Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 9*x^12 + 105*x^10 + 111*x^8 - 420*x^6 - 114*x^4 - 65*x^2 + 1)
gp: K = bnfinit(y^16 - 10*y^14 - 9*y^12 + 105*y^10 + 111*y^8 - 420*y^6 - 114*y^4 - 65*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 10*x^14 - 9*x^12 + 105*x^10 + 111*x^8 - 420*x^6 - 114*x^4 - 65*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 10*x^14 - 9*x^12 + 105*x^10 + 111*x^8 - 420*x^6 - 114*x^4 - 65*x^2 + 1)
\( x^{16} - 10x^{14} - 9x^{12} + 105x^{10} + 111x^{8} - 420x^{6} - 114x^{4} - 65x^{2} + 1 \)
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: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(105747725158992197265625\)
\(\medspace = 5^{10}\cdot 101^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.48\) | 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: | $5^{3/4}101^{1/2}\approx 33.60378443921746$ | ||
| Ramified primes: |
\(5\), \(101\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{30}a^{12}-\frac{1}{10}a^{10}+\frac{2}{15}a^{8}+\frac{1}{30}a^{6}-\frac{1}{2}a^{5}-\frac{1}{30}a^{4}-\frac{1}{10}a^{2}-\frac{2}{15}$, $\frac{1}{30}a^{13}-\frac{1}{10}a^{11}+\frac{2}{15}a^{9}+\frac{1}{30}a^{7}-\frac{1}{2}a^{6}-\frac{1}{30}a^{5}-\frac{1}{10}a^{3}-\frac{2}{15}a$, $\frac{1}{11048820}a^{14}-\frac{22697}{2209764}a^{12}-\frac{58126}{552441}a^{10}-\frac{511389}{3682940}a^{8}+\frac{2012111}{5524410}a^{6}+\frac{797491}{2762205}a^{4}-\frac{1}{2}a^{3}+\frac{874501}{5524410}a^{2}-\frac{4677647}{11048820}$, $\frac{1}{22097640}a^{15}-\frac{1}{22097640}a^{14}-\frac{22697}{4419528}a^{13}+\frac{22697}{4419528}a^{12}+\frac{436189}{2209764}a^{11}-\frac{436189}{2209764}a^{10}+\frac{1330081}{7365880}a^{9}-\frac{1330081}{7365880}a^{8}+\frac{2012111}{11048820}a^{7}-\frac{2012111}{11048820}a^{6}-\frac{1167223}{11048820}a^{5}+\frac{1167223}{11048820}a^{4}+\frac{1818353}{5524410}a^{3}+\frac{471926}{2762205}a^{2}-\frac{10202057}{22097640}a-\frac{846763}{22097640}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $5$ |
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: |
$\frac{13229}{11048820}a^{14}-\frac{126721}{11048820}a^{12}-\frac{62261}{5524410}a^{10}+\frac{818803}{11048820}a^{8}+\frac{290817}{1841470}a^{6}+\frac{269471}{5524410}a^{4}+\frac{53374}{2762205}a^{2}-\frac{6831869}{11048820}$, $\frac{1890303}{7365880}a^{15}-\frac{13229}{22097640}a^{14}-\frac{56260249}{22097640}a^{13}+\frac{126721}{22097640}a^{12}-\frac{9274203}{3682940}a^{11}+\frac{62261}{11048820}a^{10}+\frac{592645271}{22097640}a^{9}-\frac{818803}{22097640}a^{8}+\frac{338506207}{11048820}a^{7}-\frac{290817}{3682940}a^{6}-\frac{234370211}{2209764}a^{5}-\frac{269471}{11048820}a^{4}-\frac{14079101}{368294}a^{3}-\frac{26687}{2762205}a^{2}-\frac{360148757}{22097640}a+\frac{39978329}{22097640}$, $\frac{1743017}{11048820}a^{15}-\frac{17306191}{11048820}a^{13}-\frac{8490731}{5524410}a^{11}+\frac{182387077}{11048820}a^{9}+\frac{34613213}{1841470}a^{7}-\frac{362362333}{5524410}a^{5}-\frac{65306957}{2762205}a^{3}-\frac{22257835}{2209764}a-\frac{3}{2}$, $\frac{3071419}{22097640}a^{15}-\frac{13229}{22097640}a^{14}-\frac{10283673}{7365880}a^{13}+\frac{126721}{22097640}a^{12}-\frac{13071079}{11048820}a^{11}+\frac{62261}{11048820}a^{10}+\frac{322401181}{22097640}a^{9}-\frac{818803}{22097640}a^{8}+\frac{162811477}{11048820}a^{7}-\frac{290817}{3682940}a^{6}-\frac{43140695}{736588}a^{5}-\frac{269471}{11048820}a^{4}-\frac{13566407}{1104882}a^{3}-\frac{26687}{2762205}a^{2}-\frac{71726229}{7365880}a-\frac{15265771}{22097640}$, $\frac{3071419}{22097640}a^{15}-\frac{13229}{7365880}a^{14}-\frac{10283673}{7365880}a^{13}+\frac{126721}{7365880}a^{12}-\frac{13071079}{11048820}a^{11}+\frac{62261}{3682940}a^{10}+\frac{322401181}{22097640}a^{9}-\frac{818803}{7365880}a^{8}+\frac{162811477}{11048820}a^{7}-\frac{872451}{3682940}a^{6}-\frac{43140695}{736588}a^{5}-\frac{269471}{3682940}a^{4}-\frac{13566407}{1104882}a^{3}-\frac{26687}{920735}a^{2}-\frac{71726229}{7365880}a+\frac{6831869}{7365880}$, $\frac{56822}{184147}a^{15}+\frac{303973}{11048820}a^{14}-\frac{3408937}{1104882}a^{13}-\frac{3064747}{11048820}a^{12}-\frac{1024817}{368294}a^{11}-\frac{622916}{2762205}a^{10}+\frac{17910073}{552441}a^{9}+\frac{10658667}{3682940}a^{8}+\frac{18935284}{552441}a^{7}+\frac{16671257}{5524410}a^{6}-\frac{143408951}{1104882}a^{5}-\frac{32392604}{2762205}a^{4}-\frac{12975127}{368294}a^{3}-\frac{16546019}{5524410}a^{2}-\frac{22403747}{1104882}a-\frac{28948253}{11048820}$, $\frac{876049}{7365880}a^{15}-\frac{9491}{1473176}a^{14}-\frac{26249443}{22097640}a^{13}+\frac{1444009}{22097640}a^{12}-\frac{3985371}{3682940}a^{11}+\frac{130023}{3682940}a^{10}+\frac{275136149}{22097640}a^{9}-\frac{12805919}{22097640}a^{8}+\frac{146819203}{11048820}a^{7}-\frac{3211783}{11048820}a^{6}-\frac{547118407}{11048820}a^{5}+\frac{25808233}{11048820}a^{4}-\frac{12367053}{920735}a^{3}-\frac{3813531}{1841470}a^{2}-\frac{176374547}{22097640}a-\frac{895591}{22097640}$, $\frac{1005577}{11048820}a^{15}-\frac{30751}{5524410}a^{14}-\frac{10112413}{11048820}a^{13}+\frac{62622}{920735}a^{12}-\frac{4193113}{5524410}a^{11}-\frac{357031}{5524410}a^{10}+\frac{35149503}{3682940}a^{9}-\frac{847667}{1104882}a^{8}+\frac{25539724}{2762205}a^{7}+\frac{272110}{552441}a^{6}-\frac{213123067}{5524410}a^{5}+\frac{7479143}{1841470}a^{4}-\frac{16147753}{2762205}a^{3}-\frac{8896309}{2762205}a^{2}-\frac{35006447}{11048820}a+\frac{227223}{1841470}$, $\frac{13229}{11048820}a^{15}-\frac{126721}{11048820}a^{13}-\frac{62261}{5524410}a^{11}+\frac{818803}{11048820}a^{9}+\frac{290817}{1841470}a^{7}+\frac{269471}{5524410}a^{5}+\frac{53374}{2762205}a^{3}-\frac{17880689}{11048820}a$, $\frac{2361391}{22097640}a^{15}+\frac{155627}{22097640}a^{14}-\frac{7842881}{7365880}a^{13}-\frac{108853}{1473176}a^{12}-\frac{10979023}{11048820}a^{11}-\frac{59695}{2209764}a^{10}+\frac{245632081}{22097640}a^{9}+\frac{16399841}{22097640}a^{8}+\frac{135556717}{11048820}a^{7}+\frac{6032447}{11048820}a^{6}-\frac{161908263}{3682940}a^{5}-\frac{12758847}{3682940}a^{4}-\frac{38264183}{2762205}a^{3}+\frac{2444491}{5524410}a^{2}-\frac{11806149}{1473176}a-\frac{1257893}{7365880}$, $\frac{327343}{7365880}a^{15}+\frac{2945}{41304}a^{14}-\frac{8635721}{22097640}a^{13}-\frac{29545}{41304}a^{12}-\frac{3530047}{3682940}a^{11}-\frac{12887}{20652}a^{10}+\frac{97583803}{22097640}a^{9}+\frac{312055}{41304}a^{8}+\frac{117758081}{11048820}a^{7}+\frac{53545}{6884}a^{6}-\frac{163049099}{11048820}a^{5}-\frac{634735}{20652}a^{4}-\frac{55463707}{1841470}a^{3}-\frac{45730}{5163}a^{2}-\frac{4576009}{22097640}a+\frac{15571}{41304}$
| 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: | \( 1022779.0833 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 1022779.0833 \cdot 1}{2\cdot\sqrt{105747725158992197265625}}\cr\approx \mathstrut & 0.62744514619 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 9*x^12 + 105*x^10 + 111*x^8 - 420*x^6 - 114*x^4 - 65*x^2 + 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 - 10*x^14 - 9*x^12 + 105*x^10 + 111*x^8 - 420*x^6 - 114*x^4 - 65*x^2 + 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 - 10*x^14 - 9*x^12 + 105*x^10 + 111*x^8 - 420*x^6 - 114*x^4 - 65*x^2 + 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 - 10*x^14 - 9*x^12 + 105*x^10 + 111*x^8 - 420*x^6 - 114*x^4 - 65*x^2 + 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
$C_2^3:C_4$ (as 16T33):
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 $C_2^3:C_4$ |
| Character table for $C_2^3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{505}) \), 4.4.51005.1 x2, 4.4.2525.1 x2, \(\Q(\sqrt{5}, \sqrt{101})\), 8.4.325188753125.2 x2, 8.8.65037750625.1, 8.4.65037750625.2 x2 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(101\)
| 101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |