Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^14 + 6*x^12 - 3*x^10 + 22*x^8 + 42*x^6 - 7*x^4 - 21*x^2 + 1)
gp: K = bnfinit(y^16 + 6*y^14 + 6*y^12 - 3*y^10 + 22*y^8 + 42*y^6 - 7*y^4 - 21*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 6*x^14 + 6*x^12 - 3*x^10 + 22*x^8 + 42*x^6 - 7*x^4 - 21*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 6*x^14 + 6*x^12 - 3*x^10 + 22*x^8 + 42*x^6 - 7*x^4 - 21*x^2 + 1)
\( x^{16} + 6x^{14} + 6x^{12} - 3x^{10} + 22x^{8} + 42x^{6} - 7x^{4} - 21x^{2} + 1 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(503360547930148624\)
\(\medspace = 2^{4}\cdot 23^{2}\cdot 2777^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.78\) | 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: | $2\cdot 23^{1/2}2777^{1/2}\approx 505.4542511444532$ | ||
| Ramified primes: |
\(2\), \(23\), \(2777\)
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{64406}a^{14}-\frac{8127}{64406}a^{12}-\frac{7931}{32203}a^{10}-\frac{15889}{32203}a^{8}-\frac{1}{2}a^{7}+\frac{21421}{64406}a^{6}-\frac{15462}{32203}a^{4}-\frac{545}{64406}a^{2}-\frac{1}{2}a-\frac{5775}{32203}$, $\frac{1}{64406}a^{15}-\frac{8127}{64406}a^{13}-\frac{7931}{32203}a^{11}-\frac{15889}{32203}a^{9}-\frac{1}{2}a^{8}+\frac{21421}{64406}a^{7}-\frac{15462}{32203}a^{5}-\frac{545}{64406}a^{3}-\frac{1}{2}a^{2}-\frac{5775}{32203}a$
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
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: |
$\frac{9927}{64406}a^{15}+\frac{1857}{64406}a^{14}+\frac{28096}{32203}a^{13}+\frac{5684}{32203}a^{12}+\frac{42799}{64406}a^{11}+\frac{10011}{64406}a^{10}-\frac{31821}{64406}a^{9}-\frac{7925}{32203}a^{8}+\frac{235279}{64406}a^{7}+\frac{40295}{64406}a^{6}+\frac{330681}{64406}a^{5}+\frac{44345}{32203}a^{4}-\frac{48360}{32203}a^{3}-\frac{39089}{32203}a^{2}-\frac{110779}{64406}a-\frac{32779}{32203}$, $\frac{8409}{64406}a^{15}+\frac{4383}{64406}a^{14}+\frac{59229}{64406}a^{13}+\frac{14040}{32203}a^{12}+\frac{97877}{64406}a^{11}+\frac{17667}{32203}a^{10}-\frac{354}{32203}a^{9}-\frac{4999}{64406}a^{8}+\frac{73311}{32203}a^{7}+\frac{113107}{64406}a^{6}+\frac{578963}{64406}a^{5}+\frac{113978}{32203}a^{4}+\frac{183139}{64406}a^{3}+\frac{13245}{32203}a^{2}-\frac{353935}{64406}a-\frac{32737}{64406}$, $\frac{6242}{32203}a^{15}+\frac{1200}{32203}a^{14}+\frac{78591}{64406}a^{13}+\frac{5109}{32203}a^{12}+\frac{45824}{32203}a^{11}-\frac{2427}{32203}a^{10}-\frac{19999}{32203}a^{9}-\frac{5248}{32203}a^{8}+\frac{131838}{32203}a^{7}+\frac{46615}{64406}a^{6}+\frac{606205}{64406}a^{5}+\frac{10315}{64406}a^{4}-\frac{73353}{64406}a^{3}-\frac{52083}{64406}a^{2}-\frac{153598}{32203}a-\frac{12710}{32203}$, $\frac{7359}{32203}a^{15}+\frac{9927}{64406}a^{14}+\frac{85365}{64406}a^{13}+\frac{28096}{32203}a^{12}+\frac{39620}{32203}a^{11}+\frac{42799}{64406}a^{10}-\frac{24435}{64406}a^{9}-\frac{31821}{64406}a^{8}+\frac{164469}{32203}a^{7}+\frac{235279}{64406}a^{6}+\frac{565221}{64406}a^{5}+\frac{330681}{64406}a^{4}-\frac{17483}{32203}a^{3}-\frac{48360}{32203}a^{2}-\frac{109342}{32203}a-\frac{110779}{64406}$, $\frac{6242}{32203}a^{15}-\frac{657}{32203}a^{14}+\frac{78591}{64406}a^{13}-\frac{6259}{32203}a^{12}+\frac{45824}{32203}a^{11}-\frac{12438}{32203}a^{10}-\frac{19999}{32203}a^{9}+\frac{10602}{32203}a^{8}+\frac{131838}{32203}a^{7}-\frac{33975}{64406}a^{6}+\frac{606205}{64406}a^{5}-\frac{167065}{64406}a^{4}-\frac{73353}{64406}a^{3}+\frac{39867}{64406}a^{2}-\frac{153598}{32203}a+\frac{52848}{32203}$, $\frac{3069}{64406}a^{15}+\frac{4374}{32203}a^{14}+\frac{7781}{32203}a^{13}+\frac{41431}{64406}a^{12}+\frac{5229}{32203}a^{11}+\frac{1951}{64406}a^{10}+\frac{16205}{64406}a^{9}-\frac{8824}{32203}a^{8}+\frac{39566}{32203}a^{7}+\frac{113536}{32203}a^{6}+\frac{60891}{64406}a^{5}+\frac{78657}{64406}a^{4}+\frac{66357}{64406}a^{3}-\frac{65214}{32203}a^{2}+\frac{72959}{64406}a+\frac{45817}{64406}$, $\frac{19741}{64406}a^{15}-\frac{1572}{32203}a^{14}+\frac{129051}{64406}a^{13}-\frac{8947}{32203}a^{12}+\frac{171245}{64406}a^{11}-\frac{12319}{64406}a^{10}-\frac{47261}{64406}a^{9}+\frac{8163}{32203}a^{8}+\frac{200402}{32203}a^{7}-\frac{75557}{64406}a^{6}+\frac{1096489}{64406}a^{5}-\frac{46205}{32203}a^{4}+\frac{14580}{32203}a^{3}+\frac{71127}{64406}a^{2}-\frac{623167}{64406}a+\frac{26311}{32203}$, $\frac{22411}{64406}a^{15}+\frac{543}{64406}a^{14}+\frac{134783}{64406}a^{13}-\frac{575}{32203}a^{12}+\frac{134447}{64406}a^{11}-\frac{14865}{64406}a^{10}-\frac{71819}{64406}a^{9}+\frac{2677}{32203}a^{8}+\frac{498955}{64406}a^{7}+\frac{3160}{32203}a^{6}+\frac{468443}{32203}a^{5}-\frac{78375}{64406}a^{4}-\frac{170073}{64406}a^{3}-\frac{38311}{64406}a^{2}-\frac{417975}{64406}a-\frac{12134}{32203}$, $\frac{4866}{32203}a^{15}+\frac{31505}{32203}a^{13}+\frac{38302}{32203}a^{11}-\frac{25145}{32203}a^{9}+\frac{90084}{32203}a^{7}+\frac{266059}{32203}a^{5}-\frac{43527}{32203}a^{3}-\frac{201283}{32203}a$
| 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: | \( 291.344930357 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 291.344930357 \cdot 1}{2\cdot\sqrt{503360547930148624}}\cr\approx \mathstrut & 0.202132932040 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^14 + 6*x^12 - 3*x^10 + 22*x^8 + 42*x^6 - 7*x^4 - 21*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 + 6*x^14 + 6*x^12 - 3*x^10 + 22*x^8 + 42*x^6 - 7*x^4 - 21*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 + 6*x^14 + 6*x^12 - 3*x^10 + 22*x^8 + 42*x^6 - 7*x^4 - 21*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 + 6*x^14 + 6*x^12 - 3*x^10 + 22*x^8 + 42*x^6 - 7*x^4 - 21*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
$C_2^8.S_4$ (as 16T1664):
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 6144 |
| The 105 conjugacy class representatives for $C_2^8.S_4$ |
| Character table for $C_2^8.S_4$ |
Intermediate fields
| 4.4.2777.1, 8.2.30846916.1, 8.6.177369767.1, 8.4.709479068.1 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(2777\)
| $\Q_{2777}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2777}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2777}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2777}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |