Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 36*x^14 + 198*x^12 + 1618*x^10 + 142751*x^8 - 1171590*x^6 + 18391859*x^4 + 45558669*x^2 + 198274561)
gp: K = bnfinit(y^16 + 36*y^14 + 198*y^12 + 1618*y^10 + 142751*y^8 - 1171590*y^6 + 18391859*y^4 + 45558669*y^2 + 198274561, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 36*x^14 + 198*x^12 + 1618*x^10 + 142751*x^8 - 1171590*x^6 + 18391859*x^4 + 45558669*x^2 + 198274561);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 36*x^14 + 198*x^12 + 1618*x^10 + 142751*x^8 - 1171590*x^6 + 18391859*x^4 + 45558669*x^2 + 198274561)
\( x^{16} + 36 x^{14} + 198 x^{12} + 1618 x^{10} + 142751 x^{8} - 1171590 x^{6} + 18391859 x^{4} + \cdots + 198274561 \)
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: |
\(32349497931606921267167334562710001\)
\(\medspace = 31^{8}\cdot 41^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(143.50\) | 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: | $31^{1/2}41^{7/8}\approx 143.50489707977238$ | ||
| Ramified primes: |
\(31\), \(41\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{97\!\cdots\!86}a^{14}-\frac{37\!\cdots\!35}{38\!\cdots\!86}a^{12}-\frac{16\!\cdots\!71}{97\!\cdots\!86}a^{10}-\frac{16\!\cdots\!07}{97\!\cdots\!86}a^{8}-\frac{1}{2}a^{7}-\frac{42\!\cdots\!90}{48\!\cdots\!93}a^{6}-\frac{1}{2}a^{5}+\frac{27\!\cdots\!19}{48\!\cdots\!93}a^{4}-\frac{40\!\cdots\!31}{97\!\cdots\!86}a^{2}-\frac{1}{2}a+\frac{45\!\cdots\!35}{97\!\cdots\!86}$, $\frac{1}{13\!\cdots\!66}a^{15}-\frac{47\!\cdots\!20}{27\!\cdots\!83}a^{13}-\frac{32\!\cdots\!84}{68\!\cdots\!33}a^{11}-\frac{15\!\cdots\!62}{68\!\cdots\!33}a^{9}-\frac{18\!\cdots\!15}{68\!\cdots\!33}a^{7}-\frac{20\!\cdots\!55}{68\!\cdots\!33}a^{5}+\frac{75\!\cdots\!94}{68\!\cdots\!33}a^{3}+\frac{23\!\cdots\!84}{68\!\cdots\!33}a-\frac{1}{2}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}\times C_{120}$, which has order $480$ (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: | $7$ | 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: |
$\frac{85\!\cdots\!69}{10\!\cdots\!99}a^{15}-\frac{4817417021}{91\!\cdots\!27}a^{14}+\frac{24\!\cdots\!67}{82\!\cdots\!98}a^{13}-\frac{1497343291}{73\!\cdots\!54}a^{12}+\frac{29\!\cdots\!77}{20\!\cdots\!98}a^{11}-\frac{585670948947}{18\!\cdots\!54}a^{10}+\frac{88\!\cdots\!89}{10\!\cdots\!99}a^{9}+\frac{64554373616269}{18\!\cdots\!54}a^{8}+\frac{11\!\cdots\!87}{10\!\cdots\!99}a^{7}-\frac{11\!\cdots\!79}{18\!\cdots\!54}a^{6}-\frac{10\!\cdots\!97}{10\!\cdots\!99}a^{5}+\frac{109516032893907}{18\!\cdots\!54}a^{4}+\frac{34\!\cdots\!37}{20\!\cdots\!98}a^{3}-\frac{16\!\cdots\!23}{91\!\cdots\!27}a^{2}-\frac{34\!\cdots\!83}{20\!\cdots\!98}a-\frac{30\!\cdots\!92}{91\!\cdots\!27}$, $\frac{41\!\cdots\!19}{20\!\cdots\!98}a^{15}+\frac{4817417021}{91\!\cdots\!27}a^{14}+\frac{54\!\cdots\!79}{82\!\cdots\!98}a^{13}+\frac{1497343291}{73\!\cdots\!54}a^{12}+\frac{47\!\cdots\!57}{20\!\cdots\!98}a^{11}+\frac{585670948947}{18\!\cdots\!54}a^{10}+\frac{39\!\cdots\!44}{10\!\cdots\!99}a^{9}-\frac{64554373616269}{18\!\cdots\!54}a^{8}+\frac{30\!\cdots\!81}{10\!\cdots\!99}a^{7}+\frac{11\!\cdots\!79}{18\!\cdots\!54}a^{6}-\frac{33\!\cdots\!20}{10\!\cdots\!99}a^{5}-\frac{109516032893907}{18\!\cdots\!54}a^{4}+\frac{95\!\cdots\!53}{20\!\cdots\!98}a^{3}+\frac{16\!\cdots\!23}{91\!\cdots\!27}a^{2}-\frac{10\!\cdots\!83}{20\!\cdots\!98}a+\frac{51\!\cdots\!19}{18\!\cdots\!54}$, $\frac{99\!\cdots\!95}{20\!\cdots\!98}a^{15}+\frac{19269668084}{91\!\cdots\!27}a^{14}+\frac{89\!\cdots\!61}{41\!\cdots\!49}a^{13}+\frac{2994686582}{36\!\cdots\!77}a^{12}+\frac{20\!\cdots\!87}{10\!\cdots\!99}a^{11}+\frac{1171341897894}{91\!\cdots\!27}a^{10}-\frac{13\!\cdots\!77}{10\!\cdots\!99}a^{9}-\frac{129108747232538}{91\!\cdots\!27}a^{8}+\frac{31\!\cdots\!80}{10\!\cdots\!99}a^{7}+\frac{22\!\cdots\!58}{91\!\cdots\!27}a^{6}+\frac{26\!\cdots\!29}{10\!\cdots\!99}a^{5}-\frac{219032065787814}{91\!\cdots\!27}a^{4}+\frac{37\!\cdots\!79}{10\!\cdots\!99}a^{3}+\frac{65\!\cdots\!92}{91\!\cdots\!27}a^{2}+\frac{95\!\cdots\!17}{10\!\cdots\!99}a-\frac{29\!\cdots\!45}{18\!\cdots\!54}$, $\frac{14\!\cdots\!83}{13\!\cdots\!66}a^{15}-\frac{29\!\cdots\!19}{48\!\cdots\!93}a^{14}+\frac{38\!\cdots\!01}{54\!\cdots\!66}a^{13}-\frac{65\!\cdots\!34}{19\!\cdots\!43}a^{12}+\frac{11\!\cdots\!48}{68\!\cdots\!33}a^{11}-\frac{63\!\cdots\!87}{97\!\cdots\!86}a^{10}+\frac{32\!\cdots\!45}{13\!\cdots\!66}a^{9}-\frac{35\!\cdots\!77}{48\!\cdots\!93}a^{8}+\frac{45\!\cdots\!33}{13\!\cdots\!66}a^{7}-\frac{57\!\cdots\!12}{48\!\cdots\!93}a^{6}+\frac{39\!\cdots\!21}{13\!\cdots\!66}a^{5}-\frac{42\!\cdots\!57}{97\!\cdots\!86}a^{4}+\frac{86\!\cdots\!75}{13\!\cdots\!66}a^{3}+\frac{18\!\cdots\!39}{97\!\cdots\!86}a^{2}+\frac{30\!\cdots\!57}{68\!\cdots\!33}a+\frac{40\!\cdots\!99}{97\!\cdots\!86}$, $\frac{14\!\cdots\!83}{13\!\cdots\!66}a^{15}+\frac{29\!\cdots\!19}{48\!\cdots\!93}a^{14}+\frac{38\!\cdots\!01}{54\!\cdots\!66}a^{13}+\frac{65\!\cdots\!34}{19\!\cdots\!43}a^{12}+\frac{11\!\cdots\!48}{68\!\cdots\!33}a^{11}+\frac{63\!\cdots\!87}{97\!\cdots\!86}a^{10}+\frac{32\!\cdots\!45}{13\!\cdots\!66}a^{9}+\frac{35\!\cdots\!77}{48\!\cdots\!93}a^{8}+\frac{45\!\cdots\!33}{13\!\cdots\!66}a^{7}+\frac{57\!\cdots\!12}{48\!\cdots\!93}a^{6}+\frac{39\!\cdots\!21}{13\!\cdots\!66}a^{5}+\frac{42\!\cdots\!57}{97\!\cdots\!86}a^{4}+\frac{86\!\cdots\!75}{13\!\cdots\!66}a^{3}-\frac{18\!\cdots\!39}{97\!\cdots\!86}a^{2}+\frac{30\!\cdots\!57}{68\!\cdots\!33}a-\frac{40\!\cdots\!99}{97\!\cdots\!86}$, $\frac{21\!\cdots\!70}{11\!\cdots\!87}a^{15}+\frac{48\!\cdots\!11}{48\!\cdots\!93}a^{14}+\frac{87\!\cdots\!59}{92\!\cdots\!74}a^{13}+\frac{14\!\cdots\!83}{38\!\cdots\!86}a^{12}+\frac{12\!\cdots\!74}{11\!\cdots\!87}a^{11}+\frac{65\!\cdots\!53}{97\!\cdots\!86}a^{10}-\frac{83\!\cdots\!54}{11\!\cdots\!87}a^{9}-\frac{24\!\cdots\!52}{48\!\cdots\!93}a^{8}+\frac{14\!\cdots\!65}{23\!\cdots\!74}a^{7}+\frac{91\!\cdots\!37}{97\!\cdots\!86}a^{6}+\frac{55\!\cdots\!93}{23\!\cdots\!74}a^{5}+\frac{12\!\cdots\!37}{97\!\cdots\!86}a^{4}+\frac{76\!\cdots\!65}{11\!\cdots\!87}a^{3}+\frac{54\!\cdots\!99}{97\!\cdots\!86}a^{2}+\frac{22\!\cdots\!68}{11\!\cdots\!87}a-\frac{11\!\cdots\!97}{97\!\cdots\!86}$, $\frac{37\!\cdots\!06}{68\!\cdots\!33}a^{15}+\frac{59\!\cdots\!48}{27\!\cdots\!83}a^{13}-\frac{28\!\cdots\!10}{68\!\cdots\!33}a^{11}+\frac{68\!\cdots\!92}{68\!\cdots\!33}a^{9}-\frac{61\!\cdots\!02}{68\!\cdots\!33}a^{7}+\frac{50\!\cdots\!28}{68\!\cdots\!33}a^{5}+\frac{11\!\cdots\!11}{68\!\cdots\!33}a^{3}+\frac{60\!\cdots\!12}{68\!\cdots\!33}a$
| 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: | \( 81296120.0801 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 81296120.0801 \cdot 480}{2\cdot\sqrt{32349497931606921267167334562710001}}\cr\approx \mathstrut & 0.263503407244 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 36*x^14 + 198*x^12 + 1618*x^10 + 142751*x^8 - 1171590*x^6 + 18391859*x^4 + 45558669*x^2 + 198274561)
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 + 36*x^14 + 198*x^12 + 1618*x^10 + 142751*x^8 - 1171590*x^6 + 18391859*x^4 + 45558669*x^2 + 198274561, 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 + 36*x^14 + 198*x^12 + 1618*x^10 + 142751*x^8 - 1171590*x^6 + 18391859*x^4 + 45558669*x^2 + 198274561);
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 + 36*x^14 + 198*x^12 + 1618*x^10 + 142751*x^8 - 1171590*x^6 + 18391859*x^4 + 45558669*x^2 + 198274561);
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
$\OD_{16}:C_2$ (as 16T36):
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 $\OD_{16}:C_2$ |
| Character table for $\OD_{16}:C_2$ |
Intermediate fields
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(31\)
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(41\)
| 41.8.7.3 | $x^{8} + 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 41.8.7.3 | $x^{8} + 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |