Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 36*x^13 + 76*x^12 - 137*x^11 + 211*x^10 - 281*x^9 + 331*x^8 - 336*x^7 + 305*x^6 - 244*x^5 + 172*x^4 - 108*x^3 + 60*x^2 - 24*x + 11)
gp: K = bnfinit(y^16 - 4*y^15 + 14*y^14 - 36*y^13 + 76*y^12 - 137*y^11 + 211*y^10 - 281*y^9 + 331*y^8 - 336*y^7 + 305*y^6 - 244*y^5 + 172*y^4 - 108*y^3 + 60*y^2 - 24*y + 11, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 14*x^14 - 36*x^13 + 76*x^12 - 137*x^11 + 211*x^10 - 281*x^9 + 331*x^8 - 336*x^7 + 305*x^6 - 244*x^5 + 172*x^4 - 108*x^3 + 60*x^2 - 24*x + 11);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 14*x^14 - 36*x^13 + 76*x^12 - 137*x^11 + 211*x^10 - 281*x^9 + 331*x^8 - 336*x^7 + 305*x^6 - 244*x^5 + 172*x^4 - 108*x^3 + 60*x^2 - 24*x + 11)
\( x^{16} - 4 x^{15} + 14 x^{14} - 36 x^{13} + 76 x^{12} - 137 x^{11} + 211 x^{10} - 281 x^{9} + 331 x^{8} + \cdots + 11 \)
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: |
\(148333301025390625\)
\(\medspace = 5^{12}\cdot 157^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.84\) | 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: | $5^{3/4}157^{1/2}\approx 41.89646002154997$ | ||
| Ramified primes: |
\(5\), \(157\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{53553}a^{15}-\frac{8399}{53553}a^{14}-\frac{1831}{53553}a^{13}+\frac{19349}{53553}a^{12}-\frac{26381}{53553}a^{11}+\frac{8852}{53553}a^{10}-\frac{5489}{17851}a^{9}+\frac{680}{17851}a^{8}+\frac{11491}{53553}a^{7}-\frac{18328}{53553}a^{6}+\frac{23947}{53553}a^{5}-\frac{5066}{17851}a^{4}-\frac{11566}{53553}a^{3}+\frac{4873}{53553}a^{2}-\frac{12134}{53553}a+\frac{8317}{17851}$
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{8419}{53553} a^{15} + \frac{21221}{53553} a^{14} - \frac{61628}{53553} a^{13} + \frac{116101}{53553} a^{12} - \frac{143311}{53553} a^{11} + \frac{127894}{53553} a^{10} + \frac{13503}{17851} a^{9} - \frac{119706}{17851} a^{8} + \frac{670178}{53553} a^{7} - \frac{1000268}{53553} a^{6} + \frac{1034759}{53553} a^{5} - \frac{281001}{17851} a^{4} + \frac{657436}{53553} a^{3} - \frac{379060}{53553} a^{2} + \frac{191234}{53553} a - \frac{44903}{17851} \)
(order $10$)
| 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: |
$\frac{6652}{53553}a^{15}-\frac{14369}{53553}a^{14}+\frac{30272}{53553}a^{13}-\frac{31864}{53553}a^{12}-\frac{46784}{53553}a^{11}+\frac{242969}{53553}a^{10}-\frac{203894}{17851}a^{9}+\frac{381928}{17851}a^{8}-\frac{1588589}{53553}a^{7}+\frac{1896680}{53553}a^{6}-\frac{1845533}{53553}a^{5}+\frac{485633}{17851}a^{4}-\frac{1052431}{53553}a^{3}+\frac{658267}{53553}a^{2}-\frac{278762}{53553}a+\frac{40137}{17851}$, $\frac{12856}{53553}a^{15}-\frac{68249}{53553}a^{14}+\frac{238196}{53553}a^{13}-\frac{645577}{53553}a^{12}+\frac{1389391}{53553}a^{11}-\frac{2515804}{53553}a^{10}+\frac{1283691}{17851}a^{9}-\frac{1682904}{17851}a^{8}+\frac{5705740}{53553}a^{7}-\frac{5561080}{53553}a^{6}+\frac{4699099}{53553}a^{5}-\frac{1168363}{17851}a^{4}+\frac{2326964}{53553}a^{3}-\frac{1402100}{53553}a^{2}+\frac{647821}{53553}a-\frac{75542}{17851}$, $\frac{6956}{53553}a^{15}-\frac{14972}{53553}a^{14}+\frac{14960}{17851}a^{13}-\frac{76300}{53553}a^{12}+\frac{30433}{17851}a^{11}-\frac{27614}{17851}a^{10}-\frac{30287}{53553}a^{9}+\frac{70969}{17851}a^{8}-\frac{398104}{53553}a^{7}+\frac{214970}{17851}a^{6}-\frac{235463}{17851}a^{5}+\frac{674672}{53553}a^{4}-\frac{605573}{53553}a^{3}+\frac{130087}{17851}a^{2}-\frac{236639}{53553}a+\frac{136691}{53553}$, $\frac{13397}{53553}a^{15}-\frac{14084}{17851}a^{14}+\frac{122324}{53553}a^{13}-\frac{87861}{17851}a^{12}+\frac{434116}{53553}a^{11}-\frac{600883}{53553}a^{10}+\frac{654746}{53553}a^{9}-\frac{172560}{17851}a^{8}+\frac{301370}{53553}a^{7}+\frac{18140}{53553}a^{6}-\frac{196574}{53553}a^{5}+\frac{179410}{53553}a^{4}-\frac{181532}{53553}a^{3}+\frac{145282}{53553}a^{2}-\frac{20515}{17851}a+\frac{62125}{53553}$, $\frac{2978}{17851}a^{15}-\frac{44615}{53553}a^{14}+\frac{154021}{53553}a^{13}-\frac{397840}{53553}a^{12}+\frac{837896}{53553}a^{11}-\frac{1442093}{53553}a^{10}+\frac{2118182}{53553}a^{9}-\frac{868919}{17851}a^{8}+\frac{910232}{17851}a^{7}-\frac{2422865}{53553}a^{6}+\frac{1801214}{53553}a^{5}-\frac{1128839}{53553}a^{4}+\frac{223094}{17851}a^{3}-\frac{288823}{53553}a^{2}+\frac{111073}{53553}a+\frac{42499}{53553}$, $\frac{3011}{17851}a^{15}-\frac{12373}{17851}a^{14}+\frac{38520}{17851}a^{13}-\frac{95080}{17851}a^{12}+\frac{182269}{17851}a^{11}-\frac{301638}{17851}a^{10}+\frac{418514}{17851}a^{9}-\frac{498132}{17851}a^{8}+\frac{521842}{17851}a^{7}-\frac{472293}{17851}a^{6}+\frac{379099}{17851}a^{5}-\frac{294681}{17851}a^{4}+\frac{180585}{17851}a^{3}-\frac{108025}{17851}a^{2}+\frac{41225}{17851}a-\frac{7398}{17851}$, $\frac{19886}{53553}a^{15}-\frac{97813}{53553}a^{14}+\frac{326092}{53553}a^{13}-\frac{860939}{53553}a^{12}+\frac{1813424}{53553}a^{11}-\frac{3211019}{53553}a^{10}+\frac{1629052}{17851}a^{9}-\frac{2114996}{17851}a^{8}+\frac{7175477}{53553}a^{7}-\frac{7057886}{53553}a^{6}+\frac{5961149}{53553}a^{5}-\frac{1490916}{17851}a^{4}+\frac{2954074}{53553}a^{3}-\frac{1579489}{53553}a^{2}+\frac{762836}{53553}a-\frac{86908}{17851}$
| 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: | \( 374.86019186 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 374.86019186 \cdot 1}{10\cdot\sqrt{148333301025390625}}\cr\approx \mathstrut & 0.23642257283 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 36*x^13 + 76*x^12 - 137*x^11 + 211*x^10 - 281*x^9 + 331*x^8 - 336*x^7 + 305*x^6 - 244*x^5 + 172*x^4 - 108*x^3 + 60*x^2 - 24*x + 11)
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 + 14*x^14 - 36*x^13 + 76*x^12 - 137*x^11 + 211*x^10 - 281*x^9 + 331*x^8 - 336*x^7 + 305*x^6 - 244*x^5 + 172*x^4 - 108*x^3 + 60*x^2 - 24*x + 11, 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 + 14*x^14 - 36*x^13 + 76*x^12 - 137*x^11 + 211*x^10 - 281*x^9 + 331*x^8 - 336*x^7 + 305*x^6 - 244*x^5 + 172*x^4 - 108*x^3 + 60*x^2 - 24*x + 11);
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 + 14*x^14 - 36*x^13 + 76*x^12 - 137*x^11 + 211*x^10 - 281*x^9 + 331*x^8 - 336*x^7 + 305*x^6 - 244*x^5 + 172*x^4 - 108*x^3 + 60*x^2 - 24*x + 11);
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
$C_4\times S_4$ (as 16T181):
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 96 |
| The 20 conjugacy class representatives for $C_4\times S_4$ |
| Character table for $C_4\times S_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.19625.1, 8.0.385140625.1 |
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 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Minimal sibling: | 12.8.47466656328125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
|
\(157\)
| 157.4.0.1 | $x^{4} + 11 x^{2} + 136 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 157.4.0.1 | $x^{4} + 11 x^{2} + 136 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 157.8.4.1 | $x^{8} + 63428 x^{7} + 1508667344 x^{6} + 15948649432594 x^{5} + 63224804026412329 x^{4} + 2679577892469870 x^{3} + 697675392505953606 x^{2} + 8612358275803260060 x + 401322525760684908$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |