^{1/2}\approx 56.53838276009771$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{4}-\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}-\frac{1}{6}a^{3}-\frac{1}{2}a$, $\frac{1}{6}a^{8}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{9}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{2}a$, $\frac{1}{6}a^{10}+\frac{1}{6}a^{2}$, $\frac{1}{12}a^{11}-\frac{1}{12}a^{10}-\frac{1}{12}a^{9}-\frac{1}{12}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{4}a^{3}+\frac{1}{12}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{36}a^{12}+\frac{1}{18}a^{10}-\frac{1}{36}a^{8}-\frac{1}{18}a^{6}+\frac{1}{36}a^{4}-\frac{1}{4}$, $\frac{1}{36}a^{13}-\frac{1}{36}a^{11}-\frac{1}{12}a^{10}+\frac{1}{18}a^{9}-\frac{1}{12}a^{8}-\frac{1}{18}a^{7}+\frac{13}{36}a^{5}-\frac{1}{3}a^{4}-\frac{1}{4}a^{3}+\frac{1}{12}a^{2}-\frac{1}{4}$, $\frac{1}{19968876}a^{14}+\frac{82793}{19968876}a^{12}-\frac{1315729}{19968876}a^{10}+\frac{1012285}{19968876}a^{8}+\frac{1310863}{19968876}a^{6}-\frac{471103}{6656292}a^{4}+\frac{622637}{2218764}a^{2}+\frac{276485}{739588}$, $\frac{1}{19968876}a^{15}+\frac{82793}{19968876}a^{13}+\frac{87086}{4992219}a^{11}-\frac{1}{12}a^{10}-\frac{162947}{4992219}a^{9}-\frac{1}{12}a^{8}+\frac{1310863}{19968876}a^{7}-\frac{2689867}{6656292}a^{5}-\frac{1}{3}a^{4}-\frac{260359}{554691}a^{3}+\frac{1}{12}a^{2}+\frac{22897}{184897}a-\frac{1}{4}$
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 $1$ (assuming GRH)
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: | $15$ | 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: |
$\frac{56}{72351}a^{14}-\frac{2095}{72351}a^{12}+\frac{9589}{24117}a^{10}-\frac{367891}{144702}a^{8}+\frac{65674}{8039}a^{6}-\frac{966029}{72351}a^{4}+\frac{82919}{8039}a^{2}-\frac{39919}{16078}$, $\frac{5933}{4992219}a^{14}-\frac{245357}{4992219}a^{12}+\frac{3829123}{4992219}a^{10}-\frac{56638277}{9984438}a^{8}+\frac{102090812}{4992219}a^{6}-\frac{6118384}{184897}a^{4}+\frac{9847699}{554691}a^{2}+\frac{513333}{369794}$, $\frac{85243}{19968876}a^{14}-\frac{784510}{4992219}a^{12}+\frac{41770805}{19968876}a^{10}-\frac{124953055}{9984438}a^{8}+\frac{684990835}{19968876}a^{6}-\frac{118838225}{3328146}a^{4}+\frac{5199469}{2218764}a^{2}+\frac{960724}{184897}$, $\frac{59147}{9984438}a^{14}-\frac{2073341}{9984438}a^{12}+\frac{12787925}{4992219}a^{10}-\frac{136179631}{9984438}a^{8}+\frac{159481294}{4992219}a^{6}-\frac{44643349}{1664073}a^{4}-\frac{1043615}{1109382}a^{2}+\frac{21565}{184897}$, $\frac{14425}{6656292}a^{14}-\frac{515389}{6656292}a^{12}+\frac{6572723}{6656292}a^{10}-\frac{37193747}{6656292}a^{8}+\frac{97408201}{6656292}a^{6}-\frac{35086867}{2218764}a^{4}+\frac{6972245}{2218764}a^{2}+\frac{1682681}{739588}$, $\frac{61511}{19968876}a^{14}-\frac{539153}{4992219}a^{12}+\frac{26454313}{19968876}a^{10}-\frac{34157389}{4992219}a^{8}+\frac{276627587}{19968876}a^{6}-\frac{8707313}{3328146}a^{4}-\frac{11397109}{739588}a^{2}+\frac{298733}{369794}$, $\frac{85}{96468}a^{14}-\frac{3119}{144702}a^{12}+\frac{20887}{289404}a^{10}+\frac{205187}{144702}a^{8}-\frac{3046375}{289404}a^{6}+\frac{3216907}{144702}a^{4}-\frac{1311841}{96468}a^{2}-\frac{9591}{16078}$, $\frac{6031}{19968876}a^{15}+\frac{213191}{19968876}a^{14}-\frac{113002}{4992219}a^{13}-\frac{3932761}{9984438}a^{12}+\frac{11341667}{19968876}a^{11}+\frac{105501841}{19968876}a^{10}-\frac{31992983}{4992219}a^{9}-\frac{161615701}{4992219}a^{8}+\frac{701488045}{19968876}a^{7}+\frac{1905692489}{19968876}a^{6}-\frac{155383427}{1664073}a^{5}-\frac{419982863}{3328146}a^{4}+\frac{238752601}{2218764}a^{3}+\frac{127299151}{2218764}a^{2}-\frac{6682544}{184897}a-\frac{586061}{184897}$, $\frac{406}{4992219}a^{15}+\frac{19577}{1664073}a^{14}+\frac{110305}{9984438}a^{13}-\frac{78681}{184897}a^{12}-\frac{2237305}{4992219}a^{11}+\frac{9219287}{1664073}a^{10}+\frac{57056531}{9984438}a^{9}-\frac{18038911}{554691}a^{8}-\frac{307330087}{9984438}a^{7}+\frac{303017855}{3328146}a^{6}+\frac{41696983}{554691}a^{5}-\frac{378840791}{3328146}a^{4}-\frac{88492817}{1109382}a^{3}+\frac{52135001}{1109382}a^{2}+\frac{5197547}{184897}a-\frac{492083}{369794}$, $\frac{58325}{9984438}a^{15}+\frac{21733}{19968876}a^{14}-\frac{4359079}{19968876}a^{13}-\frac{316013}{9984438}a^{12}+\frac{14947583}{4992219}a^{11}+\frac{5129603}{19968876}a^{10}-\frac{381756929}{19968876}a^{9}-\frac{734498}{4992219}a^{8}+\frac{304095913}{4992219}a^{7}-\frac{87585359}{19968876}a^{6}-\frac{629022689}{6656292}a^{5}+\frac{6857012}{554691}a^{4}+\frac{70313909}{1109382}a^{3}-\frac{7676383}{739588}a^{2}-\frac{10517383}{739588}a+\frac{1049833}{369794}$, $\frac{5933}{4992219}a^{15}-\frac{50029}{9984438}a^{14}-\frac{245357}{4992219}a^{13}+\frac{3536237}{19968876}a^{12}+\frac{3829123}{4992219}a^{11}-\frac{11103889}{4992219}a^{10}-\frac{56638277}{9984438}a^{9}+\frac{247310947}{19968876}a^{8}+\frac{102090812}{4992219}a^{7}-\frac{163380548}{4992219}a^{6}-\frac{6118384}{184897}a^{5}+\frac{88806457}{2218764}a^{4}+\frac{9847699}{554691}a^{3}-\frac{20647553}{1109382}a^{2}+\frac{513333}{369794}a+\frac{307701}{739588}$, $\frac{6689}{9984438}a^{15}+\frac{62245}{9984438}a^{14}-\frac{166966}{4992219}a^{13}-\frac{4250429}{19968876}a^{12}+\frac{12390143}{19968876}a^{11}+\frac{50079061}{19968876}a^{10}-\frac{104713943}{19968876}a^{9}-\frac{61238170}{4992219}a^{8}+\frac{198386657}{9984438}a^{7}+\frac{122674724}{4992219}a^{6}-\frac{47152933}{1664073}a^{5}-\frac{10512823}{739588}a^{4}+\frac{7608713}{2218764}a^{3}-\frac{3906309}{739588}a^{2}+\frac{7347725}{739588}a+\frac{335653}{369794}$, $\frac{20635}{6656292}a^{15}-\frac{70531}{9984438}a^{14}-\frac{754313}{6656292}a^{13}+\frac{2532529}{9984438}a^{12}+\frac{1667851}{1109382}a^{11}-\frac{65143249}{19968876}a^{10}-\frac{15220954}{1664073}a^{9}+\frac{374683267}{19968876}a^{8}+\frac{20334947}{739588}a^{7}-\frac{508679329}{9984438}a^{6}-\frac{268297849}{6656292}a^{5}+\frac{67884037}{1109382}a^{4}+\frac{14369561}{554691}a^{3}-\frac{53343199}{2218764}a^{2}-\frac{2366895}{369794}a+\frac{91419}{739588}$, $\frac{66043}{19968876}a^{15}-\frac{50587}{19968876}a^{14}-\frac{2464543}{19968876}a^{13}+\frac{1886323}{19968876}a^{12}+\frac{16854859}{9984438}a^{11}-\frac{12885013}{9984438}a^{10}-\frac{106952623}{9984438}a^{9}+\frac{40784072}{4992219}a^{8}+\frac{671658085}{19968876}a^{7}-\frac{508523869}{19968876}a^{6}-\frac{333991427}{6656292}a^{5}+\frac{245116759}{6656292}a^{4}+\frac{32310719}{1109382}a^{3}-\frac{20867897}{1109382}a^{2}-\frac{356881}{369794}a+\frac{89166}{184897}$, $\frac{19387}{9984438}a^{15}+\frac{108011}{19968876}a^{14}-\frac{361277}{4992219}a^{13}-\frac{1014104}{4992219}a^{12}+\frac{19378591}{19968876}a^{11}+\frac{27881177}{9984438}a^{10}-\frac{113308225}{19968876}a^{9}-\frac{352230115}{19968876}a^{8}+\frac{65719946}{4992219}a^{7}+\frac{1064424317}{19968876}a^{6}-\frac{4169383}{3328146}a^{5}-\frac{38867273}{554691}a^{4}-\frac{20481651}{739588}a^{3}+\frac{10474777}{369794}a^{2}+\frac{11749841}{739588}a-\frac{772311}{739588}$
| 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: | \( 129617382.619 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 129617382.619 \cdot 1}{2\cdot\sqrt{134588851614250885095358464}}\cr\approx \mathstrut & 0.366107510800 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 + 608*x^12 - 4520*x^10 + 17782*x^8 - 36792*x^6 + 36288*x^4 - 12312*x^2 + 81)
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 - 40*x^14 + 608*x^12 - 4520*x^10 + 17782*x^8 - 36792*x^6 + 36288*x^4 - 12312*x^2 + 81, 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 - 40*x^14 + 608*x^12 - 4520*x^10 + 17782*x^8 - 36792*x^6 + 36288*x^4 - 12312*x^2 + 81);
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 - 40*x^14 + 608*x^12 - 4520*x^10 + 17782*x^8 - 36792*x^6 + 36288*x^4 - 12312*x^2 + 81);
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
$Q_{16}:C_2$ (as 16T50):
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 $Q_{16}:C_2$ |
| Character table for $Q_{16}:C_2$ |
Intermediate fields
| \(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{14}) \), 4.4.25088.1 x2, 4.4.7168.1 x2, \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.10070523904.1 |
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 sibling: | 16.16.222483611852129014137225216.1 |
| Minimal sibling: | 16.16.222483611852129014137225216.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.58.70 | $x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 8 x^{10} + 30 x^{8} + 16 x^{5} + 12 x^{4} + 16 x^{3} + 2$ | $16$ | $1$ | $58$ | 16T50 | $[2, 3, 7/2, 9/2]^{2}$ |
|
\(3\)
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(7\)
| 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 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |