Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1080*x^14 + 445108*x^12 + 89721256*x^10 + 9481842342*x^8 + 527214969352*x^6 + 14588368338164*x^4 + 174011003239136*x^2 + 526630603644098)
gp: K = bnfinit(y^16 + 1080*y^14 + 445108*y^12 + 89721256*y^10 + 9481842342*y^8 + 527214969352*y^6 + 14588368338164*y^4 + 174011003239136*y^2 + 526630603644098, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 1080*x^14 + 445108*x^12 + 89721256*x^10 + 9481842342*x^8 + 527214969352*x^6 + 14588368338164*x^4 + 174011003239136*x^2 + 526630603644098);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 1080*x^14 + 445108*x^12 + 89721256*x^10 + 9481842342*x^8 + 527214969352*x^6 + 14588368338164*x^4 + 174011003239136*x^2 + 526630603644098)
\( x^{16} + 1080 x^{14} + 445108 x^{12} + 89721256 x^{10} + 9481842342 x^{8} + 527214969352 x^{6} + \cdots + 526630603644098 \)
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: |
\(3245499096990365116593765338925510950912\)
\(\medspace = 2^{67}\cdot 17^{6}\cdot 977^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(294.75\) | 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: | $2^{305/64}17^{1/2}977^{1/2}\approx 3505.644036096344$ | ||
| Ramified primes: |
\(2\), \(17\), \(977\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{977}a^{10}+\frac{103}{977}a^{8}-\frac{404}{977}a^{6}+\frac{415}{977}a^{4}-\frac{301}{977}a^{2}$, $\frac{1}{977}a^{11}+\frac{103}{977}a^{9}-\frac{404}{977}a^{7}+\frac{415}{977}a^{5}-\frac{301}{977}a^{3}$, $\frac{1}{954529}a^{12}+\frac{103}{954529}a^{10}+\frac{344477}{954529}a^{8}+\frac{390238}{954529}a^{6}+\frac{100330}{954529}a^{4}+\frac{267}{977}a^{2}$, $\frac{1}{954529}a^{13}+\frac{103}{954529}a^{11}+\frac{344477}{954529}a^{9}+\frac{390238}{954529}a^{7}+\frac{100330}{954529}a^{5}+\frac{267}{977}a^{3}$, $\frac{1}{77\!\cdots\!37}a^{14}+\frac{36\!\cdots\!99}{77\!\cdots\!37}a^{12}+\frac{32\!\cdots\!42}{11\!\cdots\!91}a^{10}+\frac{22\!\cdots\!40}{77\!\cdots\!37}a^{8}+\frac{78\!\cdots\!27}{77\!\cdots\!37}a^{6}+\frac{46\!\cdots\!64}{46\!\cdots\!93}a^{4}-\frac{41\!\cdots\!98}{80\!\cdots\!53}a^{2}-\frac{12\!\cdots\!62}{48\!\cdots\!17}$, $\frac{1}{77\!\cdots\!37}a^{15}+\frac{36\!\cdots\!99}{77\!\cdots\!37}a^{13}+\frac{32\!\cdots\!42}{11\!\cdots\!91}a^{11}+\frac{22\!\cdots\!40}{77\!\cdots\!37}a^{9}+\frac{78\!\cdots\!27}{77\!\cdots\!37}a^{7}+\frac{46\!\cdots\!64}{46\!\cdots\!93}a^{5}-\frac{41\!\cdots\!98}{80\!\cdots\!53}a^{3}-\frac{12\!\cdots\!62}{48\!\cdots\!17}a$
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
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{13352508}$, which has order $427280256$ (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: |
$\frac{49\!\cdots\!78}{75\!\cdots\!43}a^{14}+\frac{50\!\cdots\!97}{75\!\cdots\!43}a^{12}+\frac{19\!\cdots\!72}{75\!\cdots\!43}a^{10}+\frac{33\!\cdots\!43}{75\!\cdots\!43}a^{8}+\frac{27\!\cdots\!48}{75\!\cdots\!43}a^{6}+\frac{98\!\cdots\!62}{77\!\cdots\!59}a^{4}+\frac{13\!\cdots\!56}{79\!\cdots\!67}a^{2}+\frac{24\!\cdots\!03}{81\!\cdots\!71}$, $\frac{76\!\cdots\!28}{11\!\cdots\!91}a^{14}+\frac{79\!\cdots\!88}{11\!\cdots\!91}a^{12}+\frac{30\!\cdots\!64}{11\!\cdots\!91}a^{10}+\frac{56\!\cdots\!05}{11\!\cdots\!91}a^{8}+\frac{50\!\cdots\!82}{11\!\cdots\!91}a^{6}+\frac{12\!\cdots\!05}{66\!\cdots\!99}a^{4}+\frac{35\!\cdots\!32}{11\!\cdots\!79}a^{2}+\frac{62\!\cdots\!79}{69\!\cdots\!31}$, $\frac{15\!\cdots\!20}{77\!\cdots\!37}a^{14}+\frac{17\!\cdots\!32}{77\!\cdots\!37}a^{12}+\frac{98\!\cdots\!82}{11\!\cdots\!91}a^{10}+\frac{13\!\cdots\!98}{77\!\cdots\!37}a^{8}+\frac{13\!\cdots\!76}{77\!\cdots\!37}a^{6}+\frac{39\!\cdots\!47}{46\!\cdots\!93}a^{4}+\frac{13\!\cdots\!17}{80\!\cdots\!53}a^{2}+\frac{40\!\cdots\!75}{48\!\cdots\!17}$, $\frac{30\!\cdots\!37}{77\!\cdots\!37}a^{14}+\frac{32\!\cdots\!42}{77\!\cdots\!37}a^{12}+\frac{18\!\cdots\!58}{11\!\cdots\!91}a^{10}+\frac{25\!\cdots\!59}{77\!\cdots\!37}a^{8}+\frac{24\!\cdots\!99}{77\!\cdots\!37}a^{6}+\frac{67\!\cdots\!57}{46\!\cdots\!93}a^{4}+\frac{20\!\cdots\!17}{80\!\cdots\!53}a^{2}+\frac{43\!\cdots\!35}{48\!\cdots\!17}$, $\frac{54\!\cdots\!39}{77\!\cdots\!37}a^{14}+\frac{57\!\cdots\!49}{77\!\cdots\!37}a^{12}+\frac{32\!\cdots\!69}{11\!\cdots\!91}a^{10}+\frac{43\!\cdots\!08}{77\!\cdots\!37}a^{8}+\frac{40\!\cdots\!46}{77\!\cdots\!37}a^{6}+\frac{11\!\cdots\!39}{46\!\cdots\!93}a^{4}+\frac{36\!\cdots\!89}{80\!\cdots\!53}a^{2}+\frac{88\!\cdots\!75}{48\!\cdots\!17}$, $\frac{20\!\cdots\!22}{77\!\cdots\!37}a^{14}+\frac{22\!\cdots\!55}{77\!\cdots\!37}a^{12}+\frac{12\!\cdots\!46}{11\!\cdots\!91}a^{10}+\frac{16\!\cdots\!35}{77\!\cdots\!37}a^{8}+\frac{16\!\cdots\!08}{77\!\cdots\!37}a^{6}+\frac{45\!\cdots\!21}{46\!\cdots\!93}a^{4}+\frac{14\!\cdots\!21}{80\!\cdots\!53}a^{2}+\frac{40\!\cdots\!05}{48\!\cdots\!17}$, $\frac{32\!\cdots\!74}{77\!\cdots\!37}a^{14}+\frac{33\!\cdots\!61}{77\!\cdots\!37}a^{12}+\frac{18\!\cdots\!18}{11\!\cdots\!91}a^{10}+\frac{22\!\cdots\!00}{77\!\cdots\!37}a^{8}+\frac{19\!\cdots\!66}{77\!\cdots\!37}a^{6}+\frac{44\!\cdots\!14}{46\!\cdots\!93}a^{4}+\frac{10\!\cdots\!03}{80\!\cdots\!53}a^{2}-\frac{98\!\cdots\!69}{48\!\cdots\!17}$
| 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: | \( 42439.7060581 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 42439.7060581 \cdot 427280256}{2\cdot\sqrt{3245499096990365116593765338925510950912}}\cr\approx \mathstrut & 0.386592475944 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 1080*x^14 + 445108*x^12 + 89721256*x^10 + 9481842342*x^8 + 527214969352*x^6 + 14588368338164*x^4 + 174011003239136*x^2 + 526630603644098)
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 + 1080*x^14 + 445108*x^12 + 89721256*x^10 + 9481842342*x^8 + 527214969352*x^6 + 14588368338164*x^4 + 174011003239136*x^2 + 526630603644098, 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 + 1080*x^14 + 445108*x^12 + 89721256*x^10 + 9481842342*x^8 + 527214969352*x^6 + 14588368338164*x^4 + 174011003239136*x^2 + 526630603644098);
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 + 1080*x^14 + 445108*x^12 + 89721256*x^10 + 9481842342*x^8 + 527214969352*x^6 + 14588368338164*x^4 + 174011003239136*x^2 + 526630603644098);
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
$(C_4\times \OD_{16}).D_4$ (as 16T937):
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 512 |
| The 32 conjugacy class representatives for $(C_4\times \OD_{16}).D_4$ |
| Character table for $(C_4\times \OD_{16}).D_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.4352.1, 8.8.9697230848.1 |
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: | 16.0.3245499096990365116593765338925510950912.3 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16$ | $16$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $16$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $16$ | $1$ | $67$ | |||
|
\(17\)
| 17.4.2.2 | $x^{4} - 272 x^{2} + 867$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(977\)
| $\Q_{977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |