Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 104*x^13 + 220*x^12 - 368*x^11 + 516*x^10 - 624*x^9 + 664*x^8 - 624*x^7 + 516*x^6 - 368*x^5 + 220*x^4 - 104*x^3 + 36*x^2 - 8*x + 1)
gp: K = bnfinit(y^16 - 8*y^15 + 36*y^14 - 104*y^13 + 220*y^12 - 368*y^11 + 516*y^10 - 624*y^9 + 664*y^8 - 624*y^7 + 516*y^6 - 368*y^5 + 220*y^4 - 104*y^3 + 36*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 36*x^14 - 104*x^13 + 220*x^12 - 368*x^11 + 516*x^10 - 624*x^9 + 664*x^8 - 624*x^7 + 516*x^6 - 368*x^5 + 220*x^4 - 104*x^3 + 36*x^2 - 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 36*x^14 - 104*x^13 + 220*x^12 - 368*x^11 + 516*x^10 - 624*x^9 + 664*x^8 - 624*x^7 + 516*x^6 - 368*x^5 + 220*x^4 - 104*x^3 + 36*x^2 - 8*x + 1)
\( x^{16} - 8 x^{15} + 36 x^{14} - 104 x^{13} + 220 x^{12} - 368 x^{11} + 516 x^{10} - 624 x^{9} + 664 x^{8} + \cdots + 1 \)
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: |
\(18014398509481984\)
\(\medspace = 2^{54}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.37\) | 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: | $2^{27/8}\approx 10.374716437208077$ | ||
| Ramified primes: |
\(2\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{3}{7}a^{11}-\frac{2}{7}a^{10}+\frac{1}{7}a^{9}-\frac{1}{7}a^{8}+\frac{1}{7}a^{7}+\frac{2}{7}a^{6}+\frac{1}{7}a^{5}-\frac{1}{7}a^{4}+\frac{1}{7}a^{3}-\frac{2}{7}a^{2}+\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{13}+\frac{3}{7}a^{11}+\frac{3}{7}a^{9}-\frac{3}{7}a^{8}-\frac{1}{7}a^{7}+\frac{2}{7}a^{6}+\frac{3}{7}a^{5}-\frac{3}{7}a^{4}+\frac{2}{7}a^{3}+\frac{2}{7}a^{2}-\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{14}-\frac{2}{7}a^{11}+\frac{2}{7}a^{10}+\frac{1}{7}a^{9}+\frac{2}{7}a^{8}-\frac{1}{7}a^{7}-\frac{3}{7}a^{6}+\frac{1}{7}a^{5}-\frac{2}{7}a^{4}-\frac{1}{7}a^{3}-\frac{2}{7}a^{2}+\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{329}a^{15}-\frac{9}{329}a^{14}-\frac{2}{329}a^{13}-\frac{8}{329}a^{12}+\frac{87}{329}a^{11}+\frac{156}{329}a^{10}-\frac{110}{329}a^{9}-\frac{13}{47}a^{8}-\frac{138}{329}a^{7}-\frac{16}{329}a^{6}-\frac{79}{329}a^{5}+\frac{134}{329}a^{4}+\frac{39}{329}a^{3}-\frac{7}{47}a^{2}-\frac{9}{329}a+\frac{48}{329}$
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: |
\( -\frac{559}{329} a^{15} + \frac{4373}{329} a^{14} - \frac{2714}{47} a^{13} + \frac{52459}{329} a^{12} - \frac{104516}{329} a^{11} + \frac{164904}{329} a^{10} - \frac{218348}{329} a^{9} + \frac{250102}{329} a^{8} - \frac{251200}{329} a^{7} + \frac{222277}{329} a^{6} - \frac{169877}{329} a^{5} + \frac{15693}{47} a^{4} - \frac{55829}{329} a^{3} + \frac{2879}{47} a^{2} - \frac{3617}{329} a + \frac{240}{329} \)
(order $16$)
| 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: |
$\frac{17}{47}a^{15}-\frac{59}{47}a^{14}+\frac{655}{329}a^{13}+\frac{2056}{329}a^{12}-\frac{9199}{329}a^{11}+\frac{20773}{329}a^{10}-\frac{33723}{329}a^{9}+\frac{45665}{329}a^{8}-\frac{51155}{329}a^{7}+\frac{51300}{329}a^{6}-\frac{44181}{329}a^{5}+\frac{32631}{329}a^{4}-\frac{19517}{329}a^{3}+\frac{8692}{329}a^{2}-\frac{1823}{329}a+\frac{17}{47}$, $\frac{559}{329}a^{15}-\frac{4373}{329}a^{14}+\frac{2714}{47}a^{13}-\frac{52459}{329}a^{12}+\frac{104516}{329}a^{11}-\frac{164904}{329}a^{10}+\frac{218348}{329}a^{9}-\frac{250102}{329}a^{8}+\frac{251200}{329}a^{7}-\frac{222277}{329}a^{6}+\frac{169877}{329}a^{5}-\frac{15693}{47}a^{4}+\frac{55829}{329}a^{3}-\frac{2879}{47}a^{2}+\frac{3946}{329}a-\frac{569}{329}$, $\frac{166}{329}a^{15}-\frac{836}{329}a^{14}+\frac{342}{47}a^{13}-\frac{2315}{329}a^{12}-\frac{2713}{329}a^{11}+\frac{14381}{329}a^{10}-\frac{30480}{329}a^{9}+\frac{47780}{329}a^{8}-\frac{8597}{47}a^{7}+\frac{65306}{329}a^{6}-\frac{61524}{329}a^{5}+\frac{49598}{329}a^{4}-\frac{33476}{329}a^{3}+\frac{17716}{329}a^{2}-\frac{6523}{329}a+\frac{158}{47}$, $\frac{1431}{329}a^{15}-\frac{9918}{329}a^{14}+\frac{40472}{329}a^{13}-\frac{102675}{329}a^{12}+\frac{193681}{329}a^{11}-\frac{292119}{329}a^{10}+\frac{376181}{329}a^{9}-\frac{418049}{329}a^{8}+\frac{412159}{329}a^{7}-\frac{354246}{329}a^{6}+\frac{265583}{329}a^{5}-\frac{165164}{329}a^{4}+\frac{82505}{329}a^{3}-\frac{4048}{47}a^{2}+\frac{6673}{329}a-\frac{543}{329}$, $\frac{10}{47}a^{15}-\frac{724}{329}a^{14}+\frac{544}{47}a^{13}-\frac{1866}{47}a^{12}+\frac{31940}{329}a^{11}-\frac{60332}{329}a^{10}+\frac{92222}{329}a^{9}-\frac{119076}{329}a^{8}+\frac{133549}{329}a^{7}-\frac{132438}{329}a^{6}+\frac{115119}{329}a^{5}-\frac{86829}{329}a^{4}+\frac{54806}{329}a^{3}-\frac{27259}{329}a^{2}+\frac{9381}{329}a-\frac{1622}{329}$, $\frac{12}{329}a^{15}-\frac{531}{329}a^{14}+\frac{3219}{329}a^{13}-\frac{11940}{329}a^{12}+\frac{27411}{329}a^{11}-\frac{47337}{329}a^{10}+\frac{66548}{329}a^{9}-\frac{80757}{329}a^{8}+\frac{84354}{329}a^{7}-\frac{78306}{329}a^{6}+\frac{8949}{47}a^{5}-\frac{42807}{329}a^{4}+\frac{23169}{329}a^{3}-\frac{9048}{329}a^{2}+\frac{1725}{329}a+\frac{12}{329}$, $\frac{669}{329}a^{15}-\frac{4423}{329}a^{14}+\frac{17368}{329}a^{13}-\frac{41495}{329}a^{12}+\frac{73431}{329}a^{11}-\frac{103423}{329}a^{10}+\frac{17808}{47}a^{9}-\frac{18285}{47}a^{8}+\frac{116076}{329}a^{7}-\frac{12742}{47}a^{6}+\frac{57223}{329}a^{5}-\frac{26303}{329}a^{4}+\frac{6351}{329}a^{3}+\frac{2986}{329}a^{2}-\frac{2637}{329}a+\frac{857}{329}$
| 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: | \( 273.623706795 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 273.623706795 \cdot 1}{16\cdot\sqrt{18014398509481984}}\cr\approx \mathstrut & 0.309501532156 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 104*x^13 + 220*x^12 - 368*x^11 + 516*x^10 - 624*x^9 + 664*x^8 - 624*x^7 + 516*x^6 - 368*x^5 + 220*x^4 - 104*x^3 + 36*x^2 - 8*x + 1)
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 + 36*x^14 - 104*x^13 + 220*x^12 - 368*x^11 + 516*x^10 - 624*x^9 + 664*x^8 - 624*x^7 + 516*x^6 - 368*x^5 + 220*x^4 - 104*x^3 + 36*x^2 - 8*x + 1, 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 + 36*x^14 - 104*x^13 + 220*x^12 - 368*x^11 + 516*x^10 - 624*x^9 + 664*x^8 - 624*x^7 + 516*x^6 - 368*x^5 + 220*x^4 - 104*x^3 + 36*x^2 - 8*x + 1);
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 + 36*x^14 - 104*x^13 + 220*x^12 - 368*x^11 + 516*x^10 - 624*x^9 + 664*x^8 - 624*x^7 + 516*x^6 - 368*x^5 + 220*x^4 - 104*x^3 + 36*x^2 - 8*x + 1);
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
$C_2^2:C_4$ (as 16T10):
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 10 conjugacy class representatives for $C_2^2 : C_4$ |
| Character table for $C_2^2 : C_4$ |
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 | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.54.6 | $x^{16} + 4 x^{14} + 8 x^{9} + 6 x^{8} + 8 x^{7} + 8 x^{6} + 12 x^{4} + 2$ | $16$ | $1$ | $54$ | $C_2^2 : C_4$ | $[2, 3, 7/2, 4]$ |