Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 24*x^13 + 35*x^12 - 40*x^11 + 34*x^10 + 12*x^9 - 58*x^8 + 20*x^7 + 34*x^6 + 8*x^5 - 41*x^4 + 24*x^3 - 6*x^2 + 4*x - 1)
gp: K = bnfinit(y^16 - 4*y^15 + 10*y^14 - 24*y^13 + 35*y^12 - 40*y^11 + 34*y^10 + 12*y^9 - 58*y^8 + 20*y^7 + 34*y^6 + 8*y^5 - 41*y^4 + 24*y^3 - 6*y^2 + 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 10*x^14 - 24*x^13 + 35*x^12 - 40*x^11 + 34*x^10 + 12*x^9 - 58*x^8 + 20*x^7 + 34*x^6 + 8*x^5 - 41*x^4 + 24*x^3 - 6*x^2 + 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 10*x^14 - 24*x^13 + 35*x^12 - 40*x^11 + 34*x^10 + 12*x^9 - 58*x^8 + 20*x^7 + 34*x^6 + 8*x^5 - 41*x^4 + 24*x^3 - 6*x^2 + 4*x - 1)
\( x^{16} - 4 x^{15} + 10 x^{14} - 24 x^{13} + 35 x^{12} - 40 x^{11} + 34 x^{10} + 12 x^{9} - 58 x^{8} + \cdots - 1 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(134166736498994446336\)
\(\medspace = 2^{52}\cdot 31^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.11\) | 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: |
\(2\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{31}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{45201515108}a^{15}-\frac{339909727}{11300378777}a^{14}+\frac{8448754417}{45201515108}a^{13}+\frac{904573711}{22600757554}a^{12}+\frac{1799812250}{11300378777}a^{11}+\frac{6875262403}{22600757554}a^{10}+\frac{6741728765}{22600757554}a^{9}-\frac{10961867771}{22600757554}a^{8}-\frac{747300824}{11300378777}a^{7}+\frac{9793330835}{22600757554}a^{6}+\frac{11202871017}{22600757554}a^{5}-\frac{10550763559}{22600757554}a^{4}-\frac{3985360939}{45201515108}a^{3}-\frac{6326521869}{22600757554}a^{2}+\frac{16221870797}{45201515108}a+\frac{4207130805}{11300378777}$
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: | $9$ | 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: |
$a$, $\frac{2973914350}{11300378777}a^{15}+\frac{22045088791}{22600757554}a^{14}-\frac{25220440699}{11300378777}a^{13}+\frac{119788717549}{22600757554}a^{12}-\frac{77565404914}{11300378777}a^{11}+\frac{74720945078}{11300378777}a^{10}-\frac{56813624911}{11300378777}a^{9}-\frac{73387553031}{11300378777}a^{8}+\frac{161097407666}{11300378777}a^{7}+\frac{30398694006}{11300378777}a^{6}-\frac{145644753226}{11300378777}a^{5}-\frac{97996293975}{11300378777}a^{4}+\frac{145961824466}{11300378777}a^{3}+\frac{5687035157}{22600757554}a^{2}+\frac{1191531769}{11300378777}a-\frac{35864449481}{22600757554}$, $\frac{7431029326}{11300378777}a^{15}+\frac{112903881381}{45201515108}a^{14}-\frac{135650211151}{22600757554}a^{13}+\frac{649901382393}{45201515108}a^{12}-\frac{222580775887}{11300378777}a^{11}+\frac{481115134549}{22600757554}a^{10}-\frac{193663035155}{11300378777}a^{9}-\frac{140802951890}{11300378777}a^{8}+\frac{410078808345}{11300378777}a^{7}-\frac{92429021755}{22600757554}a^{6}-\frac{278110137973}{11300378777}a^{5}-\frac{144318786337}{11300378777}a^{4}+\frac{287271169672}{11300378777}a^{3}-\frac{315716012385}{45201515108}a^{2}+\frac{27425278415}{22600757554}a-\frac{112122420393}{45201515108}$, $\frac{16177053593}{22600757554}a^{15}-\frac{58338443049}{22600757554}a^{14}+\frac{139032875785}{22600757554}a^{13}-\frac{167362992540}{11300378777}a^{12}+\frac{436823509535}{22600757554}a^{11}-\frac{480715274403}{22600757554}a^{10}+\frac{368974901925}{22600757554}a^{9}+\frac{337751868815}{22600757554}a^{8}-\frac{803998871817}{22600757554}a^{7}+\frac{28593993349}{22600757554}a^{6}+\frac{522654263067}{22600757554}a^{5}+\frac{339488801739}{22600757554}a^{4}-\frac{257928468905}{11300378777}a^{3}+\frac{92751877856}{11300378777}a^{2}-\frac{13091292836}{11300378777}a+\frac{46761709669}{22600757554}$, $\frac{1611954979}{11300378777}a^{15}+\frac{24532881539}{45201515108}a^{14}-\frac{30609241743}{22600757554}a^{13}+\frac{149382497865}{45201515108}a^{12}-\frac{106380901577}{22600757554}a^{11}+\frac{63771542554}{11300378777}a^{10}-\frac{114670019525}{22600757554}a^{9}-\frac{27135723573}{22600757554}a^{8}+\frac{147506015531}{22600757554}a^{7}-\frac{15106151097}{11300378777}a^{6}-\frac{86401816295}{22600757554}a^{5}-\frac{35631734161}{22600757554}a^{4}+\frac{64797172093}{22600757554}a^{3}-\frac{172194378297}{45201515108}a^{2}+\frac{37896039578}{11300378777}a-\frac{70332596795}{45201515108}$, $\frac{4504524825}{11300378777}a^{15}+\frac{65937993503}{45201515108}a^{14}-\frac{76782717283}{22600757554}a^{13}+\frac{365214412209}{45201515108}a^{12}-\frac{235264025433}{22600757554}a^{11}+\frac{118820477994}{11300378777}a^{10}-\frac{170677243099}{22600757554}a^{9}-\frac{225700320365}{22600757554}a^{8}+\frac{485790032471}{22600757554}a^{7}+\frac{17784542883}{11300378777}a^{6}-\frac{406133451923}{22600757554}a^{5}-\frac{198781144783}{22600757554}a^{4}+\frac{381673171355}{22600757554}a^{3}-\frac{110525246049}{45201515108}a^{2}-\frac{16766760720}{11300378777}a-\frac{65025555307}{45201515108}$, $\frac{9593269777}{11300378777}a^{15}+\frac{36858720718}{11300378777}a^{14}-\frac{181360816979}{22600757554}a^{13}+\frac{435623815715}{22600757554}a^{12}-\frac{305488968322}{11300378777}a^{11}+\frac{344985417798}{11300378777}a^{10}-\frac{281470463435}{11300378777}a^{9}-\frac{152377923483}{11300378777}a^{8}+\frac{532573641078}{11300378777}a^{7}-\frac{132689753216}{11300378777}a^{6}-\frac{313451414670}{11300378777}a^{5}-\frac{120089715573}{11300378777}a^{4}+\frac{344748281475}{11300378777}a^{3}-\frac{189512564632}{11300378777}a^{2}+\frac{95712357221}{22600757554}a-\frac{63685124961}{22600757554}$, $\frac{1016333}{1208209}a^{15}-\frac{3759844}{1208209}a^{14}+\frac{8983663}{1208209}a^{13}-\frac{42991567}{2416418}a^{12}+\frac{28579644}{1208209}a^{11}-\frac{30763016}{1208209}a^{10}+\frac{23317522}{1208209}a^{9}+\frac{21845700}{1208209}a^{8}-\frac{54741610}{1208209}a^{7}+\frac{4065128}{1208209}a^{6}+\frac{38406396}{1208209}a^{5}+\frac{17606536}{1208209}a^{4}-\frac{36252001}{1208209}a^{3}+\frac{13855325}{1208209}a^{2}-\frac{2001155}{1208209}a+\frac{6038323}{2416418}$, $\frac{10637259275}{45201515108}a^{15}+\frac{10207615477}{11300378777}a^{14}-\frac{95516582677}{45201515108}a^{13}+\frac{56666385298}{11300378777}a^{12}-\frac{76858748308}{11300378777}a^{11}+\frac{154498971109}{22600757554}a^{10}-\frac{125178445745}{22600757554}a^{9}-\frac{107294344029}{22600757554}a^{8}+\frac{143964459183}{11300378777}a^{7}+\frac{37586265735}{22600757554}a^{6}-\frac{281449051955}{22600757554}a^{5}-\frac{128609451765}{22600757554}a^{4}+\frac{601293998177}{45201515108}a^{3}-\frac{50761584251}{22600757554}a^{2}+\frac{4105929099}{45201515108}a-\frac{32640539523}{22600757554}$
| 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: | \( 11833.084314328344 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 11833.084314328344 \cdot 1}{2\cdot\sqrt{134166736498994446336}}\cr\approx \mathstrut & 0.502857264050413 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 24*x^13 + 35*x^12 - 40*x^11 + 34*x^10 + 12*x^9 - 58*x^8 + 20*x^7 + 34*x^6 + 8*x^5 - 41*x^4 + 24*x^3 - 6*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 - 4*x^15 + 10*x^14 - 24*x^13 + 35*x^12 - 40*x^11 + 34*x^10 + 12*x^9 - 58*x^8 + 20*x^7 + 34*x^6 + 8*x^5 - 41*x^4 + 24*x^3 - 6*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 - 4*x^15 + 10*x^14 - 24*x^13 + 35*x^12 - 40*x^11 + 34*x^10 + 12*x^9 - 58*x^8 + 20*x^7 + 34*x^6 + 8*x^5 - 41*x^4 + 24*x^3 - 6*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 - 4*x^15 + 10*x^14 - 24*x^13 + 35*x^12 - 40*x^11 + 34*x^10 + 12*x^9 - 58*x^8 + 20*x^7 + 34*x^6 + 8*x^5 - 41*x^4 + 24*x^3 - 6*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
$C_4^4.C_2\wr C_4$ (as 16T1771):
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 190 conjugacy class representatives for $C_4^4.C_2\wr C_4$ |
| Character table for $C_4^4.C_2\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.2.130023424.1 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $16$ | R | $16$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $16$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.30.39 | $x^{8} + 8 x^{7} + 16 x^{6} + 16 x^{5} + 20 x^{4} + 2$ | $8$ | $1$ | $30$ | $((C_8 : C_2):C_2):C_2$ | $[2, 3, 7/2, 4, 17/4, 19/4]$ |
| 2.8.22.2 | $x^{8} + 16 x^{7} + 88 x^{6} + 192 x^{5} + 180 x^{4} + 288 x^{3} + 432 x^{2} + 516$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ | |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |