Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 9*x^14 - 11*x^13 + 41*x^12 - 48*x^11 + 121*x^10 - 103*x^9 + 222*x^8 - 112*x^7 + 241*x^6 - 12*x^5 + 161*x^4 + 136*x^3 + 54*x^2 + 11*x + 1)
gp: K = bnfinit(y^16 - y^15 + 9*y^14 - 11*y^13 + 41*y^12 - 48*y^11 + 121*y^10 - 103*y^9 + 222*y^8 - 112*y^7 + 241*y^6 - 12*y^5 + 161*y^4 + 136*y^3 + 54*y^2 + 11*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 9*x^14 - 11*x^13 + 41*x^12 - 48*x^11 + 121*x^10 - 103*x^9 + 222*x^8 - 112*x^7 + 241*x^6 - 12*x^5 + 161*x^4 + 136*x^3 + 54*x^2 + 11*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 + 9*x^14 - 11*x^13 + 41*x^12 - 48*x^11 + 121*x^10 - 103*x^9 + 222*x^8 - 112*x^7 + 241*x^6 - 12*x^5 + 161*x^4 + 136*x^3 + 54*x^2 + 11*x + 1)
\( x^{16} - x^{15} + 9 x^{14} - 11 x^{13} + 41 x^{12} - 48 x^{11} + 121 x^{10} - 103 x^{9} + 222 x^{8} + \cdots + 1 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(119574225000000000000\)
\(\medspace = 2^{12}\cdot 3^{14}\cdot 5^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.98\) | 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: | $2\cdot 3^{7/8}5^{7/8}\approx 21.385029187241088$ | ||
| 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{3}$, $\frac{1}{21}a^{13}-\frac{1}{21}a^{11}-\frac{1}{21}a^{10}+\frac{1}{21}a^{9}+\frac{1}{21}a^{8}+\frac{10}{21}a^{7}-\frac{8}{21}a^{6}-\frac{8}{21}a^{5}-\frac{5}{21}a^{3}-\frac{1}{3}a^{2}+\frac{8}{21}a-\frac{3}{7}$, $\frac{1}{21}a^{14}-\frac{1}{21}a^{12}-\frac{1}{21}a^{11}+\frac{1}{21}a^{10}+\frac{1}{21}a^{9}+\frac{1}{7}a^{8}+\frac{2}{7}a^{7}+\frac{2}{7}a^{6}-\frac{1}{3}a^{5}+\frac{3}{7}a^{4}+\frac{1}{3}a^{3}+\frac{1}{21}a^{2}+\frac{5}{21}a-\frac{1}{3}$, $\frac{1}{10456677}a^{15}+\frac{8335}{3485559}a^{14}-\frac{6373}{497937}a^{13}-\frac{1487243}{10456677}a^{12}-\frac{10865}{105623}a^{11}-\frac{88643}{3485559}a^{10}+\frac{128182}{10456677}a^{9}-\frac{133139}{3485559}a^{8}-\frac{1228831}{3485559}a^{7}-\frac{3359008}{10456677}a^{6}-\frac{537162}{1161853}a^{5}-\frac{367072}{1161853}a^{4}-\frac{2775490}{10456677}a^{3}+\frac{680605}{3485559}a^{2}-\frac{1273886}{3485559}a-\frac{2302645}{10456677}$
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: |
\( \frac{2671523}{3485559} a^{15} - \frac{493585}{497937} a^{14} + \frac{24978755}{3485559} a^{13} - \frac{36588280}{3485559} a^{12} + \frac{10863740}{316869} a^{11} - \frac{162003104}{3485559} a^{10} + \frac{366907115}{3485559} a^{9} - \frac{125678645}{1161853} a^{8} + \frac{32920050}{165979} a^{7} - \frac{487858040}{3485559} a^{6} + \frac{763185616}{3485559} a^{5} - \frac{33782075}{497937} a^{4} + \frac{472515250}{3485559} a^{3} + \frac{235161260}{3485559} a^{2} + \frac{59808005}{3485559} a + \frac{7359727}{3485559} \)
(order $6$)
| 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: |
$\frac{2839}{4977}a^{15}-\frac{911}{1659}a^{14}+\frac{2804}{553}a^{13}-\frac{29996}{4977}a^{12}+\frac{12580}{553}a^{11}-\frac{14317}{553}a^{10}+\frac{46951}{711}a^{9}-\frac{88597}{1659}a^{8}+\frac{9345}{79}a^{7}-\frac{261613}{4977}a^{6}+\frac{204751}{1659}a^{5}+\frac{3903}{553}a^{4}+\frac{56123}{711}a^{3}+\frac{141863}{1659}a^{2}+\frac{14212}{553}a+\frac{16118}{4977}$, $\frac{5855384}{10456677}a^{15}-\frac{831728}{1161853}a^{14}+\frac{18152686}{3485559}a^{13}-\frac{79174066}{10456677}a^{12}+\frac{374009}{15089}a^{11}-\frac{116249047}{3485559}a^{10}+\frac{791671445}{10456677}a^{9}-\frac{267734542}{3485559}a^{8}+\frac{493366375}{3485559}a^{7}-\frac{1020020942}{10456677}a^{6}+\frac{537568019}{3485559}a^{5}-\frac{50755037}{1161853}a^{4}+\frac{984436960}{10456677}a^{3}+\frac{184935974}{3485559}a^{2}+\frac{39911303}{3485559}a+\frac{1067458}{1493811}$, $\frac{596411}{3485559}a^{15}-\frac{785990}{3485559}a^{14}+\frac{805286}{497937}a^{13}-\frac{2780627}{1161853}a^{12}+\frac{2481280}{316869}a^{11}-\frac{37266875}{3485559}a^{10}+\frac{4033647}{165979}a^{9}-\frac{29426959}{1161853}a^{8}+\frac{54025105}{1161853}a^{7}-\frac{117897194}{3485559}a^{6}+\frac{183758516}{3485559}a^{5}-\frac{63847655}{3485559}a^{4}+\frac{39458857}{1161853}a^{3}+\frac{46697641}{3485559}a^{2}+\frac{18619217}{3485559}a+\frac{1234949}{1161853}$, $\frac{282050}{1161853}a^{15}-\frac{1022170}{3485559}a^{14}+\frac{7832351}{3485559}a^{13}-\frac{3638757}{1161853}a^{12}+\frac{160001}{15089}a^{11}-\frac{48052729}{3485559}a^{10}+\frac{112265827}{3485559}a^{9}-\frac{109302953}{3485559}a^{8}+\frac{209584069}{3485559}a^{7}-\frac{135496085}{3485559}a^{6}+\frac{229685528}{3485559}a^{5}-\frac{52733902}{3485559}a^{4}+\frac{147215308}{3485559}a^{3}+\frac{89506729}{3485559}a^{2}+\frac{26798138}{3485559}a+\frac{364577}{1161853}$, $\frac{783572}{3485559}a^{15}-\frac{935048}{3485559}a^{14}+\frac{2400509}{1161853}a^{13}-\frac{3322899}{1161853}a^{12}+\frac{3070000}{316869}a^{11}-\frac{14550944}{1161853}a^{10}+\frac{101869066}{3485559}a^{9}-\frac{32747648}{1161853}a^{8}+\frac{62882488}{1161853}a^{7}-\frac{119348573}{3485559}a^{6}+\frac{204416249}{3485559}a^{5}-\frac{14133349}{1161853}a^{4}+\frac{42721956}{1161853}a^{3}+\frac{12064891}{497937}a^{2}+\frac{1048906}{165979}a+\frac{2671523}{3485559}$, $\frac{329536}{1493811}a^{15}-\frac{408438}{1161853}a^{14}+\frac{7437646}{3485559}a^{13}-\frac{37825175}{10456677}a^{12}+\frac{482290}{45267}a^{11}-\frac{56187325}{3485559}a^{10}+\frac{50381542}{1493811}a^{9}-\frac{45724816}{1161853}a^{8}+\frac{225482155}{3485559}a^{7}-\frac{82766830}{1493811}a^{6}+\frac{83928171}{1161853}a^{5}-\frac{41704646}{1161853}a^{4}+\frac{440455310}{10456677}a^{3}+\frac{31418920}{3485559}a^{2}-\frac{4537627}{3485559}a-\frac{8014357}{10456677}$, $\frac{423061}{950607}a^{15}-\frac{9837}{15089}a^{14}+\frac{1354049}{316869}a^{13}-\frac{6478061}{950607}a^{12}+\frac{6675602}{316869}a^{11}-\frac{3222463}{105623}a^{10}+\frac{62991907}{950607}a^{9}-\frac{23412350}{316869}a^{8}+\frac{40397585}{316869}a^{7}-\frac{13913812}{135801}a^{6}+\frac{45697399}{316869}a^{5}-\frac{950897}{15089}a^{4}+\frac{84610220}{950607}a^{3}+\frac{7881518}{316869}a^{2}+\frac{216164}{45267}a+\frac{24146}{135801}$
| 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: | \( 2507.21119771 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2507.21119771 \cdot 4}{6\cdot\sqrt{119574225000000000000}}\cr\approx \mathstrut & 0.371295614120 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 9*x^14 - 11*x^13 + 41*x^12 - 48*x^11 + 121*x^10 - 103*x^9 + 222*x^8 - 112*x^7 + 241*x^6 - 12*x^5 + 161*x^4 + 136*x^3 + 54*x^2 + 11*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 - x^15 + 9*x^14 - 11*x^13 + 41*x^12 - 48*x^11 + 121*x^10 - 103*x^9 + 222*x^8 - 112*x^7 + 241*x^6 - 12*x^5 + 161*x^4 + 136*x^3 + 54*x^2 + 11*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 - x^15 + 9*x^14 - 11*x^13 + 41*x^12 - 48*x^11 + 121*x^10 - 103*x^9 + 222*x^8 - 112*x^7 + 241*x^6 - 12*x^5 + 161*x^4 + 136*x^3 + 54*x^2 + 11*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 - x^15 + 9*x^14 - 11*x^13 + 41*x^12 - 48*x^11 + 121*x^10 - 103*x^9 + 222*x^8 - 112*x^7 + 241*x^6 - 12*x^5 + 161*x^4 + 136*x^3 + 54*x^2 + 11*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
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 $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.0.13500.1, 4.0.13500.2, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.182250000.1 |
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
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 | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(3\)
| 3.16.14.2 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34188 x^{9} + 53458 x^{8} + 68592 x^{7} + 71008 x^{6} + 56896 x^{5} + 33488 x^{4} + 14784 x^{3} + 6308 x^{2} + 2732 x + 661$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
|
\(5\)
| 5.16.14.2 | $x^{16} + 32 x^{15} + 464 x^{14} + 4032 x^{13} + 23408 x^{12} + 95872 x^{11} + 285376 x^{10} + 627456 x^{9} + 1027188 x^{8} + 1255232 x^{7} + 1144864 x^{6} + 789376 x^{5} + 469728 x^{4} + 388864 x^{3} + 473216 x^{2} + 436736 x + 184996$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ |