Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^20 + 16*x^16 + 16*x^12 + 256*x^8 - 2048*x^4 + 4096)
gp: K = bnfinit(y^24 - 8*y^20 + 16*y^16 + 16*y^12 + 256*y^8 - 2048*y^4 + 4096, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 8*x^20 + 16*x^16 + 16*x^12 + 256*x^8 - 2048*x^4 + 4096);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 8*x^20 + 16*x^16 + 16*x^12 + 256*x^8 - 2048*x^4 + 4096)
\( x^{24} - 8x^{20} + 16x^{16} + 16x^{12} + 256x^{8} - 2048x^{4} + 4096 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(14608995065922872079188435597787136\)
\(\medspace = 2^{52}\cdot 31^{4}\cdot 37^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.52\) | 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: | $2^{13/6}31^{1/2}37^{1/2}\approx 152.05943281385314$ | ||
| Ramified primes: |
\(2\), \(31\), \(37\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{9}$, $\frac{1}{8}a^{10}$, $\frac{1}{16}a^{11}-\frac{1}{4}a^{5}$, $\frac{1}{16}a^{12}$, $\frac{1}{32}a^{13}-\frac{1}{16}a^{10}-\frac{1}{8}a^{7}-\frac{1}{4}a^{4}$, $\frac{1}{32}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{64}a^{15}-\frac{1}{4}a^{3}$, $\frac{1}{64}a^{16}-\frac{1}{4}a^{4}$, $\frac{1}{128}a^{17}+\frac{1}{8}a^{5}$, $\frac{1}{256}a^{18}-\frac{1}{16}a^{10}+\frac{1}{16}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{512}a^{19}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}-\frac{3}{32}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{512}a^{20}-\frac{1}{32}a^{12}+\frac{1}{32}a^{8}-\frac{1}{4}a^{4}$, $\frac{1}{1024}a^{21}-\frac{1}{64}a^{13}-\frac{1}{16}a^{10}+\frac{1}{64}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}$, $\frac{1}{2048}a^{22}-\frac{1}{128}a^{16}+\frac{1}{128}a^{14}-\frac{7}{128}a^{10}-\frac{1}{16}a^{6}-\frac{1}{8}a^{4}$, $\frac{1}{4096}a^{23}-\frac{1}{1024}a^{20}-\frac{1}{256}a^{17}-\frac{1}{128}a^{16}+\frac{1}{256}a^{15}-\frac{1}{64}a^{14}-\frac{1}{64}a^{12}-\frac{7}{256}a^{11}-\frac{1}{16}a^{9}-\frac{1}{64}a^{8}+\frac{3}{32}a^{7}-\frac{1}{16}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$
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
$C_{4}$, which has order $4$ (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: |
\( \frac{3}{1024} a^{21} - \frac{1}{64} a^{17} - \frac{1}{64} a^{13} + \frac{3}{64} a^{9} + \frac{7}{8} a^{5} - 3 a \)
(order $8$)
| 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: |
$\frac{1}{1024}a^{21}-\frac{3}{128}a^{17}+\frac{1}{64}a^{13}+\frac{9}{64}a^{9}+\frac{1}{2}a^{5}-6a$, $\frac{5}{2048}a^{23}+\frac{1}{512}a^{21}-\frac{5}{512}a^{19}-\frac{1}{64}a^{17}-\frac{1}{128}a^{15}+\frac{1}{128}a^{11}+\frac{1}{32}a^{9}+\frac{25}{32}a^{7}+\frac{3}{4}a^{5}-2a^{3}-\frac{7}{2}a+1$, $\frac{1}{1024}a^{22}+\frac{3}{1024}a^{21}+\frac{1}{256}a^{18}-\frac{1}{64}a^{17}-\frac{1}{64}a^{14}-\frac{1}{64}a^{13}-\frac{3}{64}a^{10}+\frac{3}{64}a^{9}+\frac{3}{16}a^{6}+\frac{7}{8}a^{5}+a^{2}-3a+1$, $\frac{7}{2048}a^{22}-\frac{1}{64}a^{18}-\frac{1}{128}a^{16}-\frac{1}{128}a^{14}-\frac{1}{128}a^{10}+\frac{17}{16}a^{6}+\frac{3}{8}a^{4}-3a^{2}-2$, $\frac{5}{2048}a^{23}+\frac{1}{256}a^{21}-\frac{3}{256}a^{19}-\frac{3}{128}a^{17}-\frac{1}{128}a^{15}+\frac{5}{128}a^{11}+\frac{1}{16}a^{9}+\frac{3}{4}a^{7}+\frac{9}{8}a^{5}-\frac{11}{4}a^{3}-5a$, $\frac{7}{4096}a^{23}+\frac{3}{1024}a^{22}+\frac{1}{512}a^{21}+\frac{7}{1024}a^{20}-\frac{3}{256}a^{19}-\frac{5}{256}a^{18}-\frac{5}{256}a^{17}-\frac{5}{128}a^{16}-\frac{1}{256}a^{15}-\frac{1}{64}a^{12}+\frac{15}{256}a^{11}+\frac{3}{64}a^{10}+\frac{3}{32}a^{9}+\frac{7}{64}a^{8}+\frac{19}{32}a^{7}+\frac{17}{16}a^{6}+\frac{11}{16}a^{5}+\frac{9}{4}a^{4}-\frac{11}{4}a^{3}-\frac{17}{4}a^{2}-\frac{9}{2}a-9$, $\frac{13}{4096}a^{23}-\frac{1}{512}a^{22}+\frac{1}{512}a^{21}-\frac{3}{1024}a^{20}-\frac{3}{256}a^{19}-\frac{1}{256}a^{17}+\frac{1}{128}a^{16}-\frac{3}{256}a^{15}+\frac{1}{64}a^{14}+\frac{1}{64}a^{12}+\frac{5}{256}a^{11}+\frac{1}{32}a^{10}-\frac{1}{32}a^{9}-\frac{3}{64}a^{8}+\frac{29}{32}a^{7}-\frac{1}{2}a^{6}+\frac{7}{16}a^{5}-\frac{3}{4}a^{4}-\frac{9}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a+2$, $\frac{1}{1024}a^{21}-\frac{1}{512}a^{19}-\frac{1}{64}a^{13}+\frac{1}{32}a^{11}+\frac{1}{64}a^{9}-\frac{1}{32}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{3}+\frac{1}{2}a+1$, $\frac{1}{1024}a^{21}-\frac{1}{512}a^{19}-\frac{1}{64}a^{13}+\frac{1}{32}a^{11}+\frac{1}{64}a^{9}-\frac{1}{32}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{3}+\frac{1}{2}a-1$, $\frac{5}{2048}a^{23}-\frac{3}{2048}a^{22}+\frac{1}{512}a^{21}-\frac{5}{512}a^{19}-\frac{1}{128}a^{17}-\frac{1}{128}a^{16}-\frac{1}{128}a^{15}+\frac{1}{128}a^{14}+\frac{1}{128}a^{11}+\frac{5}{128}a^{10}+\frac{1}{32}a^{9}+\frac{25}{32}a^{7}-\frac{5}{16}a^{6}+\frac{3}{8}a^{5}+\frac{3}{8}a^{4}-2a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a-2$, $\frac{9}{4096}a^{23}-\frac{1}{256}a^{22}-\frac{1}{256}a^{21}-\frac{1}{1024}a^{20}-\frac{3}{512}a^{19}+\frac{1}{64}a^{18}+\frac{7}{256}a^{17}+\frac{3}{128}a^{16}-\frac{3}{256}a^{15}+\frac{1}{64}a^{14}-\frac{1}{64}a^{12}-\frac{7}{256}a^{11}-\frac{1}{8}a^{9}-\frac{9}{64}a^{8}+\frac{5}{8}a^{7}-\frac{5}{4}a^{6}-\frac{21}{16}a^{5}-\frac{3}{4}a^{4}-\frac{3}{4}a^{3}+\frac{11}{4}a^{2}+6a+6$
| 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: | \( 64723765.21921788 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 64723765.21921788 \cdot 4}{8\cdot\sqrt{14608995065922872079188435597787136}}\cr\approx \mathstrut & 1.01363600199058 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^20 + 16*x^16 + 16*x^12 + 256*x^8 - 2048*x^4 + 4096)
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^24 - 8*x^20 + 16*x^16 + 16*x^12 + 256*x^8 - 2048*x^4 + 4096, 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^24 - 8*x^20 + 16*x^16 + 16*x^12 + 256*x^8 - 2048*x^4 + 4096);
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^24 - 8*x^20 + 16*x^16 + 16*x^12 + 256*x^8 - 2048*x^4 + 4096);
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
$C_2^3\times S_4$ (as 24T400):
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 192 |
| The 40 conjugacy class representatives for $C_2^3\times S_4$ |
| Character table for $C_2^3\times S_4$ is not computed |
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]
Sibling fields
| Degree 24 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 | ${\href{/padicField/3.6.0.1}{6} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | R | R | ${\href{/padicField/41.3.0.1}{3} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.26.64 | $x^{12} + 2 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$ | $12$ | $1$ | $26$ | $S_3 \times C_2^2$ | $[2, 3]_{3}^{2}$ |
| 2.12.26.64 | $x^{12} + 2 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$ | $12$ | $1$ | $26$ | $S_3 \times C_2^2$ | $[2, 3]_{3}^{2}$ | |
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(37\)
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |