Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 26*x^11 + 52*x^10 - 54*x^9 - 12*x^8 + 82*x^7 - 54*x^6 - 24*x^5 + 49*x^4 - 32*x^3 + 16*x^2 - 6*x + 1)
gp: K = bnfinit(y^16 - 4*y^15 + 6*y^14 - 6*y^13 + 10*y^12 - 26*y^11 + 52*y^10 - 54*y^9 - 12*y^8 + 82*y^7 - 54*y^6 - 24*y^5 + 49*y^4 - 32*y^3 + 16*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 26*x^11 + 52*x^10 - 54*x^9 - 12*x^8 + 82*x^7 - 54*x^6 - 24*x^5 + 49*x^4 - 32*x^3 + 16*x^2 - 6*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 26*x^11 + 52*x^10 - 54*x^9 - 12*x^8 + 82*x^7 - 54*x^6 - 24*x^5 + 49*x^4 - 32*x^3 + 16*x^2 - 6*x + 1)
\( x^{16} - 4 x^{15} + 6 x^{14} - 6 x^{13} + 10 x^{12} - 26 x^{11} + 52 x^{10} - 54 x^{9} - 12 x^{8} + \cdots + 1 \)
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: | $[2, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-195238297600000000\)
\(\medspace = -\,2^{24}\cdot 5^{8}\cdot 31^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.04\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(5\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{38922761}a^{15}-\frac{10490978}{38922761}a^{14}+\frac{9526752}{38922761}a^{13}-\frac{14909679}{38922761}a^{12}-\frac{15523577}{38922761}a^{11}+\frac{17784174}{38922761}a^{10}-\frac{10321760}{38922761}a^{9}+\frac{15017570}{38922761}a^{8}+\frac{7737621}{38922761}a^{7}-\frac{11143027}{38922761}a^{6}+\frac{2309429}{38922761}a^{5}+\frac{13600278}{38922761}a^{4}+\frac{14098763}{38922761}a^{3}+\frac{5246730}{38922761}a^{2}+\frac{15062844}{38922761}a+\frac{13833234}{38922761}$
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
Trivial group, which has order $1$
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: | $8$ | 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: |
$\frac{1178724471}{38922761}a^{15}-\frac{3943152030}{38922761}a^{14}+\frac{4471397670}{38922761}a^{13}-\frac{4089135154}{38922761}a^{12}+\frac{9063182357}{38922761}a^{11}-\frac{24670246009}{38922761}a^{10}+\frac{45017869002}{38922761}a^{9}-\frac{33836890856}{38922761}a^{8}-\frac{36870815021}{38922761}a^{7}+\frac{72781430706}{38922761}a^{6}-\frac{15229430863}{38922761}a^{5}-\frac{39152400386}{38922761}a^{4}+\frac{31984264836}{38922761}a^{3}-\frac{16164566002}{38922761}a^{2}+\frac{7980389569}{38922761}a-\frac{1741766954}{38922761}$, $\frac{422093465}{38922761}a^{15}-\frac{1381099985}{38922761}a^{14}+\frac{1519516018}{38922761}a^{13}-\frac{1404192089}{38922761}a^{12}+\frac{3180892206}{38922761}a^{11}-\frac{8643669809}{38922761}a^{10}+\frac{15604769716}{38922761}a^{9}-\frac{11296038814}{38922761}a^{8}-\frac{13509039335}{38922761}a^{7}+\frac{24876550936}{38922761}a^{6}-\frac{4369713248}{38922761}a^{5}-\frac{13641924472}{38922761}a^{4}+\frac{10646584513}{38922761}a^{3}-\frac{5561080621}{38922761}a^{2}+\frac{2706644147}{38922761}a-\frac{528539140}{38922761}$, $\frac{1330078641}{38922761}a^{15}-\frac{4538131925}{38922761}a^{14}+\frac{5320832859}{38922761}a^{13}-\frac{4878167203}{38922761}a^{12}+\frac{10457401854}{38922761}a^{11}-\frac{28459144264}{38922761}a^{10}+\frac{52491414757}{38922761}a^{9}-\frac{41124033087}{38922761}a^{8}-\frac{39852499266}{38922761}a^{7}+\frac{85458112205}{38922761}a^{6}-\frac{21866179137}{38922761}a^{5}-\frac{44364382674}{38922761}a^{4}+\frac{39053230596}{38922761}a^{3}-\frac{19828455912}{38922761}a^{2}+\frac{9779946420}{38922761}a-\frac{2304652950}{38922761}$, $\frac{1851505134}{38922761}a^{15}-\frac{6237495549}{38922761}a^{14}+\frac{7165869026}{38922761}a^{13}-\frac{6566140350}{38922761}a^{12}+\frac{14347863105}{38922761}a^{11}-\frac{39060046602}{38922761}a^{10}+\frac{71578654622}{38922761}a^{9}-\frac{54675251144}{38922761}a^{8}-\frac{56958197660}{38922761}a^{7}+\frac{116052136550}{38922761}a^{6}-\frac{26551307651}{38922761}a^{5}-\frac{61552861351}{38922761}a^{4}+\frac{51997581113}{38922761}a^{3}-\frac{26257954219}{38922761}a^{2}+\frac{12868017282}{38922761}a-\frac{2894928474}{38922761}$, $a$, $\frac{563042483}{38922761}a^{15}-\frac{1845191315}{38922761}a^{14}+\frac{2036052024}{38922761}a^{13}-\frac{1890068900}{38922761}a^{12}+\frac{4265352029}{38922761}a^{11}-\frac{11552512438}{38922761}a^{10}+\frac{20883766597}{38922761}a^{9}-\frac{15200655658}{38922761}a^{8}-\frac{17867279283}{38922761}a^{7}+\frac{33172644501}{38922761}a^{6}-\frac{6044553947}{38922761}a^{5}-\frac{17879437373}{38922761}a^{4}+\frac{14209915524}{38922761}a^{3}-\frac{7587711690}{38922761}a^{2}+\frac{3723315693}{38922761}a-\frac{728445201}{38922761}$, $\frac{507761965}{38922761}a^{15}-\frac{1794868007}{38922761}a^{14}+\frac{2201271856}{38922761}a^{13}-\frac{1992473111}{38922761}a^{12}+\frac{4123839365}{38922761}a^{11}-\frac{11255080936}{38922761}a^{10}+\frac{21093928199}{38922761}a^{9}-\frac{17413625571}{38922761}a^{8}-\frac{14519940826}{38922761}a^{7}+\frac{35068312078}{38922761}a^{6}-\frac{10763068816}{38922761}a^{5}-\frac{17603042012}{38922761}a^{4}+\frac{16615457529}{38922761}a^{3}-\frac{8256855476}{38922761}a^{2}+\frac{4159376657}{38922761}a-\frac{1057383525}{38922761}$, $\frac{9232280}{38922761}a^{15}+\frac{63484648}{38922761}a^{14}-\frac{237845423}{38922761}a^{13}+\frac{207652185}{38922761}a^{12}-\frac{160882611}{38922761}a^{11}+\frac{435519376}{38922761}a^{10}-\frac{1351558726}{38922761}a^{9}+\frac{2602331336}{38922761}a^{8}-\frac{1784250690}{38922761}a^{7}-\frac{2921525126}{38922761}a^{6}+\frac{4077423640}{38922761}a^{5}+\frac{211713569}{38922761}a^{4}-\frac{2548121780}{38922761}a^{3}+\frac{1227753774}{38922761}a^{2}-\frac{676466247}{38922761}a+\frac{327419716}{38922761}$
| 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: | \( 106.032960978 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 106.032960978 \cdot 1}{2\cdot\sqrt{195238297600000000}}\cr\approx \mathstrut & 0.185544231733 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 26*x^11 + 52*x^10 - 54*x^9 - 12*x^8 + 82*x^7 - 54*x^6 - 24*x^5 + 49*x^4 - 32*x^3 + 16*x^2 - 6*x + 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 - 4*x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 26*x^11 + 52*x^10 - 54*x^9 - 12*x^8 + 82*x^7 - 54*x^6 - 24*x^5 + 49*x^4 - 32*x^3 + 16*x^2 - 6*x + 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 - 4*x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 26*x^11 + 52*x^10 - 54*x^9 - 12*x^8 + 82*x^7 - 54*x^6 - 24*x^5 + 49*x^4 - 32*x^3 + 16*x^2 - 6*x + 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 - 4*x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 26*x^11 + 52*x^10 - 54*x^9 - 12*x^8 + 82*x^7 - 54*x^6 - 24*x^5 + 49*x^4 - 32*x^3 + 16*x^2 - 6*x + 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
$C_4^4:D_4$ (as 16T1477):
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 2048 |
| The 74 conjugacy class representatives for $C_4^4:D_4$ |
| Character table for $C_4^4:D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.400.1, 8.2.4960000.1 |
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
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| 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 | $16$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $16$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $24$ | |||
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.3.2 | $x^{4} + 93$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 31.8.0.1 | $x^{8} + 25 x^{3} + 12 x^{2} + 24 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |