Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 2*x^12 - 7*x^11 + x^10 + 6*x^9 - 20*x^8 - 22*x^7 - 7*x^6 - 14*x^5 - 9*x^4 + 14*x^3 + 19*x^2 + 8*x + 1)
gp: K = bnfinit(y^16 - 4*y^14 + 2*y^12 - 7*y^11 + y^10 + 6*y^9 - 20*y^8 - 22*y^7 - 7*y^6 - 14*y^5 - 9*y^4 + 14*y^3 + 19*y^2 + 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 + 2*x^12 - 7*x^11 + x^10 + 6*x^9 - 20*x^8 - 22*x^7 - 7*x^6 - 14*x^5 - 9*x^4 + 14*x^3 + 19*x^2 + 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^14 + 2*x^12 - 7*x^11 + x^10 + 6*x^9 - 20*x^8 - 22*x^7 - 7*x^6 - 14*x^5 - 9*x^4 + 14*x^3 + 19*x^2 + 8*x + 1)
\( x^{16} - 4 x^{14} + 2 x^{12} - 7 x^{11} + x^{10} + 6 x^{9} - 20 x^{8} - 22 x^{7} - 7 x^{6} - 14 x^{5} + \cdots + 1 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(266190858984765625\)
\(\medspace = 5^{8}\cdot 31^{4}\cdot 859^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.28\) | 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: | $5^{1/2}31^{3/4}859^{1/2}\approx 860.9994112013104$ | ||
| Ramified primes: |
\(5\), \(31\), \(859\)
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{36054013}a^{15}+\frac{10396452}{36054013}a^{14}-\frac{3222361}{36054013}a^{13}+\frac{15038337}{36054013}a^{12}+\frac{5726866}{36054013}a^{11}-\frac{4832593}{36054013}a^{10}-\frac{13234340}{36054013}a^{9}+\frac{9145238}{36054013}a^{8}-\frac{9786744}{36054013}a^{7}+\frac{2936769}{36054013}a^{6}-\frac{2425339}{36054013}a^{5}-\frac{5695497}{36054013}a^{4}-\frac{13466233}{36054013}a^{3}+\frac{4442894}{36054013}a^{2}-\frac{12164752}{36054013}a+\frac{6541504}{36054013}$
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
Trivial group, which has order $1$
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: | $9$ | 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: |
$\frac{6098843}{36054013}a^{15}+\frac{10650612}{36054013}a^{14}-\frac{27936166}{36054013}a^{13}-\frac{49382141}{36054013}a^{12}+\frac{39684327}{36054013}a^{11}-\frac{7766737}{36054013}a^{10}-\frac{115073546}{36054013}a^{9}+\frac{101080738}{36054013}a^{8}-\frac{36075562}{36054013}a^{7}-\frac{452168029}{36054013}a^{6}-\frac{134247358}{36054013}a^{5}+\frac{8382775}{36054013}a^{4}-\frac{283575459}{36054013}a^{3}+\frac{83447479}{36054013}a^{2}+\frac{314289204}{36054013}a+\frac{109902748}{36054013}$, $\frac{2106}{507803}a^{15}-\frac{14039}{507803}a^{14}-\frac{12974}{507803}a^{13}+\frac{80218}{507803}a^{12}-\frac{49257}{507803}a^{11}-\frac{53132}{507803}a^{10}+\frac{263221}{507803}a^{9}-\frac{80956}{507803}a^{8}-\frac{174700}{507803}a^{7}+\frac{302777}{507803}a^{6}+\frac{226443}{507803}a^{5}+\frac{97981}{507803}a^{4}+\frac{403049}{507803}a^{3}+\frac{464489}{507803}a^{2}+\frac{201441}{507803}a-\frac{287966}{507803}$, $\frac{15777331}{36054013}a^{15}-\frac{8524083}{36054013}a^{14}-\frac{59719035}{36054013}a^{13}+\frac{30977952}{36054013}a^{12}+\frac{22981463}{36054013}a^{11}-\frac{123249507}{36054013}a^{10}+\frac{71277452}{36054013}a^{9}+\frac{80162064}{36054013}a^{8}-\frac{358295229}{36054013}a^{7}-\frac{185845320}{36054013}a^{6}+\frac{45751159}{36054013}a^{5}-\frac{197591840}{36054013}a^{4}-\frac{72478982}{36054013}a^{3}+\frac{328459145}{36054013}a^{2}+\frac{162294404}{36054013}a-\frac{22687716}{36054013}$, $\frac{13366297}{36054013}a^{15}-\frac{15777331}{36054013}a^{14}-\frac{44941105}{36054013}a^{13}+\frac{59719035}{36054013}a^{12}-\frac{4245358}{36054013}a^{11}-\frac{116545542}{36054013}a^{10}+\frac{136615804}{36054013}a^{9}+\frac{8920330}{36054013}a^{8}-\frac{347488004}{36054013}a^{7}+\frac{64236695}{36054013}a^{6}+\frac{92281241}{36054013}a^{5}-\frac{232879317}{36054013}a^{4}+\frac{77295167}{36054013}a^{3}+\frac{259607140}{36054013}a^{2}-\frac{38445489}{36054013}a-\frac{55364028}{36054013}$, $\frac{2242130}{36054013}a^{15}-\frac{20426208}{36054013}a^{14}+\frac{19558179}{36054013}a^{13}+\frac{49364158}{36054013}a^{12}-\frac{78395305}{36054013}a^{11}+\frac{46837813}{36054013}a^{10}+\frac{99089073}{36054013}a^{9}-\frac{186336550}{36054013}a^{8}+\frac{77121353}{36054013}a^{7}+\frac{289807858}{36054013}a^{6}-\frac{114875358}{36054013}a^{5}+\frac{142499938}{36054013}a^{4}+\frac{135812469}{36054013}a^{3}-\frac{105759654}{36054013}a^{2}-\frac{130621273}{36054013}a-\frac{50394945}{36054013}$, $\frac{30651650}{36054013}a^{15}+\frac{2537493}{36054013}a^{14}-\frac{136067942}{36054013}a^{13}+\frac{7181310}{36054013}a^{12}+\frac{96705215}{36054013}a^{11}-\frac{262654392}{36054013}a^{10}+\frac{50643861}{36054013}a^{9}+\frac{252614987}{36054013}a^{8}-\frac{728984090}{36054013}a^{7}-\frac{644844757}{36054013}a^{6}-\frac{43512351}{36054013}a^{5}-\frac{506163205}{36054013}a^{4}-\frac{200360782}{36054013}a^{3}+\frac{508510111}{36054013}a^{2}+\frac{521265070}{36054013}a+\frac{167774505}{36054013}$, $\frac{51810788}{36054013}a^{15}-\frac{25856344}{36054013}a^{14}-\frac{196304252}{36054013}a^{13}+\frac{102138068}{36054013}a^{12}+\frac{53930425}{36054013}a^{11}-\frac{401705718}{36054013}a^{10}+\frac{269112472}{36054013}a^{9}+\frac{184435557}{36054013}a^{8}-\frac{1156566391}{36054013}a^{7}-\frac{524137239}{36054013}a^{6}-\frac{69842375}{36054013}a^{5}-\frac{729145914}{36054013}a^{4}-\frac{47251235}{36054013}a^{3}+\frac{793847179}{36054013}a^{2}+\frac{582950504}{36054013}a+\frac{135840602}{36054013}$, $\frac{3993651}{36054013}a^{15}-\frac{444548}{36054013}a^{14}-\frac{10045843}{36054013}a^{13}-\frac{3906688}{36054013}a^{12}-\frac{11396875}{36054013}a^{11}-\frac{12762156}{36054013}a^{10}+\frac{9127997}{36054013}a^{9}-\frac{6555127}{36054013}a^{8}-\frac{54521538}{36054013}a^{7}-\frac{84191333}{36054013}a^{6}-\frac{119038265}{36054013}a^{5}-\frac{71568107}{36054013}a^{4}-\frac{43259441}{36054013}a^{3}+\frac{34540278}{36054013}a^{2}+\frac{107177623}{36054013}a+\frac{70657421}{36054013}$, $\frac{28203649}{36054013}a^{15}-\frac{30632272}{36054013}a^{14}-\frac{86675864}{36054013}a^{13}+\frac{102652987}{36054013}a^{12}-\frac{34574731}{36054013}a^{11}-\frac{185011385}{36054013}a^{10}+\frac{241088635}{36054013}a^{9}-\frac{64538083}{36054013}a^{8}-\frac{553361807}{36054013}a^{7}+\frac{32978934}{36054013}a^{6}-\frac{113967826}{36054013}a^{5}-\frac{299070912}{36054013}a^{4}+\frac{138161252}{36054013}a^{3}+\frac{281008771}{36054013}a^{2}+\frac{168992160}{36054013}a+\frac{57476990}{36054013}$
| 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: | \( 170.623593629 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 170.623593629 \cdot 1}{2\cdot\sqrt{266190858984765625}}\cr\approx \mathstrut & 0.162783998897 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 2*x^12 - 7*x^11 + x^10 + 6*x^9 - 20*x^8 - 22*x^7 - 7*x^6 - 14*x^5 - 9*x^4 + 14*x^3 + 19*x^2 + 8*x + 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 - 4*x^14 + 2*x^12 - 7*x^11 + x^10 + 6*x^9 - 20*x^8 - 22*x^7 - 7*x^6 - 14*x^5 - 9*x^4 + 14*x^3 + 19*x^2 + 8*x + 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 - 4*x^14 + 2*x^12 - 7*x^11 + x^10 + 6*x^9 - 20*x^8 - 22*x^7 - 7*x^6 - 14*x^5 - 9*x^4 + 14*x^3 + 19*x^2 + 8*x + 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 - 4*x^14 + 2*x^12 - 7*x^11 + x^10 + 6*x^9 - 20*x^8 - 22*x^7 - 7*x^6 - 14*x^5 - 9*x^4 + 14*x^3 + 19*x^2 + 8*x + 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
$C_2^6.S_4^2:C_2^2$ (as 16T1884):
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 147456 |
| The 136 conjugacy class representatives for $C_2^6.S_4^2:C_2^2$ |
| Character table for $C_2^6.S_4^2:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 8.4.16643125.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.3.1 | $x^{4} + 31$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(859\)
| $\Q_{859}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{859}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |