Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 36*x^13 + 38*x^11 + 74*x^10 - 280*x^9 + 266*x^8 - 100*x^7 + 296*x^6 - 702*x^5 + 573*x^4 + 72*x^3 - 172*x^2 + 16)
gp: K = bnfinit(y^16 - 8*y^15 + 26*y^14 - 36*y^13 + 38*y^11 + 74*y^10 - 280*y^9 + 266*y^8 - 100*y^7 + 296*y^6 - 702*y^5 + 573*y^4 + 72*y^3 - 172*y^2 + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 26*x^14 - 36*x^13 + 38*x^11 + 74*x^10 - 280*x^9 + 266*x^8 - 100*x^7 + 296*x^6 - 702*x^5 + 573*x^4 + 72*x^3 - 172*x^2 + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 26*x^14 - 36*x^13 + 38*x^11 + 74*x^10 - 280*x^9 + 266*x^8 - 100*x^7 + 296*x^6 - 702*x^5 + 573*x^4 + 72*x^3 - 172*x^2 + 16)
\( x^{16} - 8 x^{15} + 26 x^{14} - 36 x^{13} + 38 x^{11} + 74 x^{10} - 280 x^{9} + 266 x^{8} - 100 x^{7} + \cdots + 16 \)
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: |
\(63880676485490517601\)
\(\medspace = 13^{8}\cdot 23^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.29\) | 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: | $13^{1/2}23^{1/2}\approx 17.291616465790582$ | ||
| Ramified primes: |
\(13\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{16}a^{12}+\frac{1}{16}a^{9}-\frac{1}{8}a^{8}+\frac{3}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{5}{16}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{13}+\frac{1}{16}a^{10}-\frac{1}{8}a^{9}+\frac{3}{16}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{3}{16}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{48}a^{14}-\frac{1}{48}a^{13}-\frac{1}{48}a^{12}+\frac{1}{48}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{5}{48}a^{8}+\frac{1}{48}a^{7}+\frac{5}{48}a^{6}+\frac{11}{48}a^{5}-\frac{3}{16}a^{4}-\frac{1}{48}a^{3}+\frac{7}{24}a^{2}-\frac{5}{12}a-\frac{1}{3}$, $\frac{1}{89355988704}a^{15}+\frac{3458645}{347688672}a^{14}-\frac{134354537}{29785329568}a^{13}+\frac{2144124701}{89355988704}a^{12}-\frac{1184010217}{89355988704}a^{11}-\frac{3362136367}{29785329568}a^{10}-\frac{4588754755}{89355988704}a^{9}-\frac{3943486393}{89355988704}a^{8}-\frac{1184478617}{89355988704}a^{7}-\frac{6716588011}{29785329568}a^{6}-\frac{9735985139}{89355988704}a^{5}-\frac{2202002983}{89355988704}a^{4}-\frac{1262539087}{14892664784}a^{3}+\frac{1282059191}{5584749294}a^{2}-\frac{1387688197}{11169498588}a-\frac{1198932467}{5584749294}$
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
$C_{3}$, which has order $3$
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: |
$\frac{2473524167}{44677994352}a^{15}-\frac{72917455}{173844336}a^{14}+\frac{14128319863}{11169498588}a^{13}-\frac{66446490223}{44677994352}a^{12}-\frac{7701204733}{14892664784}a^{11}+\frac{12624330473}{7446332392}a^{10}+\frac{217742562583}{44677994352}a^{9}-\frac{592899053515}{44677994352}a^{8}+\frac{26093651077}{2792374647}a^{7}-\frac{124625702897}{44677994352}a^{6}+\frac{243407956749}{14892664784}a^{5}-\frac{724586946823}{22338997176}a^{4}+\frac{446621037725}{22338997176}a^{3}+\frac{197927742157}{22338997176}a^{2}-\frac{40626325241}{11169498588}a-\frac{2467650535}{1861583098}$, $\frac{1072795235}{44677994352}a^{15}-\frac{4860957}{28974056}a^{14}+\frac{19845474377}{44677994352}a^{13}-\frac{5060243361}{14892664784}a^{12}-\frac{1553874640}{2792374647}a^{11}+\frac{8219432371}{14892664784}a^{10}+\frac{113717656285}{44677994352}a^{9}-\frac{33630549613}{7446332392}a^{8}+\frac{13687258293}{14892664784}a^{7}+\frac{30432662027}{44677994352}a^{6}+\frac{19198364242}{2792374647}a^{5}-\frac{456712226105}{44677994352}a^{4}+\frac{5636740471}{5584749294}a^{3}+\frac{27329032129}{3723166196}a^{2}+\frac{5014866803}{1861583098}a-\frac{5840495183}{2792374647}$, $\frac{105658309}{89355988704}a^{15}-\frac{3791351}{347688672}a^{14}+\frac{3703804445}{89355988704}a^{13}-\frac{7253998505}{89355988704}a^{12}+\frac{2546406549}{29785329568}a^{11}-\frac{2279072327}{29785329568}a^{10}+\frac{14731341527}{89355988704}a^{9}-\frac{28208851373}{89355988704}a^{8}+\frac{42844023239}{89355988704}a^{7}-\frac{95801635171}{89355988704}a^{6}+\frac{51763998831}{29785329568}a^{5}-\frac{138399084199}{89355988704}a^{4}+\frac{4685423041}{2792374647}a^{3}-\frac{32443757539}{11169498588}a^{2}+\frac{22652465113}{11169498588}a-\frac{13476806}{930791549}$, $\frac{2271105051}{29785329568}a^{15}-\frac{205082951}{347688672}a^{14}+\frac{164656140211}{89355988704}a^{13}-\frac{212340167255}{89355988704}a^{12}-\frac{25108106089}{89355988704}a^{11}+\frac{72405713771}{29785329568}a^{10}+\frac{182672400495}{29785329568}a^{9}-\frac{1725662985833}{89355988704}a^{8}+\frac{1480677850745}{89355988704}a^{7}-\frac{607367942837}{89355988704}a^{6}+\frac{2081917053061}{89355988704}a^{5}-\frac{1439202509279}{29785329568}a^{4}+\frac{1579951631231}{44677994352}a^{3}+\frac{143733009025}{22338997176}a^{2}-\frac{18556280789}{2792374647}a-\frac{788177509}{2792374647}$, $\frac{4287774387}{29785329568}a^{15}-\frac{376199215}{347688672}a^{14}+\frac{287746059671}{89355988704}a^{13}-\frac{325142322295}{89355988704}a^{12}-\frac{150912207353}{89355988704}a^{11}+\frac{134114802063}{29785329568}a^{10}+\frac{384268986463}{29785329568}a^{9}-\frac{3026918038849}{89355988704}a^{8}+\frac{1958931986101}{89355988704}a^{7}-\frac{430962072421}{89355988704}a^{6}+\frac{3708400912037}{89355988704}a^{5}-\frac{2415026891299}{29785329568}a^{4}+\frac{1956009415675}{44677994352}a^{3}+\frac{647457348221}{22338997176}a^{2}-\frac{21641753815}{2792374647}a-\frac{10484103362}{2792374647}$, $\frac{999824353}{89355988704}a^{15}-\frac{32591291}{347688672}a^{14}+\frac{29182124981}{89355988704}a^{13}-\frac{46614014993}{89355988704}a^{12}+\frac{4961908341}{29785329568}a^{11}+\frac{12945613281}{29785329568}a^{10}+\frac{56033933975}{89355988704}a^{9}-\frac{310861358405}{89355988704}a^{8}+\frac{391658529479}{89355988704}a^{7}-\frac{209463592651}{89355988704}a^{6}+\frac{104594376111}{29785329568}a^{5}-\frac{786561803023}{89355988704}a^{4}+\frac{225177751649}{22338997176}a^{3}-\frac{49313310287}{22338997176}a^{2}-\frac{5202932807}{2792374647}a+\frac{2147143271}{1861583098}$, $\frac{3042575755}{89355988704}a^{15}-\frac{29434961}{115896224}a^{14}+\frac{66235129513}{89355988704}a^{13}-\frac{23213965517}{29785329568}a^{12}-\frac{47972644601}{89355988704}a^{11}+\frac{31892943149}{29785329568}a^{10}+\frac{294788995421}{89355988704}a^{9}-\frac{236158043787}{29785329568}a^{8}+\frac{120493000417}{29785329568}a^{7}-\frac{1211350829}{89355988704}a^{6}+\frac{900596642573}{89355988704}a^{5}-\frac{1704490984651}{89355988704}a^{4}+\frac{170910940783}{22338997176}a^{3}+\frac{16408144271}{1861583098}a^{2}-\frac{1294212393}{3723166196}a-\frac{7870595528}{2792374647}$
| 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: | \( 2823.55404905 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2823.55404905 \cdot 3}{2\cdot\sqrt{63880676485490517601}}\cr\approx \mathstrut & 1.28718674271 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 36*x^13 + 38*x^11 + 74*x^10 - 280*x^9 + 266*x^8 - 100*x^7 + 296*x^6 - 702*x^5 + 573*x^4 + 72*x^3 - 172*x^2 + 16)
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 + 26*x^14 - 36*x^13 + 38*x^11 + 74*x^10 - 280*x^9 + 266*x^8 - 100*x^7 + 296*x^6 - 702*x^5 + 573*x^4 + 72*x^3 - 172*x^2 + 16, 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 + 26*x^14 - 36*x^13 + 38*x^11 + 74*x^10 - 280*x^9 + 266*x^8 - 100*x^7 + 296*x^6 - 702*x^5 + 573*x^4 + 72*x^3 - 172*x^2 + 16);
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 + 26*x^14 - 36*x^13 + 38*x^11 + 74*x^10 - 280*x^9 + 266*x^8 - 100*x^7 + 296*x^6 - 702*x^5 + 573*x^4 + 72*x^3 - 172*x^2 + 16);
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 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{-299}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{13}, \sqrt{-23})\), 4.2.3887.1 x2, 4.0.6877.1 x2, 8.0.7992538801.1, 8.2.347501687.1 x4, 8.0.614810677.1 x4 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(23\)
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |