Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 100*x^12 - 408*x^10 + 1191*x^8 - 2040*x^6 + 2500*x^4 - 1500*x^2 + 625)
gp: K = bnfinit(y^16 - 12*y^14 + 100*y^12 - 408*y^10 + 1191*y^8 - 2040*y^6 + 2500*y^4 - 1500*y^2 + 625, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 100*x^12 - 408*x^10 + 1191*x^8 - 2040*x^6 + 2500*x^4 - 1500*x^2 + 625);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 12*x^14 + 100*x^12 - 408*x^10 + 1191*x^8 - 2040*x^6 + 2500*x^4 - 1500*x^2 + 625)
\( x^{16} - 12x^{14} + 100x^{12} - 408x^{10} + 1191x^{8} - 2040x^{6} + 2500x^{4} - 1500x^{2} + 625 \)
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: |
\(1154223326374133760000\)
\(\medspace = 2^{48}\cdot 3^{8}\cdot 5^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.72\) | 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: | $2^{3}3^{1/2}5^{1/2}\approx 30.983866769659336$ | ||
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{15}a^{8}+\frac{1}{3}a^{4}-\frac{1}{5}a^{2}-\frac{1}{3}$, $\frac{1}{15}a^{9}+\frac{1}{3}a^{5}-\frac{1}{5}a^{3}-\frac{1}{3}a$, $\frac{1}{15}a^{10}+\frac{1}{3}a^{6}-\frac{1}{5}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{15}a^{11}+\frac{1}{3}a^{7}-\frac{1}{5}a^{5}-\frac{1}{3}a^{3}$, $\frac{1}{1425}a^{12}+\frac{2}{75}a^{10}+\frac{67}{1425}a^{6}-\frac{12}{25}a^{4}-\frac{2}{15}a^{2}-\frac{14}{57}$, $\frac{1}{1425}a^{13}+\frac{2}{75}a^{11}+\frac{67}{1425}a^{7}-\frac{12}{25}a^{5}-\frac{2}{15}a^{3}-\frac{14}{57}a$, $\frac{1}{3170625}a^{14}+\frac{503}{3170625}a^{12}-\frac{434}{33375}a^{10}-\frac{96358}{3170625}a^{8}+\frac{1106821}{3170625}a^{6}-\frac{1571}{6675}a^{4}-\frac{1754}{8455}a^{2}-\frac{4342}{25365}$, $\frac{1}{3170625}a^{15}+\frac{503}{3170625}a^{13}-\frac{434}{33375}a^{11}-\frac{96358}{3170625}a^{9}+\frac{1106821}{3170625}a^{7}-\frac{1571}{6675}a^{5}-\frac{1754}{8455}a^{3}-\frac{4342}{25365}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{3}$, which has order $3$
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: |
\( -\frac{1132}{1056875} a^{14} + \frac{51787}{3170625} a^{12} - \frac{4871}{33375} a^{10} + \frac{2348393}{3170625} a^{8} - \frac{7309541}{3170625} a^{6} + \frac{29737}{6675} a^{4} - \frac{115724}{25365} a^{2} + \frac{20144}{8455} \)
(order $24$)
| 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: |
$\frac{1726}{1056875}a^{14}-\frac{54341}{3170625}a^{12}+\frac{4448}{33375}a^{10}-\frac{1364974}{3170625}a^{8}+\frac{3323888}{3170625}a^{6}-\frac{9383}{6675}a^{4}+\frac{42722}{25365}a^{2}-\frac{4497}{8455}$, $\frac{1132}{1056875}a^{14}-\frac{51787}{3170625}a^{12}+\frac{4871}{33375}a^{10}-\frac{2348393}{3170625}a^{8}+\frac{7309541}{3170625}a^{6}-\frac{29737}{6675}a^{4}+\frac{115724}{25365}a^{2}-\frac{28599}{8455}$, $\frac{2517}{1056875}a^{14}-\frac{26674}{1056875}a^{12}+\frac{6521}{33375}a^{10}-\frac{1948883}{3170625}a^{8}+\frac{4218896}{3170625}a^{6}-\frac{5329}{6675}a^{4}+\frac{13}{5073}a^{2}+\frac{4718}{25365}$, $\frac{269}{211375}a^{14}-\frac{3088}{211375}a^{12}+\frac{778}{6675}a^{10}-\frac{90277}{211375}a^{8}+\frac{652027}{634125}a^{6}-\frac{2716}{2225}a^{4}+\frac{10172}{25365}a^{2}-a-\frac{941}{1691}$, $\frac{54}{1056875}a^{15}-\frac{4}{2375}a^{14}+\frac{462}{1056875}a^{13}+\frac{119}{7125}a^{12}+\frac{2}{33375}a^{11}-\frac{2}{15}a^{10}+\frac{31754}{3170625}a^{9}+\frac{2996}{7125}a^{8}+\frac{610802}{3170625}a^{7}-\frac{8842}{7125}a^{6}-\frac{2543}{6675}a^{5}+\frac{134}{75}a^{4}+\frac{26876}{25365}a^{3}-\frac{782}{285}a^{2}+\frac{36631}{25365}a+\frac{5}{57}$, $\frac{19624}{3170625}a^{15}-\frac{4246}{3170625}a^{14}-\frac{70201}{1056875}a^{13}+\frac{44762}{3170625}a^{12}+\frac{5798}{11125}a^{11}-\frac{3986}{33375}a^{10}-\frac{1821464}{1056875}a^{9}+\frac{1393693}{3170625}a^{8}+\frac{4327993}{1056875}a^{7}-\frac{4678466}{3170625}a^{6}-\frac{1784}{445}a^{5}+\frac{15491}{6675}a^{4}+\frac{61994}{25365}a^{3}-\frac{93736}{25365}a^{2}+\frac{10929}{8455}a-\frac{7341}{8455}$, $\frac{19733}{3170625}a^{15}-\frac{2737}{3170625}a^{14}-\frac{81592}{1056875}a^{13}+\frac{22814}{3170625}a^{12}+\frac{21263}{33375}a^{11}-\frac{394}{11125}a^{10}-\frac{8357914}{3170625}a^{9}-\frac{91843}{1056875}a^{8}+\frac{23033843}{3170625}a^{7}+\frac{1189491}{1056875}a^{6}-\frac{81854}{6675}a^{5}-\frac{8643}{2225}a^{4}+\frac{109621}{8455}a^{3}+\frac{145273}{25365}a^{2}-\frac{165926}{25365}a-\frac{117596}{25365}$
| 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: | \( 42111.51791912637 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 42111.51791912637 \cdot 3}{24\cdot\sqrt{1154223326374133760000}}\cr\approx \mathstrut & 0.376361276046793 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 100*x^12 - 408*x^10 + 1191*x^8 - 2040*x^6 + 2500*x^4 - 1500*x^2 + 625)
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 - 12*x^14 + 100*x^12 - 408*x^10 + 1191*x^8 - 2040*x^6 + 2500*x^4 - 1500*x^2 + 625, 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 - 12*x^14 + 100*x^12 - 408*x^10 + 1191*x^8 - 2040*x^6 + 2500*x^4 - 1500*x^2 + 625);
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 - 12*x^14 + 100*x^12 - 408*x^10 + 1191*x^8 - 2040*x^6 + 2500*x^4 - 1500*x^2 + 625);
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
$C_2^2\times D_4$ (as 16T25):
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 20 conjugacy class representatives for $C_2^2 \times D_4$ |
| Character table for $C_2^2 \times D_4$ |
Intermediate fields
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 siblings: | deg 16, deg 16, deg 16, deg 16, deg 16, deg 16, 16.0.8906044184985600000000.3 |
| Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.48.5 | $x^{16} + 16 x^{14} + 68 x^{12} + 16 x^{11} + 56 x^{10} + 144 x^{9} + 216 x^{8} + 160 x^{7} + 272 x^{6} + 224 x^{5} + 312 x^{4} + 352 x^{3} + 304 x^{2} + 160 x + 292$ | $8$ | $2$ | $48$ | $D_4\times C_2$ | $[2, 3, 4]^{2}$ |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |