Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 + 115*x^12 - 342*x^10 + 505*x^8 - 352*x^6 + 88*x^4 + 8*x^2 - 4)
gp: K = bnfinit(y^16 - 18*y^14 + 115*y^12 - 342*y^10 + 505*y^8 - 352*y^6 + 88*y^4 + 8*y^2 - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 18*x^14 + 115*x^12 - 342*x^10 + 505*x^8 - 352*x^6 + 88*x^4 + 8*x^2 - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 18*x^14 + 115*x^12 - 342*x^10 + 505*x^8 - 352*x^6 + 88*x^4 + 8*x^2 - 4)
\( x^{16} - 18x^{14} + 115x^{12} - 342x^{10} + 505x^{8} - 352x^{6} + 88x^{4} + 8x^{2} - 4 \)
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: | $[14, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-907610275348357845090304\)
\(\medspace = -\,2^{46}\cdot 337^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.43\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(337\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{12}a^{12}-\frac{1}{12}a^{10}-\frac{1}{4}a^{8}+\frac{1}{6}a^{6}+\frac{1}{6}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{12}a^{13}-\frac{1}{12}a^{11}+\frac{5}{12}a^{7}-\frac{1}{12}a^{5}-\frac{1}{3}a^{3}-\frac{1}{6}a$, $\frac{1}{24}a^{14}-\frac{1}{24}a^{13}+\frac{1}{24}a^{11}+\frac{1}{12}a^{10}-\frac{1}{8}a^{9}-\frac{1}{24}a^{8}+\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{3}a^{5}+\frac{1}{6}a^{4}+\frac{1}{6}a^{3}-\frac{1}{6}a+\frac{1}{6}$, $\frac{1}{24}a^{15}-\frac{1}{24}a^{13}-\frac{1}{24}a^{12}-\frac{1}{8}a^{11}+\frac{1}{24}a^{10}+\frac{1}{12}a^{9}-\frac{1}{8}a^{8}+\frac{1}{12}a^{7}+\frac{1}{6}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{6}a^{2}-\frac{1}{6}$
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: | $14$ | 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: |
$a^{14}-\frac{69}{4}a^{12}+\frac{409}{4}a^{10}-\frac{1073}{4}a^{8}+\frac{637}{2}a^{6}-\frac{285}{2}a^{4}+4a^{2}+5$, $\frac{1}{6}a^{14}-\frac{11}{4}a^{12}+\frac{89}{6}a^{10}-\frac{187}{6}a^{8}+\frac{197}{12}a^{6}+\frac{115}{6}a^{4}-\frac{29}{2}a^{2}-\frac{1}{3}$, $\frac{5}{6}a^{15}-\frac{57}{4}a^{13}+\frac{995}{12}a^{11}-\frac{1253}{6}a^{9}+\frac{2689}{12}a^{7}-\frac{833}{12}a^{5}-16a^{3}+\frac{47}{6}a$, $\frac{1}{4}a^{14}-\frac{49}{12}a^{12}+\frac{131}{6}a^{10}-\frac{189}{4}a^{8}+\frac{445}{12}a^{6}-\frac{5}{3}a^{4}-\frac{25}{6}a^{2}-\frac{1}{3}$, $\frac{5}{24}a^{15}+\frac{1}{6}a^{14}-\frac{83}{24}a^{13}-\frac{67}{24}a^{12}+\frac{153}{8}a^{11}+\frac{125}{8}a^{10}-\frac{535}{12}a^{9}-\frac{877}{24}a^{8}+\frac{259}{6}a^{7}+\frac{197}{6}a^{6}-\frac{163}{12}a^{5}-\frac{19}{6}a^{4}+\frac{7}{3}a^{3}-\frac{17}{6}a^{2}-\frac{3}{2}a-\frac{3}{2}$, $\frac{19}{12}a^{15}-\frac{329}{12}a^{13}+\frac{490}{3}a^{11}-\frac{2579}{6}a^{9}+\frac{1011}{2}a^{7}-\frac{841}{4}a^{5}-\frac{41}{6}a^{3}+\frac{49}{6}a$, $\frac{13}{12}a^{15}-\frac{115}{6}a^{13}+\frac{1423}{12}a^{11}-\frac{3991}{12}a^{9}+\frac{879}{2}a^{7}-\frac{975}{4}a^{5}+\frac{199}{6}a^{3}+\frac{37}{6}a$, $\frac{1}{24}a^{15}-\frac{17}{24}a^{13}-\frac{1}{24}a^{12}+\frac{97}{24}a^{11}+\frac{19}{24}a^{10}-\frac{113}{12}a^{9}-\frac{43}{8}a^{8}+\frac{25}{4}a^{7}+\frac{197}{12}a^{6}+8a^{5}-\frac{67}{3}a^{4}-\frac{32}{3}a^{3}+\frac{35}{3}a^{2}+\frac{4}{3}a-\frac{7}{6}$, $\frac{5}{24}a^{15}-\frac{83}{24}a^{13}-\frac{1}{24}a^{12}+\frac{153}{8}a^{11}+\frac{19}{24}a^{10}-\frac{535}{12}a^{9}-\frac{43}{8}a^{8}+\frac{259}{6}a^{7}+\frac{197}{12}a^{6}-\frac{163}{12}a^{5}-\frac{67}{3}a^{4}+\frac{7}{3}a^{3}+\frac{35}{3}a^{2}-\frac{3}{2}a-\frac{13}{6}$, $\frac{5}{24}a^{15}-\frac{83}{24}a^{13}+\frac{1}{24}a^{12}+\frac{153}{8}a^{11}-\frac{19}{24}a^{10}-\frac{535}{12}a^{9}+\frac{43}{8}a^{8}+\frac{259}{6}a^{7}-\frac{197}{12}a^{6}-\frac{163}{12}a^{5}+\frac{67}{3}a^{4}+\frac{7}{3}a^{3}-\frac{35}{3}a^{2}-\frac{3}{2}a+\frac{13}{6}$, $\frac{25}{12}a^{15}+\frac{35}{24}a^{14}-\frac{291}{8}a^{13}-\frac{76}{3}a^{12}+\frac{5287}{24}a^{11}+152a^{10}-\frac{14381}{24}a^{9}-\frac{9761}{24}a^{8}+\frac{2275}{3}a^{7}+\frac{5993}{12}a^{6}-\frac{4655}{12}a^{5}-\frac{1439}{6}a^{4}+28a^{3}+\frac{71}{6}a^{2}+\frac{61}{3}a+\frac{25}{2}$, $\frac{19}{12}a^{15}-\frac{11}{8}a^{14}-\frac{659}{24}a^{13}+24a^{12}+\frac{1313}{8}a^{11}-145a^{10}-\frac{10445}{24}a^{9}+\frac{3125}{8}a^{8}+\frac{6263}{12}a^{7}-\frac{1915}{4}a^{6}-\frac{2791}{12}a^{5}+219a^{4}+\frac{29}{6}a^{3}-\frac{5}{2}a^{2}+6a-\frac{17}{2}$, $\frac{3}{2}a^{15}-\frac{1}{3}a^{14}-\frac{313}{12}a^{13}+6a^{12}+\frac{1879}{12}a^{11}-\frac{455}{12}a^{10}-\frac{1671}{4}a^{9}+\frac{1291}{12}a^{8}+\frac{1516}{3}a^{7}-\frac{1651}{12}a^{6}-\frac{1381}{6}a^{5}+\frac{373}{6}a^{4}+\frac{31}{3}a^{3}+\frac{5}{2}a^{2}+\frac{17}{3}a-\frac{10}{3}$, $\frac{2}{3}a^{15}+\frac{5}{3}a^{14}-\frac{47}{4}a^{13}-\frac{343}{12}a^{12}+\frac{217}{3}a^{11}+\frac{2009}{12}a^{10}-\frac{2423}{12}a^{9}-\frac{5141}{12}a^{8}+\frac{1591}{6}a^{7}+481a^{6}-\frac{1681}{12}a^{5}-\frac{367}{2}a^{4}+\frac{11}{2}a^{3}-\frac{23}{3}a^{2}+\frac{43}{6}a+\frac{22}{3}$
| 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: | \( 7460311.75721 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{1}\cdot 7460311.75721 \cdot 1}{2\cdot\sqrt{907610275348357845090304}}\cr\approx \mathstrut & 0.403066854765 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 + 115*x^12 - 342*x^10 + 505*x^8 - 352*x^6 + 88*x^4 + 8*x^2 - 4)
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 - 18*x^14 + 115*x^12 - 342*x^10 + 505*x^8 - 352*x^6 + 88*x^4 + 8*x^2 - 4, 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 - 18*x^14 + 115*x^12 - 342*x^10 + 505*x^8 - 352*x^6 + 88*x^4 + 8*x^2 - 4);
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 - 18*x^14 + 115*x^12 - 342*x^10 + 505*x^8 - 352*x^6 + 88*x^4 + 8*x^2 - 4);
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
$C_2^8.C_2\wr C_4$ (as 16T1776):
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 16384 |
| The 136 conjugacy class representatives for $C_2^8.C_2\wr C_4$ |
| Character table for $C_2^8.C_2\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.21568.1, 8.8.14885715968.1 |
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 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.12.1815220550696715690180608.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | $16$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.30.40 | $x^{8} + 8 x^{7} + 4 x^{4} + 18$ | $8$ | $1$ | $30$ | $((C_8 : C_2):C_2):C_2$ | $[2, 3, 7/2, 4, 17/4, 19/4]$ |
| 2.8.16.13 | $x^{8} + 8 x^{7} + 18 x^{6} + 4 x^{5} - 8 x^{4} + 24 x^{3} + 20 x^{2} - 8 x + 4$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
|
\(337\)
| $\Q_{337}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{337}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{337}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{337}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $8$ | $2$ | $4$ | $4$ |