Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 50*x^14 + 1917*x^12 - 5630*x^10 - 240908*x^8 - 707030*x^6 + 57802141*x^4 - 1883834854*x^2 + 21028770169)
gp: K = bnfinit(y^16 + 50*y^14 + 1917*y^12 - 5630*y^10 - 240908*y^8 - 707030*y^6 + 57802141*y^4 - 1883834854*y^2 + 21028770169, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 50*x^14 + 1917*x^12 - 5630*x^10 - 240908*x^8 - 707030*x^6 + 57802141*x^4 - 1883834854*x^2 + 21028770169);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 50*x^14 + 1917*x^12 - 5630*x^10 - 240908*x^8 - 707030*x^6 + 57802141*x^4 - 1883834854*x^2 + 21028770169)
\( x^{16} + 50 x^{14} + 1917 x^{12} - 5630 x^{10} - 240908 x^{8} - 707030 x^{6} + 57802141 x^{4} + \cdots + 21028770169 \)
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: |
\(419374069872350970710519002168327921\)
\(\medspace = 11^{8}\cdot 89^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(168.43\) | 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: | $11^{1/2}89^{7/8}\approx 168.42734332479344$ | ||
| Ramified primes: |
\(11\), \(89\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{6}-\frac{1}{8}a^{3}+\frac{1}{8}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{7}-\frac{1}{8}a^{4}+\frac{1}{8}a$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{8}a^{5}+\frac{1}{8}a^{2}$, $\frac{1}{32}a^{12}+\frac{1}{16}a^{6}-\frac{3}{32}$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{12}+\frac{1}{32}a^{7}-\frac{1}{32}a^{6}+\frac{29}{64}a-\frac{29}{64}$, $\frac{1}{19\!\cdots\!64}a^{14}-\frac{39\!\cdots\!85}{19\!\cdots\!64}a^{12}-\frac{65\!\cdots\!33}{11\!\cdots\!04}a^{10}-\frac{29\!\cdots\!19}{95\!\cdots\!32}a^{8}-\frac{1}{8}a^{7}+\frac{44\!\cdots\!75}{95\!\cdots\!32}a^{6}+\frac{64\!\cdots\!05}{11\!\cdots\!04}a^{4}-\frac{99\!\cdots\!31}{19\!\cdots\!64}a^{2}+\frac{1}{8}a-\frac{84\!\cdots\!55}{17\!\cdots\!24}$, $\frac{1}{55\!\cdots\!64}a^{15}-\frac{1}{38\!\cdots\!28}a^{14}+\frac{32\!\cdots\!31}{55\!\cdots\!64}a^{13}+\frac{39\!\cdots\!85}{38\!\cdots\!28}a^{12}-\frac{18\!\cdots\!25}{34\!\cdots\!04}a^{11}-\frac{84\!\cdots\!55}{23\!\cdots\!08}a^{10}+\frac{13\!\cdots\!41}{27\!\cdots\!32}a^{9}+\frac{29\!\cdots\!19}{19\!\cdots\!64}a^{8}+\frac{36\!\cdots\!55}{27\!\cdots\!32}a^{7}+\frac{19\!\cdots\!33}{19\!\cdots\!64}a^{6}+\frac{77\!\cdots\!05}{34\!\cdots\!04}a^{5}+\frac{85\!\cdots\!83}{23\!\cdots\!08}a^{4}-\frac{59\!\cdots\!39}{55\!\cdots\!64}a^{3}+\frac{99\!\cdots\!31}{38\!\cdots\!28}a^{2}-\frac{20\!\cdots\!91}{50\!\cdots\!24}a+\frac{40\!\cdots\!99}{34\!\cdots\!48}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (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: |
\( -1 \)
(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: |
$\frac{60\!\cdots\!41}{55\!\cdots\!88}a^{15}+\frac{3043344227}{29\!\cdots\!36}a^{14}-\frac{23\!\cdots\!63}{11\!\cdots\!76}a^{13}+\frac{223598469479}{58\!\cdots\!72}a^{12}+\frac{17\!\cdots\!01}{13\!\cdots\!22}a^{11}+\frac{489126209246}{36\!\cdots\!67}a^{10}-\frac{57\!\cdots\!78}{69\!\cdots\!61}a^{9}-\frac{40501298313405}{14\!\cdots\!68}a^{8}+\frac{19\!\cdots\!37}{55\!\cdots\!88}a^{7}+\frac{38686081716573}{29\!\cdots\!36}a^{6}-\frac{10\!\cdots\!37}{13\!\cdots\!22}a^{5}+\frac{18\!\cdots\!78}{36\!\cdots\!67}a^{4}+\frac{64\!\cdots\!83}{55\!\cdots\!88}a^{3}-\frac{35\!\cdots\!17}{29\!\cdots\!36}a^{2}-\frac{15\!\cdots\!89}{10\!\cdots\!16}a+\frac{22\!\cdots\!19}{58\!\cdots\!72}$, $\frac{14\!\cdots\!19}{44\!\cdots\!04}a^{15}-\frac{3043344227}{29\!\cdots\!36}a^{14}+\frac{67\!\cdots\!57}{44\!\cdots\!04}a^{13}-\frac{223598469479}{58\!\cdots\!72}a^{12}+\frac{15\!\cdots\!15}{27\!\cdots\!44}a^{11}-\frac{489126209246}{36\!\cdots\!67}a^{10}-\frac{73\!\cdots\!05}{22\!\cdots\!52}a^{9}+\frac{40501298313405}{14\!\cdots\!68}a^{8}-\frac{18\!\cdots\!15}{22\!\cdots\!52}a^{7}-\frac{38686081716573}{29\!\cdots\!36}a^{6}+\frac{14\!\cdots\!41}{27\!\cdots\!44}a^{5}-\frac{18\!\cdots\!78}{36\!\cdots\!67}a^{4}+\frac{65\!\cdots\!07}{44\!\cdots\!04}a^{3}+\frac{35\!\cdots\!17}{29\!\cdots\!36}a^{2}-\frac{43\!\cdots\!17}{40\!\cdots\!64}a+\frac{64\!\cdots\!61}{58\!\cdots\!72}$, $\frac{14\!\cdots\!19}{44\!\cdots\!04}a^{15}+\frac{3043344227}{29\!\cdots\!36}a^{14}+\frac{67\!\cdots\!57}{44\!\cdots\!04}a^{13}+\frac{223598469479}{58\!\cdots\!72}a^{12}+\frac{15\!\cdots\!15}{27\!\cdots\!44}a^{11}+\frac{489126209246}{36\!\cdots\!67}a^{10}-\frac{73\!\cdots\!05}{22\!\cdots\!52}a^{9}-\frac{40501298313405}{14\!\cdots\!68}a^{8}-\frac{18\!\cdots\!15}{22\!\cdots\!52}a^{7}+\frac{38686081716573}{29\!\cdots\!36}a^{6}+\frac{14\!\cdots\!41}{27\!\cdots\!44}a^{5}+\frac{18\!\cdots\!78}{36\!\cdots\!67}a^{4}+\frac{65\!\cdots\!07}{44\!\cdots\!04}a^{3}-\frac{35\!\cdots\!17}{29\!\cdots\!36}a^{2}-\frac{43\!\cdots\!17}{40\!\cdots\!64}a-\frac{64\!\cdots\!61}{58\!\cdots\!72}$, $\frac{20\!\cdots\!07}{16\!\cdots\!68}a^{15}+\frac{43\!\cdots\!95}{23\!\cdots\!08}a^{14}+\frac{59\!\cdots\!01}{16\!\cdots\!68}a^{13}+\frac{16\!\cdots\!17}{23\!\cdots\!08}a^{12}+\frac{48\!\cdots\!25}{10\!\cdots\!48}a^{11}+\frac{45\!\cdots\!85}{14\!\cdots\!88}a^{10}-\frac{75\!\cdots\!33}{84\!\cdots\!84}a^{9}+\frac{12\!\cdots\!47}{11\!\cdots\!04}a^{8}-\frac{13\!\cdots\!43}{84\!\cdots\!84}a^{7}+\frac{32\!\cdots\!49}{11\!\cdots\!04}a^{6}-\frac{33\!\cdots\!33}{10\!\cdots\!48}a^{5}+\frac{12\!\cdots\!99}{14\!\cdots\!88}a^{4}+\frac{26\!\cdots\!39}{16\!\cdots\!68}a^{3}+\frac{22\!\cdots\!75}{23\!\cdots\!08}a^{2}-\frac{68\!\cdots\!61}{15\!\cdots\!88}a-\frac{14\!\cdots\!61}{21\!\cdots\!28}$, $\frac{66\!\cdots\!79}{27\!\cdots\!32}a^{15}+\frac{45\!\cdots\!17}{27\!\cdots\!32}a^{13}+\frac{11\!\cdots\!29}{17\!\cdots\!52}a^{11}+\frac{80\!\cdots\!71}{13\!\cdots\!16}a^{9}-\frac{20\!\cdots\!91}{13\!\cdots\!16}a^{7}-\frac{50\!\cdots\!37}{17\!\cdots\!52}a^{5}+\frac{42\!\cdots\!35}{27\!\cdots\!32}a^{3}+\frac{77\!\cdots\!55}{25\!\cdots\!12}a$, $\frac{91\!\cdots\!81}{69\!\cdots\!08}a^{15}-\frac{40\!\cdots\!21}{19\!\cdots\!64}a^{14}+\frac{12\!\cdots\!07}{69\!\cdots\!08}a^{13}+\frac{56\!\cdots\!33}{19\!\cdots\!64}a^{12}-\frac{86\!\cdots\!69}{43\!\cdots\!88}a^{11}-\frac{28\!\cdots\!71}{11\!\cdots\!04}a^{10}+\frac{45\!\cdots\!77}{34\!\cdots\!04}a^{9}-\frac{50\!\cdots\!85}{95\!\cdots\!32}a^{8}+\frac{37\!\cdots\!27}{34\!\cdots\!04}a^{7}-\frac{35\!\cdots\!39}{95\!\cdots\!32}a^{6}-\frac{72\!\cdots\!59}{43\!\cdots\!88}a^{5}+\frac{28\!\cdots\!31}{11\!\cdots\!04}a^{4}-\frac{21\!\cdots\!31}{69\!\cdots\!08}a^{3}-\frac{69\!\cdots\!53}{19\!\cdots\!64}a^{2}+\frac{30\!\cdots\!97}{63\!\cdots\!28}a+\frac{40\!\cdots\!51}{17\!\cdots\!24}$, $\frac{24\!\cdots\!85}{69\!\cdots\!08}a^{15}-\frac{20\!\cdots\!13}{47\!\cdots\!16}a^{14}+\frac{18\!\cdots\!75}{69\!\cdots\!08}a^{13}-\frac{14\!\cdots\!27}{47\!\cdots\!16}a^{12}+\frac{64\!\cdots\!27}{43\!\cdots\!88}a^{11}-\frac{37\!\cdots\!99}{29\!\cdots\!76}a^{10}+\frac{18\!\cdots\!53}{34\!\cdots\!04}a^{9}-\frac{28\!\cdots\!93}{23\!\cdots\!08}a^{8}-\frac{31\!\cdots\!37}{34\!\cdots\!04}a^{7}+\frac{61\!\cdots\!17}{23\!\cdots\!08}a^{6}-\frac{64\!\cdots\!27}{43\!\cdots\!88}a^{5}+\frac{16\!\cdots\!79}{29\!\cdots\!76}a^{4}-\frac{89\!\cdots\!27}{69\!\cdots\!08}a^{3}-\frac{10\!\cdots\!73}{47\!\cdots\!16}a^{2}+\frac{18\!\cdots\!17}{63\!\cdots\!28}a-\frac{28\!\cdots\!09}{43\!\cdots\!56}$
| 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: | \( 59363790305.1 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 59363790305.1 \cdot 32}{2\cdot\sqrt{419374069872350970710519002168327921}}\cr\approx \mathstrut & 3.56270419131 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 50*x^14 + 1917*x^12 - 5630*x^10 - 240908*x^8 - 707030*x^6 + 57802141*x^4 - 1883834854*x^2 + 21028770169)
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 + 50*x^14 + 1917*x^12 - 5630*x^10 - 240908*x^8 - 707030*x^6 + 57802141*x^4 - 1883834854*x^2 + 21028770169, 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 + 50*x^14 + 1917*x^12 - 5630*x^10 - 240908*x^8 - 707030*x^6 + 57802141*x^4 - 1883834854*x^2 + 21028770169);
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 + 50*x^14 + 1917*x^12 - 5630*x^10 - 240908*x^8 - 707030*x^6 + 57802141*x^4 - 1883834854*x^2 + 21028770169);
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
$\OD_{16}:C_2$ (as 16T36):
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 $\OD_{16}:C_2$ |
| Character table for $\OD_{16}:C_2$ |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(89\)
| 89.8.7.3 | $x^{8} + 89$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 89.8.7.3 | $x^{8} + 89$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |