Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 5*x^14 - 31*x^12 - 50*x^10 + 166*x^8 - 50*x^6 - 31*x^4 + 5*x^2 + 1)
gp: K = bnfinit(y^16 + 5*y^14 - 31*y^12 - 50*y^10 + 166*y^8 - 50*y^6 - 31*y^4 + 5*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 5*x^14 - 31*x^12 - 50*x^10 + 166*x^8 - 50*x^6 - 31*x^4 + 5*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 5*x^14 - 31*x^12 - 50*x^10 + 166*x^8 - 50*x^6 - 31*x^4 + 5*x^2 + 1)
\( x^{16} + 5x^{14} - 31x^{12} - 50x^{10} + 166x^{8} - 50x^{6} - 31x^{4} + 5x^{2} + 1 \)
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: |
\(46803543212226806640625\)
\(\medspace = 5^{12}\cdot 61^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.12\) | 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: | $5^{3/4}61^{1/2}\approx 26.115143751039085$ | ||
| Ramified primes: |
\(5\), \(61\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{3}{8}a^{2}-\frac{3}{8}a-\frac{3}{8}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{6}-\frac{1}{8}a^{3}+\frac{1}{8}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{7}-\frac{1}{8}a^{4}+\frac{1}{8}a$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}+\frac{1}{16}a^{7}+\frac{1}{16}a^{6}+\frac{3}{16}a^{5}-\frac{3}{16}a^{4}-\frac{3}{16}a^{3}-\frac{3}{16}a^{2}+\frac{3}{16}a+\frac{3}{16}$, $\frac{1}{16}a^{12}-\frac{1}{8}a^{6}+\frac{1}{16}$, $\frac{1}{16}a^{13}-\frac{1}{8}a^{7}+\frac{1}{16}a$, $\frac{1}{224}a^{14}-\frac{1}{32}a^{13}-\frac{3}{224}a^{12}+\frac{3}{112}a^{8}+\frac{1}{16}a^{7}+\frac{3}{112}a^{6}-\frac{87}{224}a^{2}+\frac{15}{32}a-\frac{83}{224}$, $\frac{1}{224}a^{15}+\frac{1}{56}a^{13}-\frac{1}{32}a^{12}+\frac{3}{112}a^{9}-\frac{1}{28}a^{7}+\frac{1}{16}a^{6}+\frac{25}{224}a^{3}+\frac{9}{56}a-\frac{1}{32}$
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
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: |
$\frac{1}{28}a^{14}+\frac{9}{112}a^{12}-\frac{13}{8}a^{10}+\frac{61}{56}a^{8}+\frac{321}{28}a^{6}-\frac{127}{8}a^{4}+\frac{85}{56}a^{2}+\frac{123}{112}$, $\frac{1}{28}a^{15}+\frac{9}{112}a^{13}-\frac{13}{8}a^{11}+\frac{61}{56}a^{9}+\frac{321}{28}a^{7}-\frac{127}{8}a^{5}+\frac{85}{56}a^{3}+\frac{235}{112}a$, $\frac{27}{112}a^{15}+\frac{11}{56}a^{14}+\frac{17}{14}a^{13}+\frac{29}{28}a^{12}-\frac{59}{8}a^{11}-\frac{23}{4}a^{10}-\frac{675}{56}a^{9}-\frac{155}{14}a^{8}+\frac{2125}{56}a^{7}+\frac{1571}{56}a^{6}-\frac{109}{8}a^{5}-\frac{23}{4}a^{4}+\frac{59}{112}a^{3}-\frac{75}{56}a^{2}+\frac{87}{56}a-\frac{17}{56}$, $\frac{45}{112}a^{15}-\frac{23}{112}a^{14}+\frac{243}{112}a^{13}-\frac{127}{112}a^{12}-\frac{23}{2}a^{11}+\frac{23}{4}a^{10}-\frac{339}{14}a^{9}+\frac{92}{7}a^{8}+\frac{1527}{28}a^{7}-\frac{1483}{56}a^{6}-\frac{7}{2}a^{5}-\frac{9}{4}a^{4}-\frac{261}{112}a^{3}+\frac{111}{112}a^{2}-\frac{95}{112}a+\frac{61}{112}$, $\frac{27}{112}a^{15}+\frac{11}{56}a^{14}+\frac{17}{14}a^{13}+\frac{29}{28}a^{12}-\frac{59}{8}a^{11}-\frac{23}{4}a^{10}-\frac{675}{56}a^{9}-\frac{155}{14}a^{8}+\frac{2125}{56}a^{7}+\frac{1571}{56}a^{6}-\frac{109}{8}a^{5}-\frac{23}{4}a^{4}+\frac{59}{112}a^{3}-\frac{75}{56}a^{2}+\frac{87}{56}a+\frac{39}{56}$, $\frac{73}{224}a^{15}-\frac{1}{4}a^{14}+\frac{87}{56}a^{13}-\frac{43}{32}a^{12}-\frac{21}{2}a^{11}+\frac{29}{4}a^{10}-\frac{1573}{112}a^{9}+\frac{61}{4}a^{8}+\frac{1635}{28}a^{7}-\frac{573}{16}a^{6}-\frac{55}{2}a^{5}-\frac{5}{4}a^{4}-\frac{1983}{224}a^{3}+7a^{2}+\frac{83}{56}a+\frac{5}{32}$, $\frac{53}{112}a^{15}-\frac{31}{112}a^{14}+\frac{261}{112}a^{13}-\frac{145}{112}a^{12}-\frac{237}{16}a^{11}+\frac{145}{16}a^{10}-\frac{2503}{112}a^{9}+\frac{1263}{112}a^{8}+\frac{8893}{112}a^{7}-\frac{5751}{112}a^{6}-\frac{511}{16}a^{5}+\frac{419}{16}a^{4}-\frac{489}{56}a^{3}+\frac{207}{28}a^{2}+\frac{205}{56}a-\frac{83}{28}$, $\frac{1}{2}a^{15}+\frac{5}{2}a^{13}-\frac{31}{2}a^{11}-25a^{9}+83a^{7}-25a^{5}-\frac{31}{2}a^{3}+\frac{1}{2}a^{2}+\frac{5}{2}a-1$, $\frac{1}{8}a^{15}+\frac{9}{32}a^{14}+\frac{19}{32}a^{13}+\frac{51}{32}a^{12}-\frac{33}{8}a^{11}-\frac{61}{8}a^{10}-\frac{47}{8}a^{9}-\frac{303}{16}a^{8}+\frac{391}{16}a^{7}+\frac{535}{16}a^{6}-\frac{31}{8}a^{5}+\frac{45}{8}a^{4}-\frac{33}{4}a^{3}-\frac{123}{32}a^{2}-\frac{81}{32}a-\frac{17}{32}$, $\frac{83}{224}a^{15}-\frac{33}{112}a^{14}+\frac{111}{56}a^{13}-\frac{299}{224}a^{12}-\frac{43}{4}a^{11}+\frac{79}{8}a^{10}-\frac{2467}{112}a^{9}+\frac{601}{56}a^{8}+\frac{2949}{56}a^{7}-\frac{6393}{112}a^{6}-\frac{13}{4}a^{5}+\frac{273}{8}a^{4}-\frac{1789}{224}a^{3}+\frac{799}{112}a^{2}+\frac{3}{14}a-\frac{787}{224}$, $\frac{29}{224}a^{15}+\frac{1}{112}a^{14}+\frac{43}{56}a^{13}+\frac{29}{224}a^{12}-\frac{27}{8}a^{11}+\frac{1}{8}a^{10}-\frac{1117}{112}a^{9}-\frac{179}{56}a^{8}+\frac{103}{7}a^{7}-\frac{239}{112}a^{6}+\frac{83}{8}a^{5}+\frac{119}{8}a^{4}-\frac{1907}{224}a^{3}-\frac{787}{112}a^{2}-\frac{47}{56}a-\frac{159}{224}$
| 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: | \( 584759.907381 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 584759.907381 \cdot 1}{2\cdot\sqrt{46803543212226806640625}}\cr\approx \mathstrut & 0.539222077932 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 5*x^14 - 31*x^12 - 50*x^10 + 166*x^8 - 50*x^6 - 31*x^4 + 5*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 + 5*x^14 - 31*x^12 - 50*x^10 + 166*x^8 - 50*x^6 - 31*x^4 + 5*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 + 5*x^14 - 31*x^12 - 50*x^10 + 166*x^8 - 50*x^6 - 31*x^4 + 5*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 + 5*x^14 - 31*x^12 - 50*x^10 + 166*x^8 - 50*x^6 - 31*x^4 + 5*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
| \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{305}) \), 4.4.465125.1 x2, \(\Q(\sqrt{5}, \sqrt{61})\), 4.4.7625.1 x2, 8.4.216341265625.1 x2, 8.4.43268253125.1 x2, 8.8.216341265625.1 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(61\)
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |