Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 7*x^14 + 16*x^13 - 2*x^12 - 6*x^11 - 12*x^10 + 68*x^9 - 25*x^8 - 116*x^7 + 82*x^6 - 24*x^5 + 36*x^4 + 64*x^3 - 88*x^2 + 16)
gp: K = bnfinit(y^16 - 2*y^15 - 7*y^14 + 16*y^13 - 2*y^12 - 6*y^11 - 12*y^10 + 68*y^9 - 25*y^8 - 116*y^7 + 82*y^6 - 24*y^5 + 36*y^4 + 64*y^3 - 88*y^2 + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 7*x^14 + 16*x^13 - 2*x^12 - 6*x^11 - 12*x^10 + 68*x^9 - 25*x^8 - 116*x^7 + 82*x^6 - 24*x^5 + 36*x^4 + 64*x^3 - 88*x^2 + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 7*x^14 + 16*x^13 - 2*x^12 - 6*x^11 - 12*x^10 + 68*x^9 - 25*x^8 - 116*x^7 + 82*x^6 - 24*x^5 + 36*x^4 + 64*x^3 - 88*x^2 + 16)
\( x^{16} - 2 x^{15} - 7 x^{14} + 16 x^{13} - 2 x^{12} - 6 x^{11} - 12 x^{10} + 68 x^{9} - 25 x^{8} + \cdots + 16 \)
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: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(26873856000000000000\)
\(\medspace = 2^{24}\cdot 3^{8}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.38\) | 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: | $2^{3/2}3^{1/2}5^{3/4}\approx 16.3807251762544$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{20842482088}a^{15}-\frac{818375539}{20842482088}a^{14}+\frac{2109449953}{20842482088}a^{13}+\frac{666954093}{20842482088}a^{12}+\frac{636362475}{5210620522}a^{11}-\frac{398242481}{10421241044}a^{10}-\frac{2919049827}{10421241044}a^{9}-\frac{67544908}{2605310261}a^{8}+\frac{5406922911}{20842482088}a^{7}+\frac{783027317}{20842482088}a^{6}+\frac{432822077}{2605310261}a^{5}-\frac{277849925}{5210620522}a^{4}-\frac{2185674623}{5210620522}a^{3}-\frac{290617846}{2605310261}a^{2}+\frac{895137938}{2605310261}a-\frac{892808031}{2605310261}$
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 $1$ (assuming GRH)
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: | $11$ | 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: |
$\frac{80666330647}{20842482088}a^{15}-\frac{47701120141}{10421241044}a^{14}-\frac{642086638767}{20842482088}a^{13}+\frac{382405534833}{10421241044}a^{12}+\frac{115089500535}{5210620522}a^{11}-\frac{12505991469}{2605310261}a^{10}-\frac{263461440149}{5210620522}a^{9}+\frac{1156612042835}{5210620522}a^{8}+\frac{1756137861285}{20842482088}a^{7}-\frac{985139113534}{2605310261}a^{6}+\frac{38442641947}{5210620522}a^{5}-\frac{920109182241}{10421241044}a^{4}+\frac{359508852541}{5210620522}a^{3}+\frac{1571514958163}{5210620522}a^{2}-\frac{240163525379}{2605310261}a-\frac{193965754538}{2605310261}$, $\frac{3068595193}{2605310261}a^{15}-\frac{29605890863}{20842482088}a^{14}-\frac{24653578597}{2605310261}a^{13}+\frac{239482339687}{20842482088}a^{12}+\frac{76600345275}{10421241044}a^{11}-\frac{12337715185}{5210620522}a^{10}-\frac{162207118317}{10421241044}a^{9}+\frac{175191180980}{2605310261}a^{8}+\frac{66231964750}{2605310261}a^{7}-\frac{2562191754337}{20842482088}a^{6}+\frac{15523683813}{10421241044}a^{5}-\frac{58664398213}{2605310261}a^{4}+\frac{111683997277}{5210620522}a^{3}+\frac{506424162487}{5210620522}a^{2}-\frac{77465957654}{2605310261}a-\frac{64605839043}{2605310261}$, $\frac{133882256545}{20842482088}a^{15}-\frac{155376229353}{20842482088}a^{14}-\frac{1067444654713}{20842482088}a^{13}+\frac{1245886979063}{20842482088}a^{12}+\frac{388063807303}{10421241044}a^{11}-\frac{74885056995}{10421241044}a^{10}-\frac{863180889539}{10421241044}a^{9}+\frac{956139767351}{2605310261}a^{8}+\frac{3070836677855}{20842482088}a^{7}-\frac{12945332280701}{20842482088}a^{6}+\frac{63199344179}{10421241044}a^{5}-\frac{790017325029}{5210620522}a^{4}+\frac{269080201601}{2605310261}a^{3}+\frac{2602820595061}{5210620522}a^{2}-\frac{380684085876}{2605310261}a-\frac{316983173510}{2605310261}$, $\frac{2844274819}{2605310261}a^{15}-\frac{3538059571}{2605310261}a^{14}-\frac{90257600493}{10421241044}a^{13}+\frac{113269872049}{10421241044}a^{12}+\frac{63045456087}{10421241044}a^{11}-\frac{17359064081}{10421241044}a^{10}-\frac{38685329000}{2605310261}a^{9}+\frac{165149326204}{2605310261}a^{8}+\frac{104695027885}{5210620522}a^{7}-\frac{287100447683}{2605310261}a^{6}+\frac{35075205433}{10421241044}a^{5}-\frac{238095800363}{10421241044}a^{4}+\frac{59791817132}{2605310261}a^{3}+\frac{447732222063}{5210620522}a^{2}-\frac{71996336189}{2605310261}a-\frac{58489821965}{2605310261}$, $\frac{702867}{88199}a^{15}-\frac{6652205}{705592}a^{14}-\frac{22365463}{352796}a^{13}+\frac{53285463}{705592}a^{12}+\frac{3988528}{88199}a^{11}-\frac{3347031}{352796}a^{10}-\frac{36758897}{352796}a^{9}+\frac{40347905}{88199}a^{8}+\frac{30457471}{176398}a^{7}-\frac{547838547}{705592}a^{6}+\frac{2517547}{176398}a^{5}-\frac{64754059}{352796}a^{4}+\frac{12433738}{88199}a^{3}+\frac{54611710}{88199}a^{2}-\frac{16576766}{88199}a-\frac{13431573}{88199}$, $\frac{2919612583}{5210620522}a^{15}-\frac{2801977719}{5210620522}a^{14}-\frac{24240888897}{5210620522}a^{13}+\frac{11678515142}{2605310261}a^{12}+\frac{11822427934}{2605310261}a^{11}-\frac{2941066323}{2605310261}a^{10}-\frac{16463746175}{2605310261}a^{9}+\frac{156481923495}{5210620522}a^{8}+\frac{107642372253}{5210620522}a^{7}-\frac{294696380861}{5210620522}a^{6}-\frac{10855413836}{2605310261}a^{5}-\frac{58062249367}{5210620522}a^{4}+\frac{3326709010}{2605310261}a^{3}+\frac{257912396939}{5210620522}a^{2}-\frac{27182002978}{2605310261}a-\frac{31004411980}{2605310261}$, $\frac{37056704419}{20842482088}a^{15}-\frac{44582401839}{20842482088}a^{14}-\frac{294688255785}{20842482088}a^{13}+\frac{358440631359}{20842482088}a^{12}+\frac{26011072793}{2605310261}a^{11}-\frac{14356051951}{5210620522}a^{10}-\frac{239370771607}{10421241044}a^{9}+\frac{531896256231}{5210620522}a^{8}+\frac{775651626169}{20842482088}a^{7}-\frac{3682492034091}{20842482088}a^{6}+\frac{20502209078}{2605310261}a^{5}-\frac{418419264747}{10421241044}a^{4}+\frac{168323866843}{5210620522}a^{3}+\frac{735141275831}{5210620522}a^{2}-\frac{121658907444}{2605310261}a-\frac{88532253642}{2605310261}$, $\frac{122203806213}{20842482088}a^{15}-\frac{144168318477}{20842482088}a^{14}-\frac{970481099125}{20842482088}a^{13}+\frac{1152458857927}{20842482088}a^{12}+\frac{340774095567}{10421241044}a^{11}-\frac{63120791703}{10421241044}a^{10}-\frac{797325904839}{10421241044}a^{9}+\frac{1755797611207}{5210620522}a^{8}+\frac{2640267188843}{20842482088}a^{7}-\frac{11766546757257}{20842482088}a^{6}+\frac{106620999523}{10421241044}a^{5}-\frac{365977537831}{2605310261}a^{4}+\frac{265753492591}{2605310261}a^{3}+\frac{1172454099061}{2605310261}a^{2}-\frac{353502082898}{2605310261}a-\frac{283373451269}{2605310261}$, $\frac{7037559133}{20842482088}a^{15}-\frac{4908993983}{10421241044}a^{14}-\frac{53849963929}{20842482088}a^{13}+\frac{9492973551}{2605310261}a^{12}+\frac{3202250031}{2605310261}a^{11}+\frac{667069347}{10421241044}a^{10}-\frac{13428304654}{2605310261}a^{9}+\frac{107350421113}{5210620522}a^{8}+\frac{45210048499}{20842482088}a^{7}-\frac{83107663929}{2605310261}a^{6}+\frac{2897630356}{2605310261}a^{5}-\frac{25918879795}{2605310261}a^{4}+\frac{28621793770}{2605310261}a^{3}+\frac{117521136847}{5210620522}a^{2}-\frac{16517740150}{2605310261}a-\frac{17882340042}{2605310261}$, $\frac{18483949911}{2605310261}a^{15}-\frac{171195037865}{20842482088}a^{14}-\frac{147565621991}{2605310261}a^{13}+\frac{1373594809329}{20842482088}a^{12}+\frac{434771006749}{10421241044}a^{11}-\frac{21158150374}{2605310261}a^{10}-\frac{956792344103}{10421241044}a^{9}+\frac{2109603273181}{5210620522}a^{8}+\frac{428746442785}{2605310261}a^{7}-\frac{14347567134767}{20842482088}a^{6}+\frac{27585403479}{10421241044}a^{5}-\frac{428157778504}{2605310261}a^{4}+\frac{597963580015}{5210620522}a^{3}+\frac{1443648978344}{2605310261}a^{2}-\frac{416967824498}{2605310261}a-\frac{355562181367}{2605310261}$, $\frac{155277740235}{20842482088}a^{15}-\frac{91715626379}{10421241044}a^{14}-\frac{1237332490701}{20842482088}a^{13}+\frac{736221594599}{10421241044}a^{12}+\frac{447307637227}{10421241044}a^{11}-\frac{25544422149}{2605310261}a^{10}-\frac{504723669867}{5210620522}a^{9}+\frac{2222919546453}{5210620522}a^{8}+\frac{3414737285177}{20842482088}a^{7}-\frac{3813245825471}{5210620522}a^{6}+\frac{155916082527}{10421241044}a^{5}-\frac{1766836651091}{10421241044}a^{4}+\frac{676815348901}{5210620522}a^{3}+\frac{1525086688385}{2605310261}a^{2}-\frac{469802985219}{2605310261}a-\frac{374621164014}{2605310261}$
| 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: | \( 4896.43179861 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 4896.43179861 \cdot 1}{2\cdot\sqrt{26873856000000000000}}\cr\approx \mathstrut & 0.188427445259 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 7*x^14 + 16*x^13 - 2*x^12 - 6*x^11 - 12*x^10 + 68*x^9 - 25*x^8 - 116*x^7 + 82*x^6 - 24*x^5 + 36*x^4 + 64*x^3 - 88*x^2 + 16)
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 - 7*x^14 + 16*x^13 - 2*x^12 - 6*x^11 - 12*x^10 + 68*x^9 - 25*x^8 - 116*x^7 + 82*x^6 - 24*x^5 + 36*x^4 + 64*x^3 - 88*x^2 + 16, 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 - 7*x^14 + 16*x^13 - 2*x^12 - 6*x^11 - 12*x^10 + 68*x^9 - 25*x^8 - 116*x^7 + 82*x^6 - 24*x^5 + 36*x^4 + 64*x^3 - 88*x^2 + 16);
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 - 7*x^14 + 16*x^13 - 2*x^12 - 6*x^11 - 12*x^10 + 68*x^9 - 25*x^8 - 116*x^7 + 82*x^6 - 24*x^5 + 36*x^4 + 64*x^3 - 88*x^2 + 16);
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
$C_4\times D_4$ (as 16T19):
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 20 conjugacy class representatives for $C_4 \times D_4$ |
| Character table for $C_4 \times D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{15})^+\), 4.4.72000.1, 4.2.24000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 4.2.24000.2, 8.4.576000000.1, 8.4.23040000.1, 8.8.5184000000.1 |
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
| Galois closure: | deg 32 |
| Degree 16 siblings: | 16.0.6561000000000000.1, 16.0.331776000000000000.1, 16.0.26873856000000000000.11 |
| Minimal sibling: | 16.0.6561000000000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(3\)
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |