Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 244*x^14 + 20637*x^12 + 815784*x^10 + 16689063*x^8 + 188396254*x^6 + 1187687160*x^4 + 3924173404*x^2 + 5300131204)
gp: K = bnfinit(y^16 + 244*y^14 + 20637*y^12 + 815784*y^10 + 16689063*y^8 + 188396254*y^6 + 1187687160*y^4 + 3924173404*y^2 + 5300131204, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 244*x^14 + 20637*x^12 + 815784*x^10 + 16689063*x^8 + 188396254*x^6 + 1187687160*x^4 + 3924173404*x^2 + 5300131204);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 244*x^14 + 20637*x^12 + 815784*x^10 + 16689063*x^8 + 188396254*x^6 + 1187687160*x^4 + 3924173404*x^2 + 5300131204)
\( x^{16} + 244 x^{14} + 20637 x^{12} + 815784 x^{10} + 16689063 x^{8} + 188396254 x^{6} + \cdots + 5300131204 \)
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: |
\(8926609275800253364594278400000000\)
\(\medspace = 2^{38}\cdot 5^{8}\cdot 89^{6}\cdot 409^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(132.41\) | 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: | $2^{51/16}5^{1/2}89^{1/2}409^{1/2}\approx 3886.6440692752467$ | ||
| Ramified primes: |
\(2\), \(5\), \(89\), \(409\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{62\!\cdots\!52}a^{14}+\frac{92\!\cdots\!53}{31\!\cdots\!26}a^{12}+\frac{28\!\cdots\!19}{62\!\cdots\!52}a^{10}+\frac{84\!\cdots\!31}{31\!\cdots\!26}a^{8}-\frac{74\!\cdots\!07}{62\!\cdots\!52}a^{6}-\frac{12\!\cdots\!09}{15\!\cdots\!63}a^{4}+\frac{62\!\cdots\!67}{31\!\cdots\!26}a^{2}+\frac{83\!\cdots\!77}{42\!\cdots\!63}$, $\frac{1}{62\!\cdots\!52}a^{15}+\frac{92\!\cdots\!53}{31\!\cdots\!26}a^{13}+\frac{28\!\cdots\!19}{62\!\cdots\!52}a^{11}+\frac{84\!\cdots\!31}{31\!\cdots\!26}a^{9}-\frac{74\!\cdots\!07}{62\!\cdots\!52}a^{7}-\frac{12\!\cdots\!09}{15\!\cdots\!63}a^{5}+\frac{62\!\cdots\!67}{31\!\cdots\!26}a^{3}+\frac{83\!\cdots\!77}{42\!\cdots\!63}a$
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
$C_{2}\times C_{2}\times C_{2}\times C_{136716}$, which has order $1093728$ (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: |
$\frac{18400978687}{11\!\cdots\!72}a^{14}+\frac{2183371123433}{598341536783986}a^{12}+\frac{350531993892117}{11\!\cdots\!72}a^{10}+\frac{63\!\cdots\!07}{598341536783986}a^{8}+\frac{22\!\cdots\!75}{11\!\cdots\!72}a^{6}+\frac{49\!\cdots\!12}{299170768391993}a^{4}+\frac{43\!\cdots\!31}{598341536783986}a^{2}+\frac{36\!\cdots\!32}{299170768391993}$, $\frac{25\!\cdots\!41}{62\!\cdots\!52}a^{14}+\frac{14\!\cdots\!48}{15\!\cdots\!63}a^{12}+\frac{48\!\cdots\!27}{62\!\cdots\!52}a^{10}+\frac{43\!\cdots\!72}{15\!\cdots\!63}a^{8}+\frac{30\!\cdots\!37}{62\!\cdots\!52}a^{6}+\frac{13\!\cdots\!01}{31\!\cdots\!26}a^{4}+\frac{58\!\cdots\!47}{31\!\cdots\!26}a^{2}+\frac{13\!\cdots\!55}{42\!\cdots\!63}$, $\frac{25\!\cdots\!61}{62\!\cdots\!52}a^{14}+\frac{30\!\cdots\!17}{31\!\cdots\!26}a^{12}+\frac{49\!\cdots\!03}{62\!\cdots\!52}a^{10}+\frac{91\!\cdots\!01}{31\!\cdots\!26}a^{8}+\frac{33\!\cdots\!93}{62\!\cdots\!52}a^{6}+\frac{77\!\cdots\!14}{15\!\cdots\!63}a^{4}+\frac{70\!\cdots\!67}{31\!\cdots\!26}a^{2}+\frac{17\!\cdots\!83}{42\!\cdots\!63}$, $\frac{60\!\cdots\!39}{31\!\cdots\!26}a^{14}+\frac{71\!\cdots\!10}{15\!\cdots\!63}a^{12}+\frac{11\!\cdots\!25}{31\!\cdots\!26}a^{10}+\frac{21\!\cdots\!69}{15\!\cdots\!63}a^{8}+\frac{74\!\cdots\!59}{31\!\cdots\!26}a^{6}+\frac{33\!\cdots\!06}{15\!\cdots\!63}a^{4}+\frac{14\!\cdots\!94}{15\!\cdots\!63}a^{2}+\frac{69\!\cdots\!95}{42\!\cdots\!63}$, $\frac{53\!\cdots\!25}{62\!\cdots\!52}a^{14}+\frac{31\!\cdots\!73}{15\!\cdots\!63}a^{12}+\frac{98\!\cdots\!67}{62\!\cdots\!52}a^{10}+\frac{85\!\cdots\!24}{15\!\cdots\!63}a^{8}+\frac{56\!\cdots\!13}{62\!\cdots\!52}a^{6}+\frac{22\!\cdots\!07}{31\!\cdots\!26}a^{4}+\frac{81\!\cdots\!75}{31\!\cdots\!26}a^{2}+\frac{14\!\cdots\!82}{42\!\cdots\!63}$, $\frac{77\!\cdots\!25}{62\!\cdots\!52}a^{14}+\frac{45\!\cdots\!49}{15\!\cdots\!63}a^{12}+\frac{14\!\cdots\!67}{62\!\cdots\!52}a^{10}+\frac{13\!\cdots\!21}{15\!\cdots\!63}a^{8}+\frac{92\!\cdots\!45}{62\!\cdots\!52}a^{6}+\frac{40\!\cdots\!31}{31\!\cdots\!26}a^{4}+\frac{17\!\cdots\!99}{31\!\cdots\!26}a^{2}+\frac{40\!\cdots\!37}{42\!\cdots\!63}$, $\frac{69\!\cdots\!93}{62\!\cdots\!52}a^{14}+\frac{41\!\cdots\!98}{15\!\cdots\!63}a^{12}+\frac{13\!\cdots\!35}{62\!\cdots\!52}a^{10}+\frac{12\!\cdots\!58}{15\!\cdots\!63}a^{8}+\frac{85\!\cdots\!09}{62\!\cdots\!52}a^{6}+\frac{38\!\cdots\!63}{31\!\cdots\!26}a^{4}+\frac{17\!\cdots\!35}{31\!\cdots\!26}a^{2}+\frac{40\!\cdots\!64}{42\!\cdots\!63}$
| 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: | \( 21348.0660855 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 21348.0660855 \cdot 1093728}{2\cdot\sqrt{8926609275800253364594278400000000}}\cr\approx \mathstrut & 0.300146689252 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 244*x^14 + 20637*x^12 + 815784*x^10 + 16689063*x^8 + 188396254*x^6 + 1187687160*x^4 + 3924173404*x^2 + 5300131204)
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 + 244*x^14 + 20637*x^12 + 815784*x^10 + 16689063*x^8 + 188396254*x^6 + 1187687160*x^4 + 3924173404*x^2 + 5300131204, 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 + 244*x^14 + 20637*x^12 + 815784*x^10 + 16689063*x^8 + 188396254*x^6 + 1187687160*x^4 + 3924173404*x^2 + 5300131204);
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 + 244*x^14 + 20637*x^12 + 815784*x^10 + 16689063*x^8 + 188396254*x^6 + 1187687160*x^4 + 3924173404*x^2 + 5300131204);
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
$(C_2\times D_4).D_4^2$ (as 16T1127):
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 58 conjugacy class representatives for $(C_2\times D_4).D_4^2$ |
| Character table for $(C_2\times D_4).D_4^2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.2225.1, 8.8.5069440000.1 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.18.47 | $x^{8} + 8 x^{7} + 30 x^{6} + 72 x^{5} + 284 x^{4} + 208 x^{3} + 316 x^{2} + 192 x + 156$ | $4$ | $2$ | $18$ | $(C_4^2 : C_2):C_2$ | $[2, 2, 3, 7/2, 7/2]^{2}$ |
| 2.8.20.42 | $x^{8} + 4 x^{7} + 12 x^{6} + 64 x^{5} + 104 x^{4} + 56 x^{3} + 232 x^{2} + 124$ | $4$ | $2$ | $20$ | $(C_4^2 : C_2):C_2$ | $[2, 2, 3, 7/2, 7/2]^{2}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(89\)
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(409\)
| $\Q_{409}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{409}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{409}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{409}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |