Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 25*x^14 + 475*x^12 + 4495*x^10 + 21060*x^8 - 8825*x^6 + 9375*x^4 - 775*x^2 + 25)
gp: K = bnfinit(y^16 + 25*y^14 + 475*y^12 + 4495*y^10 + 21060*y^8 - 8825*y^6 + 9375*y^4 - 775*y^2 + 25, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 25*x^14 + 475*x^12 + 4495*x^10 + 21060*x^8 - 8825*x^6 + 9375*x^4 - 775*x^2 + 25);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 25*x^14 + 475*x^12 + 4495*x^10 + 21060*x^8 - 8825*x^6 + 9375*x^4 - 775*x^2 + 25)
\( x^{16} + 25x^{14} + 475x^{12} + 4495x^{10} + 21060x^{8} - 8825x^{6} + 9375x^{4} - 775x^{2} + 25 \)
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: |
\(3941366767857427978515625\)
\(\medspace = 5^{14}\cdot 71^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.45\) | 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: | $5^{7/8}71^{1/2}\approx 34.45307012928092$ | ||
| Ramified primes: |
\(5\), \(71\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{10}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{10}a^{9}-\frac{1}{2}a^{4}$, $\frac{1}{20}a^{10}-\frac{1}{4}$, $\frac{1}{40}a^{11}-\frac{1}{40}a^{10}-\frac{1}{2}a^{4}-\frac{1}{8}a-\frac{3}{8}$, $\frac{1}{80}a^{12}-\frac{1}{80}a^{10}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{16}a^{2}-\frac{1}{2}a+\frac{1}{16}$, $\frac{1}{160}a^{13}-\frac{1}{160}a^{12}-\frac{1}{160}a^{11}+\frac{1}{160}a^{10}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{3}{8}a^{4}-\frac{1}{32}a^{3}-\frac{15}{32}a^{2}+\frac{1}{32}a-\frac{1}{32}$, $\frac{1}{7708593265600}a^{14}+\frac{3984946913}{770859326560}a^{12}+\frac{31402310577}{1541718653120}a^{10}+\frac{10653854421}{385429663280}a^{8}+\frac{21241758857}{192714831640}a^{6}-\frac{74548196057}{308343730624}a^{4}-\frac{1}{2}a^{3}+\frac{49115233559}{154171865312}a^{2}-\frac{1}{2}a-\frac{73659708865}{308343730624}$, $\frac{1}{15417186531200}a^{15}-\frac{1}{15417186531200}a^{14}+\frac{3984946913}{1541718653120}a^{13}-\frac{3984946913}{1541718653120}a^{12}+\frac{31402310577}{3083437306240}a^{11}-\frac{31402310577}{3083437306240}a^{10}-\frac{27889111907}{770859326560}a^{9}+\frac{27889111907}{770859326560}a^{8}-\frac{75115656963}{385429663280}a^{7}+\frac{75115656963}{385429663280}a^{6}-\frac{74548196057}{616687461248}a^{5}-\frac{233795534567}{616687461248}a^{4}-\frac{105056631753}{308343730624}a^{3}-\frac{49115233559}{308343730624}a^{2}+\frac{234684021759}{616687461248}a-\frac{234684021759}{616687461248}$
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
$C_{14}$, which has order $14$ (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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{34262239}{9635741582} a^{14} - \frac{34374884743}{385429663280} a^{12} - \frac{653823527577}{385429663280} a^{10} - \frac{388405051507}{24089353955} a^{8} - \frac{1469026558175}{19271483164} a^{6} + \frac{480524539077}{19271483164} a^{4} - \frac{2467041119961}{77085932656} a^{2} + \frac{81124781657}{77085932656} \)
(order $10$)
| 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: |
$\frac{3559506339}{1541718653120}a^{14}+\frac{44457089837}{770859326560}a^{12}+\frac{1688793932211}{1541718653120}a^{10}+\frac{3990457898179}{385429663280}a^{8}+\frac{1864262787325}{38542966328}a^{6}-\frac{6707741976455}{308343730624}a^{4}+\frac{3133309932699}{154171865312}a^{2}-\frac{206068823651}{308343730624}$, $\frac{535518068903}{15417186531200}a^{15}-\frac{21642435497}{3083437306240}a^{14}+\frac{267946811117}{308343730624}a^{13}-\frac{270870808483}{1541718653120}a^{12}+\frac{50920950047507}{3083437306240}a^{11}-\frac{10296970017249}{3083437306240}a^{10}+\frac{120579382297643}{770859326560}a^{9}-\frac{24400335342761}{770859326560}a^{8}+\frac{282993124560021}{385429663280}a^{7}-\frac{11468894594675}{77085932656}a^{6}-\frac{181306762648799}{616687461248}a^{5}+\frac{35500011179829}{616687461248}a^{4}+\frac{98350540554855}{308343730624}a^{3}-\frac{19268094277941}{308343730624}a^{2}-\frac{11916247909251}{616687461248}a+\frac{2808923377329}{616687461248}$, $\frac{480698486503}{15417186531200}a^{15}-\frac{13532886383}{3083437306240}a^{14}+\frac{1202234516613}{1541718653120}a^{13}-\frac{33748164985}{308343730624}a^{12}+\frac{45690361826891}{3083437306240}a^{11}-\frac{6406910274887}{3083437306240}a^{10}+\frac{108150420649419}{770859326560}a^{9}-\frac{15106776765103}{770859326560}a^{8}+\frac{253612593396521}{385429663280}a^{7}-\frac{7028751363373}{77085932656}a^{6}-\frac{165929977398335}{616687461248}a^{5}+\frac{27538600946883}{616687461248}a^{4}+\frac{88482376075011}{308343730624}a^{3}-\frac{13251193893643}{308343730624}a^{2}-\frac{11883937117243}{616687461248}a+\frac{2559454218231}{616687461248}$, $\frac{4463825253}{3083437306240}a^{15}+\frac{96060880093}{15417186531200}a^{14}+\frac{59584728287}{1541718653120}a^{13}+\frac{241356282471}{1541718653120}a^{12}+\frac{2310285759213}{3083437306240}a^{11}+\frac{9186031648777}{3083437306240}a^{10}+\frac{1183904914625}{154171865312}a^{9}+\frac{21875877659801}{770859326560}a^{8}+\frac{3208763356951}{77085932656}a^{7}+\frac{51931576058091}{385429663280}a^{6}+\frac{24593453310527}{616687461248}a^{5}-\frac{23726822455333}{616687461248}a^{4}-\frac{1126031524775}{308343730624}a^{3}+\frac{15969547105745}{308343730624}a^{2}+\frac{13135898194883}{616687461248}a+\frac{1574696306343}{616687461248}$, $\frac{2151204333}{1541718653120}a^{15}+\frac{4328315111}{3854296632800}a^{14}+\frac{5436894275}{154171865312}a^{13}+\frac{5429417791}{192714831640}a^{12}+\frac{1036573920333}{1541718653120}a^{11}+\frac{413039359373}{770859326560}a^{10}+\frac{2487469992077}{385429663280}a^{9}+\frac{981590695339}{192714831640}a^{8}+\frac{1198988014907}{38542966328}a^{7}+\frac{579802483518}{24089353955}a^{6}-\frac{1308593343241}{308343730624}a^{5}-\frac{1231075638039}{154171865312}a^{4}+\frac{1473010160937}{154171865312}a^{3}+\frac{345513164777}{38542966328}a^{2}+\frac{651739665155}{308343730624}a+\frac{7071414403}{154171865312}$, $\frac{14028154601}{3083437306240}a^{15}-\frac{2415993919}{15417186531200}a^{14}+\frac{175348726867}{1541718653120}a^{13}-\frac{6052458421}{1541718653120}a^{12}+\frac{6663146213857}{3083437306240}a^{11}-\frac{230138150003}{3083437306240}a^{10}+\frac{15763098155753}{770859326560}a^{9}-\frac{545613246067}{770859326560}a^{8}+\frac{7384560191699}{77085932656}a^{7}-\frac{1284325009233}{385429663280}a^{6}-\frac{24811564086901}{616687461248}a^{5}+\frac{808547966231}{616687461248}a^{4}+\frac{13126334153093}{308343730624}a^{3}-\frac{391256985811}{308343730624}a^{2}-\frac{2478119974129}{616687461248}a+\frac{341445350819}{616687461248}$, $\frac{64301816109}{15417186531200}a^{15}-\frac{8631216569}{15417186531200}a^{14}+\frac{160023775769}{1541718653120}a^{13}-\frac{22010976001}{1541718653120}a^{12}+\frac{1214452392657}{616687461248}a^{11}-\frac{168065393269}{616687461248}a^{10}+\frac{14279314153657}{770859326560}a^{9}-\frac{2034608309941}{770859326560}a^{8}+\frac{33044543103733}{385429663280}a^{7}-\frac{4953496168853}{385429663280}a^{6}-\frac{28644274734021}{616687461248}a^{5}+\frac{594723962641}{616687461248}a^{4}+\frac{14097802842015}{308343730624}a^{3}+\frac{2073175291913}{308343730624}a^{2}-\frac{600200203365}{616687461248}a-\frac{202808539495}{616687461248}$
| 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: | \( 144974.394976 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 144974.394976 \cdot 14}{10\cdot\sqrt{3941366767857427978515625}}\cr\approx \mathstrut & 0.248333244281 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 25*x^14 + 475*x^12 + 4495*x^10 + 21060*x^8 - 8825*x^6 + 9375*x^4 - 775*x^2 + 25)
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 + 25*x^14 + 475*x^12 + 4495*x^10 + 21060*x^8 - 8825*x^6 + 9375*x^4 - 775*x^2 + 25, 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 + 25*x^14 + 475*x^12 + 4495*x^10 + 21060*x^8 - 8825*x^6 + 9375*x^4 - 775*x^2 + 25);
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 + 25*x^14 + 475*x^12 + 4495*x^10 + 21060*x^8 - 8825*x^6 + 9375*x^4 - 775*x^2 + 25);
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
| \(\Q(\sqrt{-71}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-355}) \), \(\Q(\zeta_{5})\), 4.4.630125.1, \(\Q(\sqrt{5}, \sqrt{-71})\), 8.0.393828125.1 x2, 8.4.1985287578125.1 x2, 8.0.397057515625.2 |
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: | 8.0.393828125.1, 8.4.1985287578125.1 |
| Degree 16 siblings: | 16.4.3941366767857427978515625.1, 16.0.781862084478759765625.1 |
| Minimal sibling: | 8.0.393828125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
|
\(71\)
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |