Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 + 4*x^13 + x^12 - 52*x^11 + 110*x^10 - 230*x^9 + 396*x^8 - 350*x^7 + 230*x^6 - 176*x^5 + 121*x^4 - 52*x^3 + 18*x^2 - 6*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + 2*y^14 + 4*y^13 + y^12 - 52*y^11 + 110*y^10 - 230*y^9 + 396*y^8 - 350*y^7 + 230*y^6 - 176*y^5 + 121*y^4 - 52*y^3 + 18*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 2*x^14 + 4*x^13 + x^12 - 52*x^11 + 110*x^10 - 230*x^9 + 396*x^8 - 350*x^7 + 230*x^6 - 176*x^5 + 121*x^4 - 52*x^3 + 18*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 2*x^14 + 4*x^13 + x^12 - 52*x^11 + 110*x^10 - 230*x^9 + 396*x^8 - 350*x^7 + 230*x^6 - 176*x^5 + 121*x^4 - 52*x^3 + 18*x^2 - 6*x + 1)
\( x^{16} - 2 x^{15} + 2 x^{14} + 4 x^{13} + x^{12} - 52 x^{11} + 110 x^{10} - 230 x^{9} + 396 x^{8} + \cdots + 1 \)
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: |
\(143420743705600000000\)
\(\medspace = 2^{16}\cdot 5^{8}\cdot 29^{4}\cdot 89^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.19\) | 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: | $2\cdot 5^{1/2}29^{1/2}89^{1/2}\approx 227.2003521124032$ | ||
| Ramified primes: |
\(2\), \(5\), \(29\), \(89\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{12}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a$, $\frac{1}{728}a^{14}+\frac{1}{52}a^{13}-\frac{1}{8}a^{12}-\frac{33}{364}a^{11}+\frac{16}{91}a^{10}-\frac{1}{52}a^{9}+\frac{3}{28}a^{8}-\frac{1}{182}a^{7}-\frac{3}{364}a^{6}+\frac{47}{364}a^{5}+\frac{45}{91}a^{4}+\frac{87}{364}a^{3}+\frac{13}{56}a^{2}+\frac{137}{364}a-\frac{31}{728}$, $\frac{1}{240173752}a^{15}+\frac{89303}{240173752}a^{14}-\frac{2924301}{34310536}a^{13}-\frac{18867187}{240173752}a^{12}+\frac{2411061}{120086876}a^{11}-\frac{29616807}{120086876}a^{10}+\frac{5773679}{60043438}a^{9}-\frac{589}{17155268}a^{8}+\frac{2028219}{9237452}a^{7}-\frac{888739}{8577634}a^{6}-\frac{30197415}{120086876}a^{5}+\frac{252461}{1165892}a^{4}-\frac{78415221}{240173752}a^{3}+\frac{74040643}{240173752}a^{2}-\frac{5800691}{18474904}a-\frac{117248325}{240173752}$
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: |
\( -\frac{81956143}{34310536} a^{15} + \frac{131418481}{34310536} a^{14} - \frac{110947283}{34310536} a^{13} - \frac{374020417}{34310536} a^{12} - \frac{114200861}{17155268} a^{11} + \frac{2087273425}{17155268} a^{10} - \frac{1840447767}{8577634} a^{9} + \frac{7940752877}{17155268} a^{8} - \frac{13021388629}{17155268} a^{7} + \frac{4537085817}{8577634} a^{6} - \frac{5628647241}{17155268} a^{5} + \frac{46595945}{166556} a^{4} - \frac{5939504245}{34310536} a^{3} + \frac{1717794121}{34310536} a^{2} - \frac{646345647}{34310536} a + \frac{189593689}{34310536} \)
(order $4$)
| 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: |
$\frac{1039203975}{240173752}a^{15}+\frac{1644173039}{240173752}a^{14}-\frac{194530285}{34310536}a^{13}-\frac{4776909913}{240173752}a^{12}-\frac{1494835939}{120086876}a^{11}+\frac{26461128209}{120086876}a^{10}-\frac{11517252647}{30021719}a^{9}+\frac{99498851083}{120086876}a^{8}-\frac{12520128549}{9237452}a^{7}+\frac{27703841120}{30021719}a^{6}-\frac{9745876665}{17155268}a^{5}+\frac{575275073}{1165892}a^{4}-\frac{71961604529}{240173752}a^{3}+\frac{2997532153}{34310536}a^{2}-\frac{613569123}{18474904}a+\frac{2283925965}{240173752}$, $\frac{27242947}{120086876}a^{15}-\frac{139041191}{240173752}a^{14}+\frac{2537568}{4288817}a^{13}+\frac{195282287}{240173752}a^{12}-\frac{46765533}{120086876}a^{11}-\frac{374538952}{30021719}a^{10}+\frac{3727902931}{120086876}a^{9}-\frac{7240918415}{120086876}a^{8}+\frac{6586052407}{60043438}a^{7}-\frac{13131500987}{120086876}a^{6}+\frac{603351663}{9237452}a^{5}-\frac{2146815}{41639}a^{4}+\frac{2388857337}{60043438}a^{3}-\frac{4164707331}{240173752}a^{2}+\frac{114032326}{30021719}a-\frac{408034481}{240173752}$, $a$, $\frac{92843929}{120086876}a^{15}+\frac{341306313}{240173752}a^{14}-\frac{6040066}{4288817}a^{13}-\frac{765412721}{240173752}a^{12}-\frac{166852409}{120086876}a^{11}+\frac{1186590436}{30021719}a^{10}-\frac{9481653429}{120086876}a^{9}+\frac{20379063065}{120086876}a^{8}-\frac{17191149041}{60043438}a^{7}+\frac{28898905613}{120086876}a^{6}-\frac{1521262297}{9237452}a^{5}+\frac{5181649}{41639}a^{4}-\frac{4876398661}{60043438}a^{3}+\frac{8324327773}{240173752}a^{2}-\frac{426358616}{30021719}a+\frac{792834279}{240173752}$, $\frac{9941205}{30021719}a^{15}+\frac{20970449}{34310536}a^{14}-\frac{5481867}{8577634}a^{13}-\frac{326839167}{240173752}a^{12}-\frac{66804235}{120086876}a^{11}+\frac{71884244}{4288817}a^{10}-\frac{4117100649}{120086876}a^{9}+\frac{8928518979}{120086876}a^{8}-\frac{7404032505}{60043438}a^{7}+\frac{12762867189}{120086876}a^{6}-\frac{677256743}{9237452}a^{5}+\frac{13807447}{291473}a^{4}-\frac{4116857763}{120086876}a^{3}+\frac{3826611963}{240173752}a^{2}-\frac{175799933}{60043438}a-\frac{6965653}{34310536}$, $\frac{248520075}{34310536}a^{15}-\frac{411570539}{34310536}a^{14}+\frac{357683117}{34310536}a^{13}+\frac{1114314619}{34310536}a^{12}+\frac{316347989}{17155268}a^{11}-\frac{6346553783}{17155268}a^{10}+\frac{2872541027}{4288817}a^{9}-\frac{24684852773}{17155268}a^{8}+\frac{40796047365}{17155268}a^{7}-\frac{569576372}{329909}a^{6}+\frac{18633177793}{17155268}a^{5}-\frac{150579491}{166556}a^{4}+\frac{19279281029}{34310536}a^{3}-\frac{6289739471}{34310536}a^{2}+\frac{2326251157}{34310536}a-\frac{614829847}{34310536}$, $\frac{6410749}{4288817}a^{15}+\frac{34190721}{17155268}a^{14}-\frac{28662249}{17155268}a^{13}-\frac{119100811}{17155268}a^{12}-\frac{54272623}{8577634}a^{11}+\frac{315662792}{4288817}a^{10}-\frac{984126121}{8577634}a^{9}+\frac{2301250553}{8577634}a^{8}-\frac{3609154039}{8577634}a^{7}+\frac{2190287397}{8577634}a^{6}-\frac{1718577637}{8577634}a^{5}+\frac{7043521}{41639}a^{4}-\frac{774249469}{8577634}a^{3}+\frac{518397813}{17155268}a^{2}-\frac{310124901}{17155268}a+\frac{45537153}{17155268}$
| 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: | \( 4588.812679602563 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4588.812679602563 \cdot 1}{4\cdot\sqrt{143420743705600000000}}\cr\approx \mathstrut & 0.232687632098475 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 + 4*x^13 + x^12 - 52*x^11 + 110*x^10 - 230*x^9 + 396*x^8 - 350*x^7 + 230*x^6 - 176*x^5 + 121*x^4 - 52*x^3 + 18*x^2 - 6*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 + 2*x^14 + 4*x^13 + x^12 - 52*x^11 + 110*x^10 - 230*x^9 + 396*x^8 - 350*x^7 + 230*x^6 - 176*x^5 + 121*x^4 - 52*x^3 + 18*x^2 - 6*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 + 2*x^14 + 4*x^13 + x^12 - 52*x^11 + 110*x^10 - 230*x^9 + 396*x^8 - 350*x^7 + 230*x^6 - 176*x^5 + 121*x^4 - 52*x^3 + 18*x^2 - 6*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 + 2*x^14 + 4*x^13 + x^12 - 52*x^11 + 110*x^10 - 230*x^9 + 396*x^8 - 350*x^7 + 230*x^6 - 176*x^5 + 121*x^4 - 52*x^3 + 18*x^2 - 6*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
$D_4^2:C_2^2$ (as 16T509):
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 256 |
| The 40 conjugacy class representatives for $D_4^2:C_2^2$ |
| Character table for $D_4^2:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), 4.4.725.1, 4.0.11600.1, \(\Q(i, \sqrt{5})\), deg 8, deg 8, 8.0.134560000.4 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 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 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(5\)
| 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 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(89\)
| $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |