Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 24*x^12 + 16*x^10 + 24*x^8 - 8*x^6 + 56*x^4 + 100)
gp: K = bnfinit(y^16 - 8*y^14 + 24*y^12 + 16*y^10 + 24*y^8 - 8*y^6 + 56*y^4 + 100, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 24*x^12 + 16*x^10 + 24*x^8 - 8*x^6 + 56*x^4 + 100);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 + 24*x^12 + 16*x^10 + 24*x^8 - 8*x^6 + 56*x^4 + 100)
\( x^{16} - 8x^{14} + 24x^{12} + 16x^{10} + 24x^{8} - 8x^{6} + 56x^{4} + 100 \)
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: |
\(17199267840000000000\)
\(\medspace = 2^{28}\cdot 3^{8}\cdot 5^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.93\) | 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: | $2^{7/4}3^{1/2}5^{3/4}\approx 19.480074928505935$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{8}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{45780040}a^{14}-\frac{286679}{22890020}a^{12}-\frac{106453}{22890020}a^{10}-\frac{352693}{5722505}a^{8}-\frac{1}{2}a^{7}+\frac{7211977}{22890020}a^{6}-\frac{1}{2}a^{5}-\frac{1976261}{5722505}a^{4}-\frac{1080673}{5722505}a^{2}-\frac{72865}{1144501}$, $\frac{1}{45780040}a^{15}-\frac{286679}{22890020}a^{13}-\frac{106453}{22890020}a^{11}-\frac{352693}{5722505}a^{9}+\frac{7211977}{22890020}a^{7}-\frac{1}{2}a^{6}-\frac{1976261}{5722505}a^{5}-\frac{1080673}{5722505}a^{3}-\frac{72865}{1144501}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
Class group and class number
$C_{2}$, which has order $2$
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: |
\( \frac{15093}{2289002} a^{14} - \frac{120233}{2289002} a^{12} + \frac{184275}{1144501} a^{10} + \frac{174117}{2289002} a^{8} + \frac{312254}{1144501} a^{6} - \frac{33345}{1144501} a^{4} + \frac{1033650}{1144501} a^{2} - \frac{8682}{1144501} \)
(order $4$)
| 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: |
$\frac{799}{101960}a^{15}-\frac{15093}{4578004}a^{14}-\frac{3381}{50980}a^{13}+\frac{120233}{4578004}a^{12}+\frac{21111}{101960}a^{11}-\frac{184275}{2289002}a^{10}+\frac{11159}{101960}a^{9}-\frac{174117}{4578004}a^{8}-\frac{1737}{50980}a^{7}-\frac{156127}{1144501}a^{6}-\frac{26781}{50980}a^{5}+\frac{33345}{2289002}a^{4}+\frac{367}{50980}a^{3}-\frac{516825}{1144501}a^{2}+\frac{2649}{10196}a+\frac{4341}{1144501}$, $\frac{4177}{22890020}a^{15}+\frac{23321}{9156008}a^{14}-\frac{96047}{45780040}a^{13}-\frac{76735}{9156008}a^{12}+\frac{1082331}{45780040}a^{11}-\frac{41936}{1144501}a^{10}-\frac{2951493}{22890020}a^{9}+\frac{375234}{1144501}a^{8}+\frac{8045723}{22890020}a^{7}+\frac{1515663}{4578004}a^{6}+\frac{4692629}{22890020}a^{5}-\frac{743547}{4578004}a^{4}+\frac{2960087}{22890020}a^{3}+\frac{218475}{2289002}a^{2}-\frac{823857}{2289002}a-\frac{440303}{2289002}$, $\frac{19722}{5722505}a^{14}-\frac{96596}{5722505}a^{12}-\frac{176353}{22890020}a^{10}+\frac{4018209}{11445010}a^{8}+\frac{1271971}{11445010}a^{6}+\frac{1296904}{5722505}a^{4}+\frac{1268959}{11445010}a^{2}+\frac{164305}{1144501}$, $\frac{15093}{4578004}a^{15}+\frac{146753}{22890020}a^{14}-\frac{120233}{4578004}a^{13}-\frac{2308613}{45780040}a^{12}+\frac{184275}{2289002}a^{11}+\frac{6288669}{45780040}a^{10}+\frac{174117}{4578004}a^{9}+\frac{4692663}{22890020}a^{8}+\frac{156127}{1144501}a^{7}-\frac{1800643}{22890020}a^{6}-\frac{33345}{2289002}a^{5}-\frac{5359529}{22890020}a^{4}+\frac{516825}{1144501}a^{3}+\frac{6267903}{22890020}a^{2}-\frac{4341}{1144501}a+\frac{806493}{2289002}$, $\frac{114909}{22890020}a^{15}-\frac{244001}{45780040}a^{14}-\frac{794357}{22890020}a^{13}+\frac{1845623}{45780040}a^{12}+\frac{3509147}{45780040}a^{11}-\frac{5345989}{45780040}a^{10}+\frac{2444397}{11445010}a^{9}-\frac{893499}{11445010}a^{8}+\frac{4394511}{22890020}a^{7}-\frac{1952483}{5722505}a^{6}+\frac{1130179}{11445010}a^{5}-\frac{6061771}{22890020}a^{4}+\frac{11605459}{22890020}a^{3}-\frac{2338383}{22890020}a^{2}-\frac{494439}{1144501}a+\frac{454331}{1144501}$, $\frac{208119}{45780040}a^{15}-\frac{120239}{9156008}a^{14}-\frac{509671}{22890020}a^{13}+\frac{974827}{9156008}a^{12}-\frac{427599}{45780040}a^{11}-\frac{720709}{2289002}a^{10}+\frac{20824859}{45780040}a^{9}-\frac{628353}{2289002}a^{8}+\frac{5049823}{22890020}a^{7}+\frac{37173}{4578004}a^{6}+\frac{1722739}{22890020}a^{5}+\frac{1383529}{4578004}a^{4}+\frac{6397207}{22890020}a^{3}-\frac{735687}{1144501}a^{2}+\frac{5911765}{4578004}a-\frac{846601}{1144501}$, $\frac{45233}{45780040}a^{15}-\frac{104649}{45780040}a^{14}-\frac{154877}{22890020}a^{13}+\frac{876417}{45780040}a^{12}+\frac{598817}{45780040}a^{11}-\frac{758619}{11445010}a^{10}+\frac{2129263}{45780040}a^{9}+\frac{972533}{22890020}a^{8}+\frac{2235611}{22890020}a^{7}-\frac{7886643}{22890020}a^{6}-\frac{9575337}{22890020}a^{5}+\frac{15349261}{22890020}a^{4}+\frac{15391119}{22890020}a^{3}-\frac{2517538}{5722505}a^{2}-\frac{2392163}{4578004}a+\frac{22945}{2289002}$
| 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: | \( 2463.52568788 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2463.52568788 \cdot 2}{4\cdot\sqrt{17199267840000000000}}\cr\approx \mathstrut & 0.721458021554 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 24*x^12 + 16*x^10 + 24*x^8 - 8*x^6 + 56*x^4 + 100)
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 - 8*x^14 + 24*x^12 + 16*x^10 + 24*x^8 - 8*x^6 + 56*x^4 + 100, 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 - 8*x^14 + 24*x^12 + 16*x^10 + 24*x^8 - 8*x^6 + 56*x^4 + 100);
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 - 8*x^14 + 24*x^12 + 16*x^10 + 24*x^8 - 8*x^6 + 56*x^4 + 100);
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
$C_2^3:C_4$ (as 16T33):
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 $C_2^3:C_4$ |
| Character table for $C_2^3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.2.3600.1 x2, 4.0.2880.1 x2, \(\Q(i, \sqrt{5})\), 8.0.115200000.1 x2, 8.0.207360000.5, 8.0.829440000.2 x2 |
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 | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $8$ | $2$ | $28$ | |||
|
\(3\)
| 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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |