Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 4*x^13 + 28*x^12 - 24*x^11 + 141*x^10 - 218*x^9 + 489*x^8 - 664*x^7 + 568*x^6 - 440*x^5 + 148*x^4 - 288*x^3 + 768*x^2 - 576*x + 144)
gp: K = bnfinit(y^16 + 8*y^14 - 4*y^13 + 28*y^12 - 24*y^11 + 141*y^10 - 218*y^9 + 489*y^8 - 664*y^7 + 568*y^6 - 440*y^5 + 148*y^4 - 288*y^3 + 768*y^2 - 576*y + 144, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 8*x^14 - 4*x^13 + 28*x^12 - 24*x^11 + 141*x^10 - 218*x^9 + 489*x^8 - 664*x^7 + 568*x^6 - 440*x^5 + 148*x^4 - 288*x^3 + 768*x^2 - 576*x + 144);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 8*x^14 - 4*x^13 + 28*x^12 - 24*x^11 + 141*x^10 - 218*x^9 + 489*x^8 - 664*x^7 + 568*x^6 - 440*x^5 + 148*x^4 - 288*x^3 + 768*x^2 - 576*x + 144)
\( x^{16} + 8 x^{14} - 4 x^{13} + 28 x^{12} - 24 x^{11} + 141 x^{10} - 218 x^{9} + 489 x^{8} - 664 x^{7} + \cdots + 144 \)
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: |
\(968265199641600000000\)
\(\medspace = 2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.49\) | 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: | $2\cdot 3^{1/2}5^{1/2}7^{1/2}\approx 20.493901531919196$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{6}a^{4}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{6}a^{5}-\frac{1}{3}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{1608}a^{12}-\frac{1}{12}a^{11}+\frac{1}{268}a^{10}-\frac{32}{201}a^{9}+\frac{1}{6}a^{8}+\frac{2}{67}a^{7}+\frac{79}{536}a^{6}-\frac{13}{201}a^{5}+\frac{515}{1608}a^{4}+\frac{145}{804}a^{3}+\frac{391}{804}a^{2}-\frac{19}{134}a+\frac{51}{134}$, $\frac{1}{4824}a^{13}-\frac{1}{4824}a^{12}+\frac{35}{1206}a^{11}+\frac{137}{2412}a^{10}-\frac{137}{1206}a^{9}-\frac{323}{1206}a^{8}-\frac{1955}{4824}a^{7}+\frac{329}{4824}a^{6}-\frac{419}{1608}a^{5}+\frac{109}{536}a^{4}-\frac{413}{1206}a^{3}-\frac{347}{804}a^{2}-\frac{32}{201}a-\frac{17}{134}$, $\frac{1}{9648}a^{14}-\frac{1}{4824}a^{12}+\frac{1}{804}a^{11}-\frac{79}{2412}a^{10}+\frac{61}{1206}a^{9}+\frac{2381}{9648}a^{8}+\frac{209}{1608}a^{7}-\frac{2185}{9648}a^{6}-\frac{275}{804}a^{5}-\frac{61}{4824}a^{4}-\frac{176}{603}a^{3}-\frac{127}{268}a^{2}-\frac{125}{402}a+\frac{33}{67}$, $\frac{1}{22474639728}a^{15}-\frac{248399}{5618659932}a^{14}-\frac{156535}{1872886644}a^{13}+\frac{1575253}{11237319864}a^{12}-\frac{29001841}{624295548}a^{11}-\frac{145443173}{1872886644}a^{10}+\frac{376936477}{2497182192}a^{9}-\frac{5612596903}{11237319864}a^{8}-\frac{10857099515}{22474639728}a^{7}-\frac{192864377}{416197032}a^{6}+\frac{7155442}{20965149}a^{5}+\frac{2292729865}{11237319864}a^{4}+\frac{379102421}{1872886644}a^{3}-\frac{775112947}{1872886644}a^{2}-\frac{947809}{52024629}a+\frac{87796993}{312147774}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
\( -1 \)
(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: |
$\frac{14848669}{2809329966}a^{15}+\frac{36736547}{22474639728}a^{14}+\frac{161933891}{3745773288}a^{13}-\frac{94388689}{11237319864}a^{12}+\frac{46016021}{312147774}a^{11}-\frac{164612311}{1872886644}a^{10}+\frac{225759953}{312147774}a^{9}-\frac{21269027113}{22474639728}a^{8}+\frac{13230571013}{5618659932}a^{7}-\frac{7273600963}{2497182192}a^{6}+\frac{26094343153}{11237319864}a^{5}-\frac{22031558485}{11237319864}a^{4}+\frac{647894575}{1872886644}a^{3}-\frac{2587141829}{1872886644}a^{2}+\frac{76104262}{17341543}a-\frac{318548090}{156073887}$, $\frac{4224451}{208098516}a^{15}+\frac{14812385}{1248591096}a^{14}+\frac{8830021}{52024629}a^{13}+\frac{22773203}{1248591096}a^{12}+\frac{120083837}{208098516}a^{11}-\frac{98077657}{624295548}a^{10}+\frac{1708116185}{624295548}a^{9}-\frac{3573088859}{1248591096}a^{8}+\frac{425005361}{52024629}a^{7}-\frac{1370604763}{156073887}a^{6}+\frac{614494663}{104049258}a^{5}-\frac{6716497711}{1248591096}a^{4}-\frac{945500711}{624295548}a^{3}-\frac{402755081}{69366172}a^{2}+\frac{17740123}{1552974}a-\frac{152181375}{34683086}$, $\frac{207055955}{22474639728}a^{15}+\frac{164924945}{22474639728}a^{14}+\frac{294103367}{3745773288}a^{13}+\frac{306551477}{11237319864}a^{12}+\frac{42622210}{156073887}a^{11}+\frac{184889}{27953532}a^{10}+\frac{3170140727}{2497182192}a^{9}-\frac{22229526721}{22474639728}a^{8}+\frac{79362060155}{22474639728}a^{7}-\frac{2497284823}{832394064}a^{6}+\frac{25186852963}{11237319864}a^{5}-\frac{19143544405}{11237319864}a^{4}-\frac{1304474693}{1872886644}a^{3}-\frac{6997116173}{1872886644}a^{2}+\frac{499049153}{104049258}a-\frac{70204109}{156073887}$, $\frac{9131915}{3745773288}a^{15}-\frac{20152991}{7491546576}a^{14}+\frac{8761919}{624295548}a^{13}-\frac{141605213}{3745773288}a^{12}+\frac{6201947}{208098516}a^{11}-\frac{102260747}{624295548}a^{10}+\frac{33172649}{138732344}a^{9}-\frac{7159839263}{7491546576}a^{8}+\frac{464932783}{468221661}a^{7}-\frac{2192900281}{832394064}a^{6}+\frac{1038508439}{936443322}a^{5}-\frac{4461138953}{3745773288}a^{4}+\frac{27489827}{156073887}a^{3}-\frac{383667955}{624295548}a^{2}+\frac{96150965}{34683086}a-\frac{85525741}{52024629}$, $\frac{6810989}{11237319864}a^{15}-\frac{42628039}{11237319864}a^{14}-\frac{2209703}{3745773288}a^{13}-\frac{394916075}{11237319864}a^{12}-\frac{2580293}{312147774}a^{11}-\frac{214351769}{1872886644}a^{10}+\frac{60421429}{1248591096}a^{9}-\frac{6776822593}{11237319864}a^{8}+\frac{2300784877}{5618659932}a^{7}-\frac{223664888}{156073887}a^{6}+\frac{10046454701}{11237319864}a^{5}-\frac{1766374805}{11237319864}a^{4}+\frac{34465727}{468221661}a^{3}+\frac{533105093}{1872886644}a^{2}+\frac{5901802}{17341543}a-\frac{156080471}{312147774}$, $\frac{65165557}{11237319864}a^{15}+\frac{52718363}{5618659932}a^{14}+\frac{109168327}{1872886644}a^{13}+\frac{710638469}{11237319864}a^{12}+\frac{47947333}{208098516}a^{11}+\frac{335312057}{1872886644}a^{10}+\frac{1221246281}{1248591096}a^{9}+\frac{220462244}{1404664983}a^{8}+\frac{28619587123}{11237319864}a^{7}-\frac{57701179}{416197032}a^{6}+\frac{10879493537}{5618659932}a^{5}-\frac{2413276909}{11237319864}a^{4}+\frac{458776501}{1872886644}a^{3}-\frac{3571255595}{1872886644}a^{2}+\frac{145342097}{104049258}a-\frac{17238775}{312147774}$, $\frac{1352159}{239091912}a^{15}+\frac{1047389}{239091912}a^{14}+\frac{2005523}{39848652}a^{13}+\frac{1878893}{119545956}a^{12}+\frac{1206263}{6641442}a^{11}-\frac{76433}{9962163}a^{10}+\frac{7300721}{8855256}a^{9}-\frac{155723173}{239091912}a^{8}+\frac{586968863}{239091912}a^{7}-\frac{20161451}{8855256}a^{6}+\frac{254017093}{119545956}a^{5}-\frac{214884799}{119545956}a^{4}+\frac{85949}{9962163}a^{3}-\frac{19147276}{9962163}a^{2}+\frac{2699219}{1106907}a-\frac{1163785}{3320721}$
| 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: | \( 7388.19103958 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7388.19103958 \cdot 4}{2\cdot\sqrt{968265199641600000000}}\cr\approx \mathstrut & 1.15347954536 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 4*x^13 + 28*x^12 - 24*x^11 + 141*x^10 - 218*x^9 + 489*x^8 - 664*x^7 + 568*x^6 - 440*x^5 + 148*x^4 - 288*x^3 + 768*x^2 - 576*x + 144)
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^14 - 4*x^13 + 28*x^12 - 24*x^11 + 141*x^10 - 218*x^9 + 489*x^8 - 664*x^7 + 568*x^6 - 440*x^5 + 148*x^4 - 288*x^3 + 768*x^2 - 576*x + 144, 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^14 - 4*x^13 + 28*x^12 - 24*x^11 + 141*x^10 - 218*x^9 + 489*x^8 - 664*x^7 + 568*x^6 - 440*x^5 + 148*x^4 - 288*x^3 + 768*x^2 - 576*x + 144);
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^14 - 4*x^13 + 28*x^12 - 24*x^11 + 141*x^10 - 218*x^9 + 489*x^8 - 664*x^7 + 568*x^6 - 440*x^5 + 148*x^4 - 288*x^3 + 768*x^2 - 576*x + 144);
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
$D_4:C_2^2$ (as 16T23):
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 17 conjugacy class representatives for $Q_8 : C_2^2$ |
| Character table for $Q_8 : C_2^2$ |
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 | R | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |