Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 28*x^13 + 18*x^12 - 48*x^11 - 64*x^10 - 64*x^9 - 36*x^8 + 52*x^7 + 212*x^6 + 348*x^5 + 394*x^4 + 264*x^3 + 96*x^2 + 16*x + 1)
gp: K = bnfinit(y^16 - 4*y^15 - 4*y^14 + 28*y^13 + 18*y^12 - 48*y^11 - 64*y^10 - 64*y^9 - 36*y^8 + 52*y^7 + 212*y^6 + 348*y^5 + 394*y^4 + 264*y^3 + 96*y^2 + 16*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 4*x^14 + 28*x^13 + 18*x^12 - 48*x^11 - 64*x^10 - 64*x^9 - 36*x^8 + 52*x^7 + 212*x^6 + 348*x^5 + 394*x^4 + 264*x^3 + 96*x^2 + 16*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - 4*x^14 + 28*x^13 + 18*x^12 - 48*x^11 - 64*x^10 - 64*x^9 - 36*x^8 + 52*x^7 + 212*x^6 + 348*x^5 + 394*x^4 + 264*x^3 + 96*x^2 + 16*x + 1)
\( x^{16} - 4 x^{15} - 4 x^{14} + 28 x^{13} + 18 x^{12} - 48 x^{11} - 64 x^{10} - 64 x^{9} - 36 x^{8} + \cdots + 1 \)
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: |
\(1073741824000000000000\)
\(\medspace = 2^{42}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.63\) | 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: | $2^{21/8}5^{3/4}\approx 20.626770754424918$ | ||
| Ramified primes: |
\(2\), \(5\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{979}a^{14}-\frac{335}{979}a^{13}+\frac{476}{979}a^{12}+\frac{125}{979}a^{11}-\frac{299}{979}a^{10}+\frac{380}{979}a^{9}-\frac{338}{979}a^{8}+\frac{34}{89}a^{7}-\frac{129}{979}a^{6}+\frac{467}{979}a^{5}+\frac{70}{979}a^{4}-\frac{405}{979}a^{3}+\frac{202}{979}a^{2}+\frac{12}{89}a+\frac{269}{979}$, $\frac{1}{795411144341}a^{15}-\frac{163759324}{795411144341}a^{14}-\frac{26569349077}{795411144341}a^{13}+\frac{109342547599}{795411144341}a^{12}-\frac{216510611003}{795411144341}a^{11}+\frac{9116421885}{25658424011}a^{10}+\frac{331718017756}{795411144341}a^{9}-\frac{79620497577}{795411144341}a^{8}-\frac{36381718326}{795411144341}a^{7}-\frac{33586186580}{72310104031}a^{6}-\frac{193931336002}{795411144341}a^{5}+\frac{247012420016}{795411144341}a^{4}-\frac{129511966279}{795411144341}a^{3}-\frac{27442799508}{795411144341}a^{2}-\frac{135705866203}{795411144341}a+\frac{231755432600}{795411144341}$
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
$C_{4}$, which has order $4$
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: |
\( -\frac{567417922}{564521749} a^{15} + \frac{2296378330}{564521749} a^{14} + \frac{2191276870}{564521749} a^{13} - \frac{16127816755}{564521749} a^{12} - \frac{9489149920}{564521749} a^{11} + \frac{920357388}{18210379} a^{10} + \frac{35002721360}{564521749} a^{9} + \frac{33233943350}{564521749} a^{8} + \frac{17749389390}{564521749} a^{7} - \frac{2876148910}{51320159} a^{6} - \frac{118991529606}{564521749} a^{5} - \frac{189948096905}{564521749} a^{4} - \frac{209219214200}{564521749} a^{3} - \frac{132938490340}{564521749} a^{2} - \frac{40818620340}{564521749} a - \frac{3978716187}{564521749} \)
(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: |
$\frac{728283826264}{795411144341}a^{15}-\frac{3162003091680}{795411144341}a^{14}-\frac{1814222059850}{795411144341}a^{13}+\frac{20927786767256}{795411144341}a^{12}+\frac{5936837700472}{795411144341}a^{11}-\frac{1176738300358}{25658424011}a^{10}-\frac{34117093375862}{795411144341}a^{9}-\frac{35715631089196}{795411144341}a^{8}-\frac{14719097984906}{795411144341}a^{7}+\frac{3809619103342}{72310104031}a^{6}+\frac{139790712538932}{795411144341}a^{5}+\frac{206641831024811}{795411144341}a^{4}+\frac{219695412089864}{795411144341}a^{3}+\frac{121633605769078}{795411144341}a^{2}+\frac{33734112271610}{795411144341}a+\frac{3346209734632}{795411144341}$, $\frac{273665321878}{795411144341}a^{15}-\frac{1107770215330}{795411144341}a^{14}-\frac{1035333028509}{795411144341}a^{13}+\frac{7669620275435}{795411144341}a^{12}+\frac{4618235424623}{795411144341}a^{11}-\frac{425578501714}{25658424011}a^{10}-\frac{17235687183648}{795411144341}a^{9}-\frac{16790057870414}{795411144341}a^{8}-\frac{8916688800725}{795411144341}a^{7}+\frac{1289808454371}{72310104031}a^{6}+\frac{58030304319513}{795411144341}a^{5}+\frac{93390724925066}{795411144341}a^{4}+\frac{104144264889051}{795411144341}a^{3}+\frac{68070540723363}{795411144341}a^{2}+\frac{23827127214092}{795411144341}a+\frac{2521716083851}{795411144341}$, $a$, $\frac{267263854159}{795411144341}a^{15}-\frac{1099021608777}{795411144341}a^{14}-\frac{961157845617}{795411144341}a^{13}+\frac{7654735307687}{795411144341}a^{12}+\frac{4027752233634}{795411144341}a^{11}-\frac{447066328747}{25658424011}a^{10}-\frac{15608811958412}{795411144341}a^{9}-\frac{14002004247619}{795411144341}a^{8}-\frac{8118835688817}{795411144341}a^{7}+\frac{1391538492200}{72310104031}a^{6}+\frac{55598590854426}{795411144341}a^{5}+\frac{84742370988149}{795411144341}a^{4}+\frac{94563965596052}{795411144341}a^{3}+\frac{55318796209750}{795411144341}a^{2}+\frac{16286526434764}{795411144341}a+\frac{1468887985461}{795411144341}$, $\frac{220956842259}{795411144341}a^{15}-\frac{907960251459}{795411144341}a^{14}-\frac{823872817354}{795411144341}a^{13}+\frac{6454307316227}{795411144341}a^{12}+\frac{3349994563580}{795411144341}a^{11}-\frac{393176402189}{25658424011}a^{10}-\frac{12834104632463}{795411144341}a^{9}-\frac{10444772070871}{795411144341}a^{8}-\frac{5852990766626}{795411144341}a^{7}+\frac{1243128997313}{72310104031}a^{6}+\frac{46038296348569}{795411144341}a^{5}+\frac{68656076862330}{795411144341}a^{4}+\frac{73120661873405}{795411144341}a^{3}+\frac{40708902475024}{795411144341}a^{2}+\frac{7694129867980}{795411144341}a-\frac{787141884176}{795411144341}$, $\frac{526963406715}{795411144341}a^{15}-\frac{2464670348464}{795411144341}a^{14}-\frac{433726583859}{795411144341}a^{13}+\frac{15049909060463}{795411144341}a^{12}-\frac{847485200068}{795411144341}a^{11}-\frac{790728319170}{25658424011}a^{10}-\frac{16393352370986}{795411144341}a^{9}-\frac{23353257881680}{795411144341}a^{8}-\frac{4467750564932}{795411144341}a^{7}+\frac{2719664169171}{72310104031}a^{6}+\frac{90622934288727}{795411144341}a^{5}+\frac{122688994879200}{795411144341}a^{4}+\frac{127112075164927}{795411144341}a^{3}+\frac{58712631002042}{795411144341}a^{2}+\frac{16713862095778}{795411144341}a+\frac{1934788477239}{795411144341}$, $\frac{24531375603}{72310104031}a^{15}-\frac{91000709115}{72310104031}a^{14}-\frac{134763013389}{72310104031}a^{13}+\frac{696958760887}{72310104031}a^{12}+\frac{647191756660}{72310104031}a^{11}-\frac{41726997379}{2332584001}a^{10}-\frac{1870192517241}{72310104031}a^{9}-\frac{1606460809095}{72310104031}a^{8}-\frac{1183951669885}{72310104031}a^{7}+\frac{1338528434201}{72310104031}a^{6}+\frac{5656562550508}{72310104031}a^{5}+\frac{9510179832022}{72310104031}a^{4}+\frac{985031775454}{6573645821}a^{3}+\frac{7522897167956}{72310104031}a^{2}+\frac{2614448560626}{72310104031}a+\frac{343695004310}{72310104031}$
| 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: | \( 10024.846831 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10024.846831 \cdot 4}{4\cdot\sqrt{1073741824000000000000}}\cr\approx \mathstrut & 0.74313336006 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 28*x^13 + 18*x^12 - 48*x^11 - 64*x^10 - 64*x^9 - 36*x^8 + 52*x^7 + 212*x^6 + 348*x^5 + 394*x^4 + 264*x^3 + 96*x^2 + 16*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 - 4*x^15 - 4*x^14 + 28*x^13 + 18*x^12 - 48*x^11 - 64*x^10 - 64*x^9 - 36*x^8 + 52*x^7 + 212*x^6 + 348*x^5 + 394*x^4 + 264*x^3 + 96*x^2 + 16*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 - 4*x^15 - 4*x^14 + 28*x^13 + 18*x^12 - 48*x^11 - 64*x^10 - 64*x^9 - 36*x^8 + 52*x^7 + 212*x^6 + 348*x^5 + 394*x^4 + 264*x^3 + 96*x^2 + 16*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 - 4*x^15 - 4*x^14 + 28*x^13 + 18*x^12 - 48*x^11 - 64*x^10 - 64*x^9 - 36*x^8 + 52*x^7 + 212*x^6 + 348*x^5 + 394*x^4 + 264*x^3 + 96*x^2 + 16*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
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 32 |
| The 11 conjugacy class representatives for $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{10}) \), 4.0.320.1, 4.0.1280.1, \(\Q(i, \sqrt{10})\), 8.0.163840000.2 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $16$ | $1$ | $42$ | |||
|
\(5\)
| 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.4 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |