Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 35*x^14 + 486*x^12 - 3428*x^10 + 13207*x^8 - 28366*x^6 + 33276*x^4 - 19631*x^2 + 4489)
gp: K = bnfinit(y^16 - 35*y^14 + 486*y^12 - 3428*y^10 + 13207*y^8 - 28366*y^6 + 33276*y^4 - 19631*y^2 + 4489, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 35*x^14 + 486*x^12 - 3428*x^10 + 13207*x^8 - 28366*x^6 + 33276*x^4 - 19631*x^2 + 4489);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 35*x^14 + 486*x^12 - 3428*x^10 + 13207*x^8 - 28366*x^6 + 33276*x^4 - 19631*x^2 + 4489)
\( x^{16} - 35x^{14} + 486x^{12} - 3428x^{10} + 13207x^{8} - 28366x^{6} + 33276x^{4} - 19631x^{2} + 4489 \)
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: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(49535258684630680881135616\)
\(\medspace = 2^{16}\cdot 17^{14}\cdot 67^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(40.36\) | 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: | $2^{15/8}17^{7/8}67^{1/2}\approx 358.1883442427048$ | ||
| Ramified primes: |
\(2\), \(17\), \(67\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{210951778}a^{14}+\frac{9792417}{210951778}a^{12}-\frac{48909045}{210951778}a^{10}-\frac{21753398}{105475889}a^{8}+\frac{52417052}{105475889}a^{6}-\frac{78940899}{210951778}a^{4}-\frac{84471747}{210951778}a^{2}+\frac{905713}{3148534}$, $\frac{1}{210951778}a^{15}+\frac{9792417}{210951778}a^{13}-\frac{48909045}{210951778}a^{11}-\frac{21753398}{105475889}a^{9}+\frac{52417052}{105475889}a^{7}-\frac{78940899}{210951778}a^{5}-\frac{84471747}{210951778}a^{3}+\frac{905713}{3148534}a$
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: | $15$ | 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: |
$\frac{3702365}{210951778}a^{14}-\frac{125359765}{210951778}a^{12}+\frac{1658891997}{210951778}a^{10}-\frac{5435761656}{105475889}a^{8}+\frac{18635747342}{105475889}a^{6}-\frac{66478615211}{210951778}a^{4}+\frac{57432065193}{210951778}a^{2}-\frac{280125767}{3148534}$, $\frac{1815433}{210951778}a^{14}-\frac{62215833}{210951778}a^{12}+\frac{829538273}{210951778}a^{10}-\frac{2718508624}{105475889}a^{8}+\frac{9213641282}{105475889}a^{6}-\frac{31988814221}{210951778}a^{4}+\frac{26232628567}{210951778}a^{2}-\frac{117286383}{3148534}$, $\frac{100931}{105475889}a^{14}-\frac{6755295}{210951778}a^{12}+\frac{43736083}{105475889}a^{10}-\frac{551811901}{210951778}a^{8}+\frac{1765478135}{210951778}a^{6}-\frac{2718292821}{210951778}a^{4}+\frac{747494414}{105475889}a^{2}-\frac{1608973}{3148534}$, $\frac{674979}{105475889}a^{14}-\frac{46975773}{210951778}a^{12}+\frac{320213055}{105475889}a^{10}-\frac{4321602369}{210951778}a^{8}+\frac{15079209343}{210951778}a^{6}-\frac{26230032787}{210951778}a^{4}+\frac{10149780205}{105475889}a^{2}-\frac{79134317}{3148534}$, $\frac{517155}{210951778}a^{14}-\frac{8222461}{105475889}a^{12}+\frac{193870559}{210951778}a^{10}-\frac{1011630457}{210951778}a^{8}+\frac{2034342899}{210951778}a^{6}+\frac{39321386}{105475889}a^{4}-\frac{4045456437}{210951778}a^{2}+\frac{17702806}{1574267}$, $\frac{517155}{210951778}a^{14}-\frac{8222461}{105475889}a^{12}+\frac{193870559}{210951778}a^{10}-\frac{1011630457}{210951778}a^{8}+\frac{2034342899}{210951778}a^{6}+\frac{39321386}{105475889}a^{4}-\frac{4045456437}{210951778}a^{2}+\frac{19277073}{1574267}$, $\frac{497573}{210951778}a^{14}-\frac{8194207}{105475889}a^{12}+\frac{209717629}{210951778}a^{10}-\frac{1306734305}{210951778}a^{8}+\frac{4054187869}{210951778}a^{6}-\frac{2825500100}{105475889}a^{4}+\frac{2335955359}{210951778}a^{2}+\frac{1970164}{1574267}$, $\frac{2087049}{105475889}a^{15}-\frac{2352525}{210951778}a^{14}-\frac{70572874}{105475889}a^{13}+\frac{83113565}{210951778}a^{12}+\frac{933115214}{105475889}a^{11}-\frac{1153848361}{210951778}a^{10}-\frac{6115074914}{105475889}a^{9}+\frac{3996033378}{105475889}a^{8}+\frac{20991273857}{105475889}a^{7}-\frac{14591719637}{105475889}a^{6}-\frac{37524163757}{105475889}a^{5}+\frac{55364782401}{210951778}a^{4}+\frac{32325903480}{105475889}a^{3}-\frac{49736678883}{210951778}a^{2}-\frac{153333605}{1574267}a+\frac{240524147}{3148534}$, $\frac{2142203}{105475889}a^{15}+\frac{12274785}{210951778}a^{14}-\frac{137790561}{210951778}a^{13}-\frac{202161583}{105475889}a^{12}+\frac{844032570}{105475889}a^{11}+\frac{5137826103}{210951778}a^{10}-\frac{9798908049}{210951778}a^{9}-\frac{31653971325}{210951778}a^{8}+\frac{27618128549}{210951778}a^{7}+\frac{98735738511}{210951778}a^{6}-\frac{36905774415}{210951778}a^{5}-\frac{77281636174}{105475889}a^{4}+\frac{11567912328}{105475889}a^{3}+\frac{114724462995}{210951778}a^{2}-\frac{89735381}{3148534}a-\frac{235491093}{1574267}$, $\frac{2862959}{105475889}a^{15}-\frac{167615}{105475889}a^{14}-\frac{97438408}{105475889}a^{13}+\frac{14142437}{210951778}a^{12}+\frac{1297064352}{105475889}a^{11}-\frac{118418961}{105475889}a^{10}-\frac{8551782049}{105475889}a^{9}+\frac{2003237607}{210951778}a^{8}+\frac{29413617596}{105475889}a^{7}-\frac{8990678575}{210951778}a^{6}-\frac{51998326561}{105475889}a^{5}+\frac{20657675359}{210951778}a^{4}+\frac{43223178099}{105475889}a^{3}-\frac{11004530744}{105475889}a^{2}-\frac{193705250}{1574267}a+\frac{118480257}{3148534}$, $\frac{4918955}{210951778}a^{15}+\frac{2708264}{105475889}a^{14}-\frac{82474198}{105475889}a^{13}-\frac{90668405}{105475889}a^{12}+\frac{2149952351}{210951778}a^{11}+\frac{1179834990}{105475889}a^{10}-\frac{13741699975}{210951778}a^{9}-\frac{7524217140}{105475889}a^{8}+\frac{45125503587}{210951778}a^{7}+\frac{24589845728}{105475889}a^{6}-\frac{37384632730}{105475889}a^{5}-\frac{40210132830}{105475889}a^{4}+\frac{57217552943}{210951778}a^{3}+\frac{29776754151}{105475889}a^{2}-\frac{121194400}{1574267}a-\frac{117649969}{1574267}$, $\frac{548208}{105475889}a^{15}-\frac{924369}{210951778}a^{14}-\frac{19507808}{105475889}a^{13}+\frac{14304109}{105475889}a^{12}+\frac{274097774}{105475889}a^{11}-\frac{336236243}{210951778}a^{10}-\frac{1936787445}{105475889}a^{9}+\frac{1857740361}{210951778}a^{8}+\frac{7313730765}{105475889}a^{7}-\frac{4802131965}{210951778}a^{6}-\frac{14674582086}{105475889}a^{5}+\frac{2362901489}{105475889}a^{4}+\frac{14391548088}{105475889}a^{3}+\frac{529386611}{210951778}a^{2}-\frac{78690045}{1574267}a-\frac{15998950}{1574267}$, $\frac{2109760}{105475889}a^{15}+\frac{1460621}{105475889}a^{14}-\frac{141804687}{210951778}a^{13}-\frac{48016888}{105475889}a^{12}+\frac{926381278}{105475889}a^{11}+\frac{608547421}{105475889}a^{10}-\frac{11883153769}{210951778}a^{9}-\frac{3728883489}{105475889}a^{8}+\frac{39305837583}{210951778}a^{7}+\frac{11456448959}{105475889}a^{6}-\frac{66399972439}{210951778}a^{5}-\frac{17166257723}{105475889}a^{4}+\frac{26693304378}{105475889}a^{3}+\frac{11554261876}{105475889}a^{2}-\frac{238423087}{3148534}a-\frac{39717012}{1574267}$, $\frac{9791}{105475889}a^{14}-\frac{28254}{105475889}a^{12}-\frac{7923535}{105475889}a^{10}+\frac{147551924}{105475889}a^{8}-\frac{1009922485}{105475889}a^{6}+\frac{2864821486}{105475889}a^{4}-\frac{3190705898}{105475889}a^{2}+a+\frac{15732642}{1574267}$, $\frac{592181}{105475889}a^{15}-\frac{2949166}{105475889}a^{14}-\frac{45743819}{210951778}a^{13}+\frac{197647401}{210951778}a^{12}+\frac{355514588}{105475889}a^{11}-\frac{1288208923}{105475889}a^{10}-\frac{5671996655}{210951778}a^{9}+\frac{16522636295}{210951778}a^{8}+\frac{24529372101}{210951778}a^{7}-\frac{55021591759}{210951778}a^{6}-\frac{55936901785}{210951778}a^{5}+\frac{95360549739}{210951778}a^{4}+\frac{30386659409}{105475889}a^{3}-\frac{40902191472}{105475889}a^{2}-\frac{361386701}{3148534}a+\frac{408115587}{3148534}$
| 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: | \( 40729869.2956 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 40729869.2956 \cdot 1}{2\cdot\sqrt{49535258684630680881135616}}\cr\approx \mathstrut & 0.189629427421 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 35*x^14 + 486*x^12 - 3428*x^10 + 13207*x^8 - 28366*x^6 + 33276*x^4 - 19631*x^2 + 4489)
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 - 35*x^14 + 486*x^12 - 3428*x^10 + 13207*x^8 - 28366*x^6 + 33276*x^4 - 19631*x^2 + 4489, 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 - 35*x^14 + 486*x^12 - 3428*x^10 + 13207*x^8 - 28366*x^6 + 33276*x^4 - 19631*x^2 + 4489);
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 - 35*x^14 + 486*x^12 - 3428*x^10 + 13207*x^8 - 28366*x^6 + 33276*x^4 - 19631*x^2 + 4489);
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
$C_2^7:C_8$ (as 16T1194):
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 1024 |
| The 40 conjugacy class representatives for $C_2^7:C_8$ |
| Character table for $C_2^7:C_8$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
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: | 16.16.49535258684630680881135616.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.10 | $x^{8} - 6 x^{7} + 36 x^{6} - 60 x^{5} + 88 x^{4} + 136 x^{3} + 336 x^{2} + 400 x + 144$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ |
| 2.8.8.9 | $x^{8} + 8 x^{7} + 56 x^{6} + 216 x^{5} + 680 x^{4} + 1296 x^{3} + 2016 x^{2} + 1728 x + 1296$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
|
\(17\)
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
|
\(67\)
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |