Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 8*x^12 - 88*x^10 + 127*x^8 + 616*x^6 + 392*x^4 + 1372*x^2 + 2401)
gp: K = bnfinit(y^16 - 4*y^14 + 8*y^12 - 88*y^10 + 127*y^8 + 616*y^6 + 392*y^4 + 1372*y^2 + 2401, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 + 8*x^12 - 88*x^10 + 127*x^8 + 616*x^6 + 392*x^4 + 1372*x^2 + 2401);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^14 + 8*x^12 - 88*x^10 + 127*x^8 + 616*x^6 + 392*x^4 + 1372*x^2 + 2401)
\( x^{16} - 4x^{14} + 8x^{12} - 88x^{10} + 127x^{8} + 616x^{6} + 392x^{4} + 1372x^{2} + 2401 \)
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: |
\(4434064330598872252416\)
\(\medspace = 2^{48}\cdot 3^{8}\cdot 7^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.54\) | 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: | $2^{3}3^{1/2}7^{1/2}\approx 36.66060555964672$ | ||
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{21}a^{10}+\frac{1}{7}a^{8}+\frac{1}{21}a^{6}+\frac{1}{7}a^{4}+\frac{8}{21}a^{2}-\frac{1}{3}$, $\frac{1}{21}a^{11}+\frac{1}{7}a^{9}+\frac{1}{21}a^{7}+\frac{1}{7}a^{5}+\frac{8}{21}a^{3}-\frac{1}{3}a$, $\frac{1}{24255}a^{12}+\frac{38}{1617}a^{10}+\frac{3}{49}a^{8}+\frac{8522}{24255}a^{6}+\frac{17}{49}a^{4}+\frac{79}{231}a^{2}-\frac{172}{495}$, $\frac{1}{24255}a^{13}+\frac{38}{1617}a^{11}+\frac{3}{49}a^{9}+\frac{8522}{24255}a^{7}+\frac{17}{49}a^{5}+\frac{79}{231}a^{3}-\frac{172}{495}a$, $\frac{1}{57217545}a^{14}+\frac{73}{3814503}a^{12}-\frac{14447}{3814503}a^{10}+\frac{1362182}{57217545}a^{8}-\frac{457536}{1271501}a^{6}-\frac{8032}{25949}a^{4}-\frac{425272}{1167705}a^{2}+\frac{2275}{11121}$, $\frac{1}{57217545}a^{15}+\frac{73}{3814503}a^{13}-\frac{14447}{3814503}a^{11}+\frac{1362182}{57217545}a^{9}-\frac{457536}{1271501}a^{7}-\frac{8032}{25949}a^{5}-\frac{425272}{1167705}a^{3}+\frac{2275}{11121}a$
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
$C_{4}$, which has order $4$
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: |
\( \frac{2}{77847} a^{14} + \frac{23}{166815} a^{12} - \frac{4}{11121} a^{10} - \frac{281}{77847} a^{8} - \frac{1124}{166815} a^{6} - \frac{35}{3707} a^{4} + \frac{47651}{77847} a^{2} - \frac{2744}{166815} \)
(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: |
$\frac{19324}{19072515}a^{14}-\frac{116041}{19072515}a^{12}+\frac{23258}{1271501}a^{10}-\frac{2325787}{19072515}a^{8}+\frac{6535498}{19072515}a^{6}+\frac{13920}{181643}a^{4}+\frac{28142}{389235}a^{2}+\frac{102121}{55605}$, $\frac{39307}{57217545}a^{14}-\frac{14044}{5201595}a^{12}+\frac{20918}{3814503}a^{10}-\frac{3381841}{57217545}a^{8}+\frac{4606457}{57217545}a^{6}+\frac{220151}{544929}a^{4}+\frac{26716}{106155}a^{2}-\frac{17476}{166815}$, $\frac{1490}{3814503}a^{14}-\frac{164146}{57217545}a^{12}+\frac{37778}{3814503}a^{10}-\frac{209345}{3814503}a^{8}+\frac{10615618}{57217545}a^{6}-\frac{3725}{25949}a^{4}+\frac{19637}{77847}a^{2}+\frac{75556}{166815}$, $\frac{1124}{1733865}a^{15}+\frac{7424}{57217545}a^{14}-\frac{5336}{1733865}a^{13}-\frac{61168}{57217545}a^{12}+\frac{928}{115591}a^{11}+\frac{4640}{1271501}a^{10}-\frac{122537}{1733865}a^{9}-\frac{1043072}{57217545}a^{8}+\frac{260768}{1733865}a^{7}+\frac{2640019}{57217545}a^{6}+\frac{3480}{16513}a^{5}-\frac{3712}{77847}a^{4}+\frac{15292}{35385}a^{3}-\frac{21344}{166815}a^{2}+\frac{1856}{5055}a+\frac{135343}{166815}$, $\frac{10894}{19072515}a^{15}+\frac{18136}{57217545}a^{14}-\frac{4439}{1271501}a^{13}-\frac{171349}{57217545}a^{12}+\frac{11580}{1271501}a^{11}+\frac{11335}{1271501}a^{10}-\frac{1141372}{19072515}a^{9}-\frac{2548108}{57217545}a^{8}+\frac{216932}{1271501}a^{7}+\frac{12098407}{57217545}a^{6}+\frac{43425}{181643}a^{5}-\frac{9068}{77847}a^{4}-\frac{22474}{55605}a^{3}-\frac{52141}{166815}a^{2}+\frac{1544}{3707}a+\frac{19819}{166815}$, $\frac{18254}{19072515}a^{15}-\frac{1231}{2724645}a^{14}-\frac{25432}{5201595}a^{13}+\frac{2077}{2724645}a^{12}+\frac{59627}{3814503}a^{11}+\frac{8}{7077}a^{10}-\frac{2175452}{19072515}a^{9}+\frac{23671}{908215}a^{8}+\frac{15483686}{57217545}a^{7}+\frac{774}{11795}a^{6}+\frac{17665}{181643}a^{5}-\frac{267247}{544929}a^{4}+\frac{9024}{11795}a^{3}-\frac{62742}{129745}a^{2}+\frac{393572}{166815}a-\frac{6254}{5055}$, $\frac{2864}{3814503}a^{15}-\frac{20981}{19072515}a^{14}-\frac{22912}{5201595}a^{13}+\frac{14647}{3814503}a^{12}+\frac{64037}{3814503}a^{11}-\frac{2392}{346773}a^{10}-\frac{402392}{3814503}a^{9}+\frac{1844998}{19072515}a^{8}+\frac{16722896}{57217545}a^{7}-\frac{29398}{346773}a^{6}-\frac{7160}{25949}a^{5}-\frac{382087}{544929}a^{4}+\frac{5728}{7077}a^{3}-\frac{356438}{389235}a^{2}+\frac{402392}{166815}a-\frac{3644}{1011}$
| 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: | \( 52534.22926879539 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 52534.22926879539 \cdot 4}{24\cdot\sqrt{4434064330598872252416}}\cr\approx \mathstrut & 0.319395701297026 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 8*x^12 - 88*x^10 + 127*x^8 + 616*x^6 + 392*x^4 + 1372*x^2 + 2401)
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 - 4*x^14 + 8*x^12 - 88*x^10 + 127*x^8 + 616*x^6 + 392*x^4 + 1372*x^2 + 2401, 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 - 4*x^14 + 8*x^12 - 88*x^10 + 127*x^8 + 616*x^6 + 392*x^4 + 1372*x^2 + 2401);
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 - 4*x^14 + 8*x^12 - 88*x^10 + 127*x^8 + 616*x^6 + 392*x^4 + 1372*x^2 + 2401);
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, deg 16 |
| 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 | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.48.3 | $x^{16} + 8 x^{14} + 8 x^{13} + 84 x^{12} + 144 x^{11} + 160 x^{10} + 160 x^{9} + 360 x^{8} + 384 x^{7} + 432 x^{6} + 432 x^{5} + 600 x^{4} + 416 x^{3} + 352 x^{2} + 288 x + 324$ | $8$ | $2$ | $48$ | $D_4\times C_2$ | $[2, 3, 4]^{2}$ |
|
\(3\)
| 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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |