Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 52*x^13 + 36*x^12 + 68*x^11 - 204*x^10 + 228*x^9 - 61*x^8 - 188*x^7 + 348*x^6 - 340*x^5 + 228*x^4 - 108*x^3 + 36*x^2 - 8*x + 1)
gp: K = bnfinit(y^16 - 8*y^15 + 28*y^14 - 52*y^13 + 36*y^12 + 68*y^11 - 204*y^10 + 228*y^9 - 61*y^8 - 188*y^7 + 348*y^6 - 340*y^5 + 228*y^4 - 108*y^3 + 36*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 28*x^14 - 52*x^13 + 36*x^12 + 68*x^11 - 204*x^10 + 228*x^9 - 61*x^8 - 188*x^7 + 348*x^6 - 340*x^5 + 228*x^4 - 108*x^3 + 36*x^2 - 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 28*x^14 - 52*x^13 + 36*x^12 + 68*x^11 - 204*x^10 + 228*x^9 - 61*x^8 - 188*x^7 + 348*x^6 - 340*x^5 + 228*x^4 - 108*x^3 + 36*x^2 - 8*x + 1)
\( x^{16} - 8 x^{15} + 28 x^{14} - 52 x^{13} + 36 x^{12} + 68 x^{11} - 204 x^{10} + 228 x^{9} - 61 x^{8} + \cdots + 1 \)
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: |
\(10737418240000000000\)
\(\medspace = 2^{40}\cdot 5^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.47\) | 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: | $2^{5/2}5^{3/4}\approx 18.914832180063517$ | ||
| Ramified primes: |
\(2\), \(5\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{148428799}a^{15}-\frac{2468841}{148428799}a^{14}-\frac{72498354}{148428799}a^{13}+\frac{47664303}{148428799}a^{12}+\frac{38251631}{148428799}a^{11}+\frac{43874401}{148428799}a^{10}+\frac{18744395}{148428799}a^{9}+\frac{53153815}{148428799}a^{8}-\frac{61777669}{148428799}a^{7}-\frac{6666356}{148428799}a^{6}+\frac{37592178}{148428799}a^{5}+\frac{7706111}{148428799}a^{4}+\frac{57031188}{148428799}a^{3}-\frac{29659518}{148428799}a^{2}+\frac{17391860}{148428799}a+\frac{33504131}{148428799}$
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 $1$ (assuming GRH)
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: |
\( \frac{175532702}{148428799} a^{15} - \frac{1297965238}{148428799} a^{14} + \frac{4112541718}{148428799} a^{13} - \frac{6529376787}{148428799} a^{12} + \frac{2092499934}{148428799} a^{11} + \frac{13413244316}{148428799} a^{10} - \frac{27147820104}{148428799} a^{9} + \frac{21915168372}{148428799} a^{8} + \frac{3841511330}{148428799} a^{7} - \frac{29497121382}{148428799} a^{6} + \frac{39987674200}{148428799} a^{5} - \frac{33155090166}{148428799} a^{4} + \frac{20181582432}{148428799} a^{3} - \frac{8416673086}{148428799} a^{2} + \frac{2653701410}{148428799} a - \frac{464281868}{148428799} \)
(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: |
$\frac{43507777}{148428799}a^{15}-\frac{166275328}{148428799}a^{14}-\frac{67892335}{148428799}a^{13}+\frac{1579851886}{148428799}a^{12}-\frac{3938908484}{148428799}a^{11}+\frac{3296453916}{148428799}a^{10}+\frac{5256299878}{148428799}a^{9}-\frac{13876421688}{148428799}a^{8}+\frac{12524400496}{148428799}a^{7}+\frac{1042438524}{148428799}a^{6}-\frac{13930857298}{148428799}a^{5}+\frac{18591622354}{148428799}a^{4}-\frac{13263807327}{148428799}a^{3}+\frac{6479556207}{148428799}a^{2}-\frac{1577795629}{148428799}a+\frac{265621190}{148428799}$, $\frac{69236324}{148428799}a^{15}-\frac{495591300}{148428799}a^{14}+\frac{1514218001}{148428799}a^{13}-\frac{2302699449}{148428799}a^{12}+\frac{615479354}{148428799}a^{11}+\frac{4752393212}{148428799}a^{10}-\frac{9041532420}{148428799}a^{9}+\frac{7366162220}{148428799}a^{8}+\frac{338484736}{148428799}a^{7}-\frac{8399415286}{148428799}a^{6}+\frac{13461645686}{148428799}a^{5}-\frac{13315216542}{148428799}a^{4}+\frac{9640723702}{148428799}a^{3}-\frac{4751197224}{148428799}a^{2}+\frac{1713721662}{148428799}a-\frac{437177965}{148428799}$, $\frac{106296378}{148428799}a^{15}-\frac{802373938}{148428799}a^{14}+\frac{2598323717}{148428799}a^{13}-\frac{4226677338}{148428799}a^{12}+\frac{1477020580}{148428799}a^{11}+\frac{8660851104}{148428799}a^{10}-\frac{18106287684}{148428799}a^{9}+\frac{14549006152}{148428799}a^{8}+\frac{3503026594}{148428799}a^{7}-\frac{21097706096}{148428799}a^{6}+\frac{26526028514}{148428799}a^{5}-\frac{19839873624}{148428799}a^{4}+\frac{10540858730}{148428799}a^{3}-\frac{3665475862}{148428799}a^{2}+\frac{939979748}{148428799}a-\frac{175532702}{148428799}$, $\frac{184326527}{148428799}a^{15}-\frac{1321671132}{148428799}a^{14}+\frac{4034580825}{148428799}a^{13}-\frac{6044119892}{148428799}a^{12}+\frac{1100883975}{148428799}a^{11}+\frac{14069215923}{148428799}a^{10}-\frac{25665182307}{148428799}a^{9}+\frac{18553983525}{148428799}a^{8}+\frac{6969210099}{148428799}a^{7}-\frac{30014150434}{148428799}a^{6}+\frac{36916954260}{148428799}a^{5}-\frac{27975702669}{148428799}a^{4}+\frac{15063596241}{148428799}a^{3}-\frac{5637511465}{148428799}a^{2}+\frac{1553474717}{148428799}a-\frac{384360205}{148428799}$, $\frac{18123462}{148428799}a^{15}-\frac{84588992}{148428799}a^{14}+\frac{102297453}{148428799}a^{13}+\frac{229446494}{148428799}a^{12}-\frac{1052030868}{148428799}a^{11}+\frac{1589306206}{148428799}a^{10}+\frac{33789019}{148428799}a^{9}-\frac{2780821461}{148428799}a^{8}+\frac{5106478707}{148428799}a^{7}-\frac{3086554427}{148428799}a^{6}-\frac{1581808071}{148428799}a^{5}+\frac{5940327574}{148428799}a^{4}-\frac{6012484278}{148428799}a^{3}+\frac{3677741571}{148428799}a^{2}-\frac{1347877889}{148428799}a+\frac{205758844}{148428799}$, $\frac{86549312}{148428799}a^{15}-\frac{617378977}{148428799}a^{14}+\frac{1880959115}{148428799}a^{13}-\frac{2820763636}{148428799}a^{12}+\frac{528027718}{148428799}a^{11}+\frac{6516228573}{148428799}a^{10}-\frac{11836677363}{148428799}a^{9}+\frac{8485624184}{148428799}a^{8}+\frac{3371556291}{148428799}a^{7}-\frac{13315614956}{148428799}a^{6}+\frac{15938827149}{148428799}a^{5}-\frac{12585499225}{148428799}a^{4}+\frac{8592052068}{148428799}a^{3}-\frac{4114695749}{148428799}a^{2}+\frac{1295400761}{148428799}a-\frac{300282959}{148428799}$, $\frac{182986474}{148428799}a^{15}-\frac{1718669867}{148428799}a^{14}+\frac{6944404901}{148428799}a^{13}-\frac{15019146216}{148428799}a^{12}+\frac{14538618896}{148428799}a^{11}+\frac{12100101374}{148428799}a^{10}-\frac{58493473850}{148428799}a^{9}+\frac{77305172648}{148428799}a^{8}-\frac{32285814635}{148428799}a^{7}-\frac{52343810030}{148428799}a^{6}+\frac{109683110544}{148428799}a^{5}-\frac{109275015577}{148428799}a^{4}+\frac{69523567218}{148428799}a^{3}-\frac{30769127468}{148428799}a^{2}+\frac{8211763474}{148428799}a-\frac{1546212824}{148428799}$
| 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: | \( 2042.10948725 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2042.10948725 \cdot 1}{4\cdot\sqrt{10737418240000000000}}\cr\approx \mathstrut & 0.378449593906 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 52*x^13 + 36*x^12 + 68*x^11 - 204*x^10 + 228*x^9 - 61*x^8 - 188*x^7 + 348*x^6 - 340*x^5 + 228*x^4 - 108*x^3 + 36*x^2 - 8*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 - 8*x^15 + 28*x^14 - 52*x^13 + 36*x^12 + 68*x^11 - 204*x^10 + 228*x^9 - 61*x^8 - 188*x^7 + 348*x^6 - 340*x^5 + 228*x^4 - 108*x^3 + 36*x^2 - 8*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 - 8*x^15 + 28*x^14 - 52*x^13 + 36*x^12 + 68*x^11 - 204*x^10 + 228*x^9 - 61*x^8 - 188*x^7 + 348*x^6 - 340*x^5 + 228*x^4 - 108*x^3 + 36*x^2 - 8*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 - 8*x^15 + 28*x^14 - 52*x^13 + 36*x^12 + 68*x^11 - 204*x^10 + 228*x^9 - 61*x^8 - 188*x^7 + 348*x^6 - 340*x^5 + 228*x^4 - 108*x^3 + 36*x^2 - 8*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
$C_2^3:C_4$ (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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.2.1600.1 x2, 4.0.1280.1 x2, \(\Q(i, \sqrt{5})\), 8.0.204800000.2 x2, 8.0.40960000.2, 8.0.655360000.4 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 | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 $16$ | $8$ | $2$ | $40$ | |||
|
\(5\)
| 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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |