Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 9*x^14 + 56*x^12 + 279*x^10 + 1111*x^8 + 6975*x^6 + 35000*x^4 + 140625*x^2 + 390625)
gp: K = bnfinit(y^16 + 9*y^14 + 56*y^12 + 279*y^10 + 1111*y^8 + 6975*y^6 + 35000*y^4 + 140625*y^2 + 390625, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 9*x^14 + 56*x^12 + 279*x^10 + 1111*x^8 + 6975*x^6 + 35000*x^4 + 140625*x^2 + 390625);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 9*x^14 + 56*x^12 + 279*x^10 + 1111*x^8 + 6975*x^6 + 35000*x^4 + 140625*x^2 + 390625)
\( x^{16} + 9x^{14} + 56x^{12} + 279x^{10} + 1111x^{8} + 6975x^{6} + 35000x^{4} + 140625x^{2} + 390625 \)
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: |
\(271737008656000000000000\)
\(\medspace = 2^{16}\cdot 5^{12}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.15\) | 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: | $2\cdot 5^{3/4}19^{1/2}\approx 29.149714088647936$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(380=2^{2}\cdot 5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{380}(1,·)$, $\chi_{380}(267,·)$, $\chi_{380}(77,·)$, $\chi_{380}(341,·)$, $\chi_{380}(343,·)$, $\chi_{380}(153,·)$, $\chi_{380}(37,·)$, $\chi_{380}(227,·)$, $\chi_{380}(229,·)$, $\chi_{380}(39,·)$, $\chi_{380}(303,·)$, $\chi_{380}(113,·)$, $\chi_{380}(379,·)$, $\chi_{380}(151,·)$, $\chi_{380}(189,·)$, $\chi_{380}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{7}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{27775}a^{10}-\frac{6}{25}a^{8}-\frac{4}{25}a^{6}-\frac{11}{25}a^{4}+\frac{1}{25}a^{2}+\frac{279}{1111}$, $\frac{1}{138875}a^{11}-\frac{6}{125}a^{9}+\frac{46}{125}a^{7}-\frac{61}{125}a^{5}+\frac{51}{125}a^{3}-\frac{1943}{5555}a$, $\frac{1}{694375}a^{12}+\frac{9}{694375}a^{10}+\frac{121}{625}a^{8}-\frac{11}{625}a^{6}+\frac{1}{625}a^{4}+\frac{279}{27775}a^{2}+\frac{56}{1111}$, $\frac{1}{3471875}a^{13}+\frac{9}{3471875}a^{11}+\frac{121}{3125}a^{9}-\frac{11}{3125}a^{7}+\frac{1}{3125}a^{5}+\frac{279}{138875}a^{3}+\frac{56}{5555}a$, $\frac{1}{17359375}a^{14}+\frac{9}{17359375}a^{12}+\frac{56}{17359375}a^{10}-\frac{3136}{15625}a^{8}+\frac{1}{15625}a^{6}+\frac{279}{694375}a^{4}+\frac{56}{27775}a^{2}+\frac{9}{1111}$, $\frac{1}{86796875}a^{15}+\frac{9}{86796875}a^{13}+\frac{56}{86796875}a^{11}-\frac{3136}{78125}a^{9}-\frac{31249}{78125}a^{7}+\frac{1389029}{3471875}a^{5}-\frac{55494}{138875}a^{3}+\frac{2231}{5555}a$
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
$C_{2}\times C_{8}$, which has order $16$ (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: |
\( \frac{9}{86796875} a^{15} + \frac{56}{86796875} a^{13} + \frac{279}{86796875} a^{11} + \frac{1}{78125} a^{9} + \frac{284}{78125} a^{7} + \frac{56}{138875} a^{5} + \frac{9}{5555} a^{3} + \frac{5}{1111} a \)
(order $20$)
| 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: |
$\frac{9}{694375}a^{14}+\frac{1}{27775}a^{12}+\frac{559}{694375}a^{4}-\frac{3024}{27775}a^{2}-1$, $\frac{9}{694375}a^{14}+\frac{1}{5555}a^{11}+\frac{559}{694375}a^{4}+\frac{2531}{5555}a$, $\frac{9}{694375}a^{14}-\frac{4}{138875}a^{13}+\frac{559}{694375}a^{4}-\frac{15679}{138875}a^{3}$, $\frac{843}{86796875}a^{15}+\frac{509}{17359375}a^{14}+\frac{10737}{86796875}a^{13}+\frac{4431}{17359375}a^{12}+\frac{31808}{86796875}a^{11}+\frac{279}{17359375}a^{10}+\frac{2}{78125}a^{9}+\frac{1}{15625}a^{8}+\frac{568}{78125}a^{7}+\frac{284}{15625}a^{6}+\frac{236308}{3471875}a^{5}+\frac{15959}{138875}a^{4}+\frac{32367}{138875}a^{3}+\frac{6832}{27775}a^{2}-\frac{443}{5555}a-\frac{1086}{1111}$, $\frac{449}{17359375}a^{15}+\frac{189}{3471875}a^{14}+\frac{376}{1578125}a^{13}+\frac{491}{3471875}a^{12}+\frac{12499}{17359375}a^{11}-\frac{56}{3471875}a^{10}+\frac{56}{15625}a^{9}+\frac{11}{3125}a^{8}+\frac{279}{15625}a^{7}-\frac{1}{3125}a^{6}+\frac{314164}{3471875}a^{5}+\frac{78118}{694375}a^{4}+\frac{8578}{12625}a^{3}+\frac{15399}{27775}a^{2}+\frac{10024}{5555}a-\frac{1156}{1111}$, $\frac{2227}{86796875}a^{15}-\frac{216}{17359375}a^{14}+\frac{9943}{86796875}a^{13}+\frac{3181}{17359375}a^{12}+\frac{30687}{86796875}a^{11}+\frac{15904}{17359375}a^{10}+\frac{278}{78125}a^{9}+\frac{1}{15625}a^{8}+\frac{827}{78125}a^{7}+\frac{284}{15625}a^{6}+\frac{311364}{3471875}a^{5}+\frac{841}{694375}a^{4}+\frac{61991}{138875}a^{3}+\frac{2576}{5555}a^{2}-\frac{643}{5555}a+\frac{4778}{1111}$, $\frac{2547}{17359375}a^{14}+\frac{15848}{17359375}a^{12}-\frac{1}{1111}a^{11}+\frac{78957}{17359375}a^{10}+\frac{283}{15625}a^{8}+\frac{2247}{15625}a^{6}+\frac{15848}{27775}a^{4}+\frac{2547}{1111}a^{2}-\frac{5864}{1111}a+\frac{7075}{1111}$
| 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: | \( 123279.066823 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 123279.066823 \cdot 16}{20\cdot\sqrt{271737008656000000000000}}\cr\approx \mathstrut & 0.459561752581 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 9*x^14 + 56*x^12 + 279*x^10 + 1111*x^8 + 6975*x^6 + 35000*x^4 + 140625*x^2 + 390625)
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 + 9*x^14 + 56*x^12 + 279*x^10 + 1111*x^8 + 6975*x^6 + 35000*x^4 + 140625*x^2 + 390625, 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 + 9*x^14 + 56*x^12 + 279*x^10 + 1111*x^8 + 6975*x^6 + 35000*x^4 + 140625*x^2 + 390625);
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 + 9*x^14 + 56*x^12 + 279*x^10 + 1111*x^8 + 6975*x^6 + 35000*x^4 + 140625*x^2 + 390625);
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
$C_2^2\times C_4$ (as 16T2):
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)
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4\times C_2^2$ |
| Character table for $C_4\times C_2^2$ |
Intermediate fields
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(5\)
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(19\)
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |