Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 52*x^14 + 778*x^12 - 4009*x^10 + 6565*x^8 - 4009*x^6 + 778*x^4 - 52*x^2 + 1)
gp: K = bnfinit(y^16 - 52*y^14 + 778*y^12 - 4009*y^10 + 6565*y^8 - 4009*y^6 + 778*y^4 - 52*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 52*x^14 + 778*x^12 - 4009*x^10 + 6565*x^8 - 4009*x^6 + 778*x^4 - 52*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 52*x^14 + 778*x^12 - 4009*x^10 + 6565*x^8 - 4009*x^6 + 778*x^4 - 52*x^2 + 1)
\( x^{16} - 52x^{14} + 778x^{12} - 4009x^{10} + 6565x^{8} - 4009x^{6} + 778x^{4} - 52x^{2} + 1 \)
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: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1622628503115275885009765625\)
\(\medspace = 3^{12}\cdot 5^{14}\cdot 29^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.19\) | 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: | $3^{3/4}5^{7/8}29^{1/2}\approx 50.19248452708817$ | ||
| Ramified primes: |
\(3\), \(5\), \(29\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6}a^{8}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}+\frac{1}{6}$, $\frac{1}{6}a^{9}-\frac{1}{3}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{3}a^{3}+\frac{1}{6}a$, $\frac{1}{6}a^{10}-\frac{1}{2}a^{7}-\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{3}a^{4}-\frac{1}{2}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{11}-\frac{1}{6}a^{7}-\frac{1}{2}a^{6}+\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{2706}a^{12}+\frac{86}{1353}a^{10}+\frac{34}{1353}a^{8}-\frac{1}{2}a^{7}-\frac{225}{902}a^{6}-\frac{383}{2706}a^{4}-\frac{1}{2}a^{3}-\frac{365}{1353}a^{2}-\frac{1}{2}a-\frac{901}{2706}$, $\frac{1}{2706}a^{13}+\frac{86}{1353}a^{11}+\frac{34}{1353}a^{9}-\frac{225}{902}a^{7}+\frac{485}{1353}a^{5}+\frac{623}{2706}a^{3}-\frac{1}{2}a^{2}-\frac{901}{2706}a-\frac{1}{2}$, $\frac{1}{51414}a^{14}+\frac{2}{25707}a^{12}-\frac{3121}{51414}a^{10}+\frac{3235}{51414}a^{8}-\frac{1}{2}a^{7}-\frac{6949}{51414}a^{6}-\frac{1}{2}a^{5}+\frac{16259}{51414}a^{4}-\frac{1}{2}a^{3}+\frac{2325}{8569}a^{2}-\frac{1}{2}a+\frac{10289}{25707}$, $\frac{1}{51414}a^{15}+\frac{2}{25707}a^{13}-\frac{3121}{51414}a^{11}+\frac{3235}{51414}a^{9}-\frac{6949}{51414}a^{7}-\frac{1}{2}a^{6}-\frac{4724}{25707}a^{5}-\frac{3919}{17138}a^{3}-\frac{1}{2}a^{2}+\frac{10289}{25707}a-\frac{1}{2}$
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 $2$ (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: |
$\frac{223}{1353}a^{14}-\frac{11593}{1353}a^{12}+\frac{173336}{1353}a^{10}-\frac{891569}{1353}a^{8}+\frac{1450414}{1353}a^{6}-\frac{866401}{1353}a^{4}+\frac{49942}{451}a^{2}-\frac{4849}{1353}$, $\frac{706}{1353}a^{15}-\frac{223}{2706}a^{14}-\frac{24427}{902}a^{13}+\frac{11593}{2706}a^{12}+\frac{181842}{451}a^{11}-\frac{86668}{1353}a^{10}-\frac{5547029}{2706}a^{9}+\frac{891569}{2706}a^{8}+\frac{1443795}{451}a^{7}-\frac{725207}{1353}a^{6}-\frac{1527025}{902}a^{5}+\frac{866401}{2706}a^{4}+\frac{442169}{2706}a^{3}-\frac{24971}{451}a^{2}+\frac{903}{451}a+\frac{1748}{1353}$, $\frac{1816}{1353}a^{15}+\frac{223}{2706}a^{14}-\frac{62841}{902}a^{13}-\frac{11593}{2706}a^{12}+\frac{936035}{902}a^{11}+\frac{86668}{1353}a^{10}-\frac{7151299}{1353}a^{9}-\frac{891569}{2706}a^{8}+\frac{7526795}{902}a^{7}+\frac{725207}{1353}a^{6}-\frac{4265709}{902}a^{5}-\frac{866401}{2706}a^{4}+\frac{1976657}{2706}a^{3}+\frac{24971}{451}a^{2}-\frac{25431}{902}a-\frac{7555}{2706}$, $a$, $\frac{12163}{51414}a^{15}+\frac{148}{2337}a^{14}-\frac{317275}{25707}a^{13}-\frac{7559}{2337}a^{12}+\frac{233353}{1254}a^{11}+\frac{216317}{4674}a^{10}-\frac{50212501}{51414}a^{9}-\frac{493811}{2337}a^{8}+\frac{14308608}{8569}a^{7}+\frac{174388}{779}a^{6}-\frac{51990475}{51414}a^{5}-\frac{149966}{2337}a^{4}+\frac{7698565}{51414}a^{3}+\frac{8988}{779}a^{2}-\frac{192497}{25707}a-\frac{635}{4674}$, $\frac{2015}{17138}a^{15}-\frac{14318}{25707}a^{14}-\frac{52552}{8569}a^{13}+\frac{743692}{25707}a^{12}+\frac{2377415}{25707}a^{11}-\frac{11095717}{25707}a^{10}-\frac{25055209}{51414}a^{9}+\frac{113508949}{51414}a^{8}+\frac{22242557}{25707}a^{7}-\frac{30253313}{8569}a^{6}-\frac{17228647}{25707}a^{5}+\frac{52518050}{25707}a^{4}+\frac{5649637}{25707}a^{3}-\frac{16853791}{51414}a^{2}-\frac{397906}{25707}a+\frac{549325}{51414}$, $\frac{2015}{17138}a^{15}+\frac{14318}{25707}a^{14}-\frac{52552}{8569}a^{13}-\frac{743692}{25707}a^{12}+\frac{2377415}{25707}a^{11}+\frac{11095717}{25707}a^{10}-\frac{25055209}{51414}a^{9}-\frac{113508949}{51414}a^{8}+\frac{22242557}{25707}a^{7}+\frac{30253313}{8569}a^{6}-\frac{17228647}{25707}a^{5}-\frac{52518050}{25707}a^{4}+\frac{5649637}{25707}a^{3}+\frac{16853791}{51414}a^{2}-\frac{397906}{25707}a-\frac{549325}{51414}$, $\frac{8069}{25707}a^{15}+\frac{6142}{25707}a^{14}-\frac{140779}{8569}a^{13}-\frac{105570}{8569}a^{12}+\frac{156532}{627}a^{11}+\frac{1547301}{8569}a^{10}-\frac{22895685}{17138}a^{9}-\frac{22669198}{25707}a^{8}+\frac{20646701}{8569}a^{7}+\frac{20936503}{17138}a^{6}-\frac{82156643}{51414}a^{5}-\frac{10155865}{17138}a^{4}+\frac{7762369}{25707}a^{3}+\frac{2345209}{25707}a^{2}-\frac{241513}{17138}a-\frac{31817}{8569}$, $\frac{8133}{17138}a^{15}+\frac{15925}{51414}a^{14}-\frac{1267117}{51414}a^{13}-\frac{826583}{51414}a^{12}+\frac{3149675}{8569}a^{11}+\frac{6155641}{25707}a^{10}-\frac{48284569}{25707}a^{9}-\frac{31344758}{25707}a^{8}+\frac{76981006}{25707}a^{7}+\frac{16474380}{8569}a^{6}-\frac{29524827}{17138}a^{5}-\frac{55651243}{51414}a^{4}+\frac{14007097}{51414}a^{3}+\frac{8313709}{51414}a^{2}-\frac{84768}{8569}a-\frac{22475}{4674}$, $\frac{18158}{25707}a^{15}+\frac{3730}{25707}a^{14}-\frac{628403}{17138}a^{13}-\frac{194099}{25707}a^{12}+\frac{28085917}{51414}a^{11}+\frac{5817607}{51414}a^{10}-\frac{143066621}{51414}a^{9}-\frac{15043087}{25707}a^{8}+\frac{225268177}{51414}a^{7}+\frac{16518639}{17138}a^{6}-\frac{123592793}{51414}a^{5}-\frac{14647054}{25707}a^{4}+\frac{5140453}{17138}a^{3}+\frac{2183314}{25707}a^{2}-\frac{184249}{25707}a-\frac{4670}{2337}$, $\frac{15442}{25707}a^{15}+\frac{331}{8569}a^{14}-\frac{1601683}{51414}a^{13}-\frac{52078}{25707}a^{12}+\frac{3967772}{8569}a^{11}+\frac{1589965}{51414}a^{10}-\frac{120567343}{51414}a^{9}-\frac{2864379}{17138}a^{8}+\frac{93415945}{25707}a^{7}+\frac{192472}{627}a^{6}-\frac{33852315}{17138}a^{5}-\frac{5112697}{25707}a^{4}+\frac{2280424}{8569}a^{3}+\frac{301372}{8569}a^{2}-\frac{5718}{779}a-\frac{8295}{8569}$, $\frac{30275}{25707}a^{15}+\frac{127}{2706}a^{14}-\frac{142504}{2337}a^{13}-\frac{6923}{2706}a^{12}+\frac{23206427}{25707}a^{11}+\frac{115069}{2706}a^{10}-\frac{77539993}{17138}a^{9}-\frac{370373}{1353}a^{8}+\frac{174447704}{25707}a^{7}+\frac{1876147}{2706}a^{6}-\frac{179545463}{51414}a^{5}-\frac{766912}{1353}a^{4}+\frac{20599313}{51414}a^{3}+\frac{148892}{1353}a^{2}-\frac{492787}{51414}a-\frac{1412}{451}$, $\frac{27601}{25707}a^{15}+\frac{2311}{25707}a^{14}-\frac{1434790}{25707}a^{13}-\frac{123566}{25707}a^{12}+\frac{42896189}{51414}a^{11}+\frac{1313791}{17138}a^{10}-\frac{73479081}{17138}a^{9}-\frac{23427089}{51414}a^{8}+\frac{178291600}{25707}a^{7}+\frac{52085029}{51414}a^{6}-\frac{102890537}{25707}a^{5}-\frac{12782005}{17138}a^{4}+\frac{30234481}{51414}a^{3}+\frac{5876777}{51414}a^{2}-\frac{745211}{51414}a-\frac{83588}{25707}$, $\frac{4715}{1254}a^{15}-\frac{9721}{25707}a^{14}-\frac{304429}{1558}a^{13}+\frac{337275}{17138}a^{12}+\frac{150074647}{51414}a^{11}-\frac{689470}{2337}a^{10}-\frac{385058243}{25707}a^{9}+\frac{13088776}{8569}a^{8}+\frac{1243580027}{51414}a^{7}-\frac{11826269}{4674}a^{6}-\frac{731659265}{51414}a^{5}+\frac{39669742}{25707}a^{4}+\frac{123712057}{51414}a^{3}-\frac{7249916}{25707}a^{2}-\frac{5423161}{51414}a+\frac{773105}{51414}$, $\frac{218419}{51414}a^{15}+\frac{17117}{17138}a^{14}-\frac{138241}{627}a^{13}-\frac{1330069}{25707}a^{12}+\frac{84397376}{25707}a^{11}+\frac{19716296}{25707}a^{10}-\frac{429511096}{25707}a^{9}-\frac{99188432}{25707}a^{8}+\frac{676499557}{25707}a^{7}+\frac{100658551}{17138}a^{6}-\frac{764579393}{51414}a^{5}-\frac{81903703}{25707}a^{4}+\frac{38790463}{17138}a^{3}+\frac{11804935}{25707}a^{2}-\frac{578649}{8569}a-\frac{502993}{51414}$
| 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: | \( 205402316.371 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 205402316.371 \cdot 2}{2\cdot\sqrt{1622628503115275885009765625}}\cr\approx \mathstrut & 0.334176354699 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 52*x^14 + 778*x^12 - 4009*x^10 + 6565*x^8 - 4009*x^6 + 778*x^4 - 52*x^2 + 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 - 52*x^14 + 778*x^12 - 4009*x^10 + 6565*x^8 - 4009*x^6 + 778*x^4 - 52*x^2 + 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 - 52*x^14 + 778*x^12 - 4009*x^10 + 6565*x^8 - 4009*x^6 + 778*x^4 - 52*x^2 + 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 - 52*x^14 + 778*x^12 - 4009*x^10 + 6565*x^8 - 4009*x^6 + 778*x^4 - 52*x^2 + 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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.4.946125.1, \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.40281863203125.1 x2, 8.8.47897578125.1 x2, 8.8.895152515625.1 |
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: | 8.8.47897578125.1, 8.8.40281863203125.1 |
| Degree 16 siblings: | deg 16, 16.16.1929403689792242431640625.1 |
| Minimal sibling: | 8.8.47897578125.1 |
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 | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.6.3 | $x^{8} - 6 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
| 3.8.6.3 | $x^{8} - 6 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ | |
|
\(5\)
| 5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
|
\(29\)
| 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |