Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 22*x^14 + 206*x^12 + 1112*x^10 + 4217*x^8 + 12912*x^6 + 29016*x^4 + 31136*x^2 + 13456)
gp: K = bnfinit(y^16 + 22*y^14 + 206*y^12 + 1112*y^10 + 4217*y^8 + 12912*y^6 + 29016*y^4 + 31136*y^2 + 13456, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 22*x^14 + 206*x^12 + 1112*x^10 + 4217*x^8 + 12912*x^6 + 29016*x^4 + 31136*x^2 + 13456);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 22*x^14 + 206*x^12 + 1112*x^10 + 4217*x^8 + 12912*x^6 + 29016*x^4 + 31136*x^2 + 13456)
\( x^{16} + 22x^{14} + 206x^{12} + 1112x^{10} + 4217x^{8} + 12912x^{6} + 29016x^{4} + 31136x^{2} + 13456 \)
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: |
\(19826723432390077223796736\)
\(\medspace = 2^{32}\cdot 389^{4}\cdot 449^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(38.11\) | 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: | $2^{25/12}389^{1/2}449^{1/2}\approx 1771.1029052870458$ | ||
| Ramified primes: |
\(2\), \(389\), \(449\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{14}a^{10}+\frac{3}{7}a^{8}+\frac{2}{7}a^{6}+\frac{5}{14}a^{2}-\frac{1}{7}$, $\frac{1}{28}a^{11}-\frac{2}{7}a^{9}-\frac{5}{14}a^{7}+\frac{5}{28}a^{3}-\frac{1}{14}a$, $\frac{1}{56}a^{12}-\frac{1}{28}a^{10}+\frac{13}{28}a^{8}+\frac{3}{7}a^{6}-\frac{23}{56}a^{4}-\frac{3}{14}$, $\frac{1}{56}a^{13}+\frac{5}{28}a^{9}+\frac{1}{14}a^{7}-\frac{23}{56}a^{5}+\frac{5}{28}a^{3}-\frac{2}{7}a$, $\frac{1}{31793451760}a^{14}-\frac{3929379}{2270960840}a^{12}+\frac{26772217}{15896725880}a^{10}-\frac{150232868}{397418147}a^{8}+\frac{10636729617}{31793451760}a^{6}-\frac{346453494}{1987090735}a^{4}+\frac{2437892847}{7948362940}a^{2}+\frac{839431632}{1987090735}$, $\frac{1}{1844020202080}a^{15}+\frac{139970363}{65857864360}a^{13}-\frac{3947409253}{922010101040}a^{11}-\frac{30407292497}{92201010104}a^{9}-\frac{847786467903}{1844020202080}a^{7}-\frac{80268166617}{922010101040}a^{5}-\frac{37303921853}{461005050520}a^{3}+\frac{7640135469}{230502525260}a$
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
$C_{2}\times C_{56}$, which has order $112$ (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: |
$\frac{212335581}{368804040416}a^{15}+\frac{277312121}{23050252526}a^{13}+\frac{19512811019}{184402020208}a^{11}+\frac{48900200141}{92201010104}a^{9}+\frac{694743026205}{368804040416}a^{7}+\frac{1013454073265}{184402020208}a^{5}+\frac{1022456213587}{92201010104}a^{3}+\frac{305990825751}{46100505052}a$, $\frac{8229}{23674672}a^{15}+\frac{88489}{11837336}a^{13}+\frac{793705}{11837336}a^{11}+\frac{1005269}{2959334}a^{9}+\frac{28678277}{23674672}a^{7}+\frac{21276193}{5918668}a^{5}+\frac{43562467}{5918668}a^{3}+\frac{6548411}{1479667}a+1$, $\frac{8152117}{184402020208}a^{15}+\frac{45970123}{46100505052}a^{13}+\frac{916381693}{92201010104}a^{11}+\frac{2469665671}{46100505052}a^{9}+\frac{35917384813}{184402020208}a^{7}+\frac{7905652713}{13171572872}a^{5}+\frac{4411020325}{3292893218}a^{3}+\frac{9525638800}{11525126263}a$, $\frac{212335581}{368804040416}a^{15}-\frac{531765}{3179345176}a^{14}+\frac{277312121}{23050252526}a^{13}-\frac{11055265}{3179345176}a^{12}+\frac{19512811019}{184402020208}a^{11}-\frac{1599611}{56774021}a^{10}+\frac{48900200141}{92201010104}a^{9}-\frac{186624731}{1589672588}a^{8}+\frac{694743026205}{368804040416}a^{7}-\frac{1054659797}{3179345176}a^{6}+\frac{1013454073265}{184402020208}a^{5}-\frac{2633869267}{3179345176}a^{4}+\frac{1022456213587}{92201010104}a^{3}-\frac{385211123}{397418147}a^{2}+\frac{305990825751}{46100505052}a+\frac{2380443003}{794836294}$, $\frac{49726}{397418147}a^{14}+\frac{6195349}{3179345176}a^{12}+\frac{20274967}{1589672588}a^{10}+\frac{85108089}{1589672588}a^{8}+\frac{78635335}{397418147}a^{6}+\frac{1903086237}{3179345176}a^{4}+\frac{41888414}{56774021}a^{2}-\frac{149649421}{794836294}$, $\frac{121690529}{92201010104}a^{15}+\frac{531765}{3179345176}a^{14}+\frac{2568876203}{92201010104}a^{13}+\frac{11055265}{3179345176}a^{12}+\frac{2839295302}{11525126263}a^{11}+\frac{1599611}{56774021}a^{10}+\frac{57055804941}{46100505052}a^{9}+\frac{186624731}{1589672588}a^{8}+\frac{404363257593}{92201010104}a^{7}+\frac{1054659797}{3179345176}a^{6}+\frac{1177641520989}{92201010104}a^{5}+\frac{2633869267}{3179345176}a^{4}+\frac{592592558105}{23050252526}a^{3}+\frac{385211123}{397418147}a^{2}+\frac{354446533849}{23050252526}a-\frac{3175279297}{794836294}$, $\frac{24898087}{184402020208}a^{15}-\frac{531765}{3179345176}a^{14}+\frac{61730335}{23050252526}a^{13}-\frac{11055265}{3179345176}a^{12}+\frac{2064235155}{92201010104}a^{11}-\frac{1599611}{56774021}a^{10}+\frac{719258573}{6585786436}a^{9}-\frac{186624731}{1589672588}a^{8}+\frac{73474225759}{184402020208}a^{7}-\frac{1054659797}{3179345176}a^{6}+\frac{111913517839}{92201010104}a^{5}-\frac{2633869267}{3179345176}a^{4}+\frac{57412434145}{23050252526}a^{3}-\frac{385211123}{397418147}a^{2}+\frac{17252080430}{11525126263}a+\frac{790770415}{794836294}$
| 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: | \( 42689.08699965526 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 42689.08699965526 \cdot 112}{2\cdot\sqrt{19826723432390077223796736}}\cr\approx \mathstrut & 1.30412243483984 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 22*x^14 + 206*x^12 + 1112*x^10 + 4217*x^8 + 12912*x^6 + 29016*x^4 + 31136*x^2 + 13456)
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 + 22*x^14 + 206*x^12 + 1112*x^10 + 4217*x^8 + 12912*x^6 + 29016*x^4 + 31136*x^2 + 13456, 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 + 22*x^14 + 206*x^12 + 1112*x^10 + 4217*x^8 + 12912*x^6 + 29016*x^4 + 31136*x^2 + 13456);
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 + 22*x^14 + 206*x^12 + 1112*x^10 + 4217*x^8 + 12912*x^6 + 29016*x^4 + 31136*x^2 + 13456);
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
$C_2^5:S_4$ (as 16T1045):
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 768 |
| The 40 conjugacy class representatives for $C_2^5:S_4$ |
| Character table for $C_2^5:S_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.6224.1, 8.0.17393441024.1, 8.0.4452720902144.1, 8.8.9916973056.1 |
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
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.16.58 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{2} + 4 x + 2$ | $8$ | $1$ | $16$ | $S_4\times C_2$ | $[4/3, 4/3, 3]_{3}^{2}$ |
| 2.8.16.58 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{2} + 4 x + 2$ | $8$ | $1$ | $16$ | $S_4\times C_2$ | $[4/3, 4/3, 3]_{3}^{2}$ | |
|
\(389\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
|
\(449\)
| $\Q_{449}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{449}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{449}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{449}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |