Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 14*x^13 - 6*x^12 - 21*x^11 + 45*x^10 + 43*x^9 + 120*x^8 + 96*x^7 + 142*x^6 + 99*x^5 + 101*x^4 + 58*x^3 + 61*x^2 + 5*x + 25)
gp: K = bnfinit(y^16 + 2*y^14 - 14*y^13 - 6*y^12 - 21*y^11 + 45*y^10 + 43*y^9 + 120*y^8 + 96*y^7 + 142*y^6 + 99*y^5 + 101*y^4 + 58*y^3 + 61*y^2 + 5*y + 25, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 2*x^14 - 14*x^13 - 6*x^12 - 21*x^11 + 45*x^10 + 43*x^9 + 120*x^8 + 96*x^7 + 142*x^6 + 99*x^5 + 101*x^4 + 58*x^3 + 61*x^2 + 5*x + 25);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 2*x^14 - 14*x^13 - 6*x^12 - 21*x^11 + 45*x^10 + 43*x^9 + 120*x^8 + 96*x^7 + 142*x^6 + 99*x^5 + 101*x^4 + 58*x^3 + 61*x^2 + 5*x + 25)
\( x^{16} + 2 x^{14} - 14 x^{13} - 6 x^{12} - 21 x^{11} + 45 x^{10} + 43 x^{9} + 120 x^{8} + 96 x^{7} + \cdots + 25 \)
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: |
\(1257791680575160641\)
\(\medspace = 3^{8}\cdot 61^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.53\) | 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: | $3^{1/2}61^{1/2}\approx 13.527749258468683$ | ||
| Ramified primes: |
\(3\), \(61\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{15}a^{12}+\frac{2}{15}a^{11}+\frac{1}{3}a^{9}-\frac{4}{15}a^{8}-\frac{4}{15}a^{7}+\frac{1}{15}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{4}{15}a+\frac{1}{3}$, $\frac{1}{15}a^{13}+\frac{1}{15}a^{11}+\frac{2}{5}a^{9}+\frac{4}{15}a^{8}+\frac{4}{15}a^{7}+\frac{1}{5}a^{6}+\frac{7}{15}a^{4}+\frac{7}{15}a^{3}-\frac{1}{5}a^{2}+\frac{2}{15}a-\frac{1}{3}$, $\frac{1}{75}a^{14}-\frac{2}{75}a^{13}+\frac{2}{75}a^{12}+\frac{1}{75}a^{10}-\frac{8}{75}a^{9}+\frac{7}{75}a^{8}+\frac{7}{25}a^{7}+\frac{2}{5}a^{6}+\frac{32}{75}a^{5}+\frac{28}{75}a^{4}-\frac{2}{15}a^{3}-\frac{7}{25}a^{2}-\frac{7}{15}a+\frac{1}{3}$, $\frac{1}{9872393925}a^{15}-\frac{19826417}{9872393925}a^{14}+\frac{203152627}{9872393925}a^{13}+\frac{40241498}{1974478785}a^{12}-\frac{468608088}{3290797975}a^{11}+\frac{138242302}{9872393925}a^{10}-\frac{3917269453}{9872393925}a^{9}-\frac{2680251434}{9872393925}a^{8}+\frac{153332831}{658159595}a^{7}+\frac{2051900287}{9872393925}a^{6}-\frac{3533256202}{9872393925}a^{5}-\frac{49302958}{1974478785}a^{4}-\frac{3003669316}{9872393925}a^{3}+\frac{294097111}{658159595}a^{2}-\frac{97023181}{1974478785}a+\frac{20342232}{131631919}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
\( -\frac{2760391}{1974478785} a^{15} - \frac{71545502}{1974478785} a^{14} - \frac{2499763}{394895757} a^{13} - \frac{71438177}{1974478785} a^{12} + \frac{68160385}{131631919} a^{11} + \frac{627927379}{1974478785} a^{10} + \frac{976367809}{1974478785} a^{9} - \frac{1231248489}{658159595} a^{8} - \frac{4229184002}{1974478785} a^{7} - \frac{7617247754}{1974478785} a^{6} - \frac{1740737122}{658159595} a^{5} - \frac{7292411429}{1974478785} a^{4} - \frac{5255462869}{1974478785} a^{3} - \frac{5761925512}{1974478785} a^{2} - \frac{647306611}{658159595} a - \frac{111879376}{131631919} \)
(order $6$)
| 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: |
$\frac{209790577}{9872393925}a^{15}+\frac{8551433}{394895757}a^{14}+\frac{89113947}{3290797975}a^{13}-\frac{818856734}{3290797975}a^{12}-\frac{4837951873}{9872393925}a^{11}-\frac{3726052612}{9872393925}a^{10}+\frac{5205755177}{9872393925}a^{9}+\frac{1660434946}{658159595}a^{8}+\frac{10924036238}{3290797975}a^{7}+\frac{40985338559}{9872393925}a^{6}+\frac{21617152834}{9872393925}a^{5}+\frac{12499341157}{9872393925}a^{4}+\frac{2294709938}{9872393925}a^{3}+\frac{6968601926}{9872393925}a^{2}+\frac{686448157}{1974478785}a+\frac{201515350}{394895757}$, $\frac{60479603}{9872393925}a^{15}-\frac{102697496}{9872393925}a^{14}-\frac{222371609}{9872393925}a^{13}-\frac{16423231}{131631919}a^{12}+\frac{337089856}{3290797975}a^{11}+\frac{3902665286}{9872393925}a^{10}+\frac{10214244496}{9872393925}a^{9}-\frac{1886556242}{9872393925}a^{8}-\frac{203392527}{131631919}a^{7}-\frac{36305535134}{9872393925}a^{6}-\frac{25633136746}{9872393925}a^{5}-\frac{4428506698}{1974478785}a^{4}-\frac{2986819006}{3290797975}a^{3}-\frac{266181185}{394895757}a^{2}-\frac{1427564284}{1974478785}a-\frac{112420221}{131631919}$, $\frac{69707023}{3290797975}a^{15}-\frac{657188651}{9872393925}a^{14}+\frac{20360323}{3290797975}a^{13}-\frac{3797310416}{9872393925}a^{12}+\frac{7257198869}{9872393925}a^{11}+\frac{385034974}{658159595}a^{10}+\frac{19222183142}{9872393925}a^{9}-\frac{16670429012}{9872393925}a^{8}-\frac{32381242823}{9872393925}a^{7}-\frac{58694657482}{9872393925}a^{6}-\frac{46406626259}{9872393925}a^{5}-\frac{13701643523}{3290797975}a^{4}-\frac{9878277463}{3290797975}a^{3}-\frac{10664291749}{3290797975}a^{2}-\frac{447728573}{658159595}a-\frac{559963727}{394895757}$, $\frac{50345183}{3290797975}a^{15}-\frac{172390366}{9872393925}a^{14}+\frac{389283899}{9872393925}a^{13}-\frac{933342172}{3290797975}a^{12}+\frac{1347355904}{9872393925}a^{11}-\frac{659515852}{1974478785}a^{10}+\frac{14173390252}{9872393925}a^{9}+\frac{1330830811}{3290797975}a^{8}+\frac{4952973549}{3290797975}a^{7}-\frac{19792493582}{9872393925}a^{6}-\frac{5152184718}{3290797975}a^{5}-\frac{32019463289}{9872393925}a^{4}-\frac{3933565643}{3290797975}a^{3}-\frac{25493734937}{9872393925}a^{2}+\frac{37733593}{1974478785}a-\frac{384542399}{394895757}$, $\frac{195988622}{9872393925}a^{15}-\frac{28788337}{1974478785}a^{14}+\frac{204847766}{9872393925}a^{13}-\frac{2813761087}{9872393925}a^{12}+\frac{274077002}{9872393925}a^{11}-\frac{586415717}{9872393925}a^{10}+\frac{10087594222}{9872393925}a^{9}+\frac{429186457}{658159595}a^{8}+\frac{11626188704}{9872393925}a^{7}+\frac{2899099789}{9872393925}a^{6}-\frac{4493903996}{9872393925}a^{5}-\frac{7987571996}{3290797975}a^{4}-\frac{7994201469}{3290797975}a^{3}-\frac{7280341878}{3290797975}a^{2}-\frac{1255471676}{1974478785}a-\frac{529018535}{394895757}$, $\frac{222923024}{9872393925}a^{15}-\frac{299461141}{9872393925}a^{14}+\frac{123575044}{9872393925}a^{13}-\frac{3440119531}{9872393925}a^{12}+\frac{2145169994}{9872393925}a^{11}+\frac{527177543}{1974478785}a^{10}+\frac{4780114699}{3290797975}a^{9}+\frac{1798298323}{9872393925}a^{8}-\frac{11235322813}{9872393925}a^{7}-\frac{6869472904}{3290797975}a^{6}-\frac{20295569429}{9872393925}a^{5}-\frac{4642873424}{9872393925}a^{4}-\frac{3357518044}{9872393925}a^{3}-\frac{3183247687}{9872393925}a^{2}+\frac{199578038}{658159595}a+\frac{170570158}{394895757}$, $\frac{3794219}{9872393925}a^{15}+\frac{294042149}{9872393925}a^{14}-\frac{158531641}{9872393925}a^{13}+\frac{430667524}{9872393925}a^{12}-\frac{1335459837}{3290797975}a^{11}-\frac{59993228}{1974478785}a^{10}-\frac{927355281}{3290797975}a^{9}+\frac{3748161396}{3290797975}a^{8}+\frac{11068708352}{9872393925}a^{7}+\frac{7371255201}{3290797975}a^{6}+\frac{6077208047}{3290797975}a^{5}+\frac{20174718461}{9872393925}a^{4}+\frac{13634596546}{9872393925}a^{3}+\frac{17601859198}{9872393925}a^{2}+\frac{271528318}{658159595}a+\frac{206034284}{394895757}$
| 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: | \( 809.594531745 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 809.594531745 \cdot 1}{6\cdot\sqrt{1257791680575160641}}\cr\approx \mathstrut & 0.292247566606 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 14*x^13 - 6*x^12 - 21*x^11 + 45*x^10 + 43*x^9 + 120*x^8 + 96*x^7 + 142*x^6 + 99*x^5 + 101*x^4 + 58*x^3 + 61*x^2 + 5*x + 25)
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^14 - 14*x^13 - 6*x^12 - 21*x^11 + 45*x^10 + 43*x^9 + 120*x^8 + 96*x^7 + 142*x^6 + 99*x^5 + 101*x^4 + 58*x^3 + 61*x^2 + 5*x + 25, 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^14 - 14*x^13 - 6*x^12 - 21*x^11 + 45*x^10 + 43*x^9 + 120*x^8 + 96*x^7 + 142*x^6 + 99*x^5 + 101*x^4 + 58*x^3 + 61*x^2 + 5*x + 25);
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^14 - 14*x^13 - 6*x^12 - 21*x^11 + 45*x^10 + 43*x^9 + 120*x^8 + 96*x^7 + 142*x^6 + 99*x^5 + 101*x^4 + 58*x^3 + 61*x^2 + 5*x + 25);
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 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{-183}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{61})\), 4.2.11163.1 x2, 4.0.549.1 x2, 8.0.1121513121.2, 8.2.373837707.1 x4, 8.0.18385461.1 x4 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(61\)
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |