Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 2*x^13 - 3*x^12 + 6*x^11 - 8*x^10 - 10*x^9 + 48*x^8 - 90*x^7 + 122*x^6 - 114*x^5 + 83*x^4 - 44*x^3 + 17*x^2 - 4*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + 3*y^14 - 2*y^13 - 3*y^12 + 6*y^11 - 8*y^10 - 10*y^9 + 48*y^8 - 90*y^7 + 122*y^6 - 114*y^5 + 83*y^4 - 44*y^3 + 17*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 3*x^14 - 2*x^13 - 3*x^12 + 6*x^11 - 8*x^10 - 10*x^9 + 48*x^8 - 90*x^7 + 122*x^6 - 114*x^5 + 83*x^4 - 44*x^3 + 17*x^2 - 4*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 3*x^14 - 2*x^13 - 3*x^12 + 6*x^11 - 8*x^10 - 10*x^9 + 48*x^8 - 90*x^7 + 122*x^6 - 114*x^5 + 83*x^4 - 44*x^3 + 17*x^2 - 4*x + 1)
\( x^{16} - 2 x^{15} + 3 x^{14} - 2 x^{13} - 3 x^{12} + 6 x^{11} - 8 x^{10} - 10 x^{9} + 48 x^{8} + \cdots + 1 \)
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: |
\(748709478400000000\)
\(\medspace = 2^{26}\cdot 5^{8}\cdot 13^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.10\) | 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: | $2^{51/16}5^{1/2}13^{1/2}\approx 73.44965995393171$ | ||
| Ramified primes: |
\(2\), \(5\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{119858}a^{15}-\frac{3229}{59929}a^{14}-\frac{17733}{119858}a^{13}+\frac{9928}{59929}a^{12}-\frac{1104}{59929}a^{11}-\frac{4124}{59929}a^{10}+\frac{16064}{59929}a^{9}+\frac{27910}{59929}a^{8}+\frac{19567}{59929}a^{7}+\frac{5735}{59929}a^{6}+\frac{11023}{59929}a^{5}-\frac{28822}{59929}a^{4}-\frac{9343}{119858}a^{3}+\frac{14895}{59929}a^{2}+\frac{47867}{119858}a-\frac{17716}{59929}$
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
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: |
\( -1 \)
(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: |
$\frac{2091}{119858}a^{15}-\frac{19653}{119858}a^{14}+\frac{8174}{59929}a^{13}-\frac{11901}{119858}a^{12}-\frac{31162}{59929}a^{11}+\frac{72913}{119858}a^{10}-\frac{30345}{59929}a^{9}-\frac{82001}{119858}a^{8}+\frac{222806}{59929}a^{7}-\frac{647049}{119858}a^{6}+\frac{336002}{59929}a^{5}+\frac{103473}{119858}a^{4}-\frac{598649}{119858}a^{3}+\frac{401868}{59929}a^{2}-\frac{265447}{59929}a+\frac{51895}{59929}$, $\frac{38179}{119858}a^{15}-\frac{72005}{119858}a^{14}+\frac{54782}{59929}a^{13}-\frac{9813}{59929}a^{12}-\frac{79458}{59929}a^{11}+\frac{266219}{119858}a^{10}-\frac{125788}{59929}a^{9}-\frac{582679}{119858}a^{8}+\frac{932443}{59929}a^{7}-\frac{3344097}{119858}a^{6}+\frac{1823549}{59929}a^{5}-\frac{2414769}{119858}a^{4}+\frac{1069733}{119858}a^{3}+\frac{69853}{59929}a^{2}-\frac{70260}{59929}a+\frac{18989}{119858}$, $a$, $\frac{79483}{119858}a^{15}-\frac{34629}{59929}a^{14}+\frac{58041}{119858}a^{13}+\frac{104091}{119858}a^{12}-\frac{192963}{59929}a^{11}+\frac{83667}{59929}a^{10}-\frac{32362}{59929}a^{9}-\frac{740411}{59929}a^{8}+\frac{1404749}{59929}a^{7}-\frac{1363336}{59929}a^{6}+\frac{818135}{59929}a^{5}+\frac{646147}{59929}a^{4}-\frac{2366661}{119858}a^{3}+\frac{1200470}{59929}a^{2}-\frac{1238313}{119858}a+\frac{361415}{119858}$, $\frac{42655}{119858}a^{15}-\frac{92235}{119858}a^{14}+\frac{41326}{59929}a^{13}-\frac{18877}{119858}a^{12}-\frac{106784}{59929}a^{11}+\frac{264435}{119858}a^{10}-\frac{78195}{59929}a^{9}-\frac{636289}{119858}a^{8}+\frac{1137853}{59929}a^{7}-\frac{3303601}{119858}a^{6}+\frac{1661143}{59929}a^{5}-\frac{2135323}{119858}a^{4}+\frac{841191}{119858}a^{3}-\frac{21033}{59929}a^{2}+\frac{22892}{59929}a-\frac{31219}{59929}$, $\frac{80609}{59929}a^{15}-\frac{239043}{119858}a^{14}+\frac{166898}{59929}a^{13}-\frac{71357}{59929}a^{12}-\frac{295187}{59929}a^{11}+\frac{637807}{119858}a^{10}-\frac{445286}{59929}a^{9}-\frac{2087111}{119858}a^{8}+\frac{3365928}{59929}a^{7}-\frac{10606197}{119858}a^{6}+\frac{6803354}{59929}a^{5}-\frac{10547937}{119858}a^{4}+\frac{3533667}{59929}a^{3}-\frac{1511145}{59929}a^{2}+\frac{401841}{59929}a-\frac{143541}{119858}$, $\frac{27561}{59929}a^{15}+\frac{192}{59929}a^{14}+\frac{23493}{119858}a^{13}+\frac{39517}{59929}a^{12}-\frac{86682}{59929}a^{11}-\frac{84791}{119858}a^{10}-\frac{31096}{59929}a^{9}-\frac{983471}{119858}a^{8}+\frac{509393}{59929}a^{7}-\frac{660829}{119858}a^{6}+\frac{229391}{59929}a^{5}+\frac{762089}{119858}a^{4}-\frac{47439}{59929}a^{3}+\frac{329425}{119858}a^{2}+\frac{150549}{119858}a-\frac{56523}{119858}$
| 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: | \( 301.521901775 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 301.521901775 \cdot 1}{2\cdot\sqrt{748709478400000000}}\cr\approx \mathstrut & 0.423224839795 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 2*x^13 - 3*x^12 + 6*x^11 - 8*x^10 - 10*x^9 + 48*x^8 - 90*x^7 + 122*x^6 - 114*x^5 + 83*x^4 - 44*x^3 + 17*x^2 - 4*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 - 2*x^15 + 3*x^14 - 2*x^13 - 3*x^12 + 6*x^11 - 8*x^10 - 10*x^9 + 48*x^8 - 90*x^7 + 122*x^6 - 114*x^5 + 83*x^4 - 44*x^3 + 17*x^2 - 4*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 - 2*x^15 + 3*x^14 - 2*x^13 - 3*x^12 + 6*x^11 - 8*x^10 - 10*x^9 + 48*x^8 - 90*x^7 + 122*x^6 - 114*x^5 + 83*x^4 - 44*x^3 + 17*x^2 - 4*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 - 2*x^15 + 3*x^14 - 2*x^13 - 3*x^12 + 6*x^11 - 8*x^10 - 10*x^9 + 48*x^8 - 90*x^7 + 122*x^6 - 114*x^5 + 83*x^4 - 44*x^3 + 17*x^2 - 4*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
$C_2^2.C_2\wr C_2^3$ (as 16T1701):
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 8192 |
| The 119 conjugacy class representatives for $C_2^2.C_2\wr C_2^3$ |
| Character table for $C_2^2.C_2\wr C_2^3$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.400.1, 8.0.27040000.1 |
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: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.3 | $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.4.4.3 | $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ | |
| 2.8.18.11 | $x^{8} + 8 x^{7} + 10 x^{6} - 16 x^{5} + 80 x^{4} + 96 x^{3} + 100 x^{2} + 144 x + 84$ | $4$ | $2$ | $18$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 3, 7/2, 7/2]^{2}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(13\)
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |