Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^22 + 32*x^20 - 168*x^18 + 880*x^16 - 4608*x^14 + 24128*x^12 - 18432*x^10 + 14080*x^8 - 10752*x^6 + 8192*x^4 - 6144*x^2 + 4096)
gp: K = bnfinit(y^24 - 6*y^22 + 32*y^20 - 168*y^18 + 880*y^16 - 4608*y^14 + 24128*y^12 - 18432*y^10 + 14080*y^8 - 10752*y^6 + 8192*y^4 - 6144*y^2 + 4096, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 6*x^22 + 32*x^20 - 168*x^18 + 880*x^16 - 4608*x^14 + 24128*x^12 - 18432*x^10 + 14080*x^8 - 10752*x^6 + 8192*x^4 - 6144*x^2 + 4096);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 6*x^22 + 32*x^20 - 168*x^18 + 880*x^16 - 4608*x^14 + 24128*x^12 - 18432*x^10 + 14080*x^8 - 10752*x^6 + 8192*x^4 - 6144*x^2 + 4096)
\( x^{24} - 6 x^{22} + 32 x^{20} - 168 x^{18} + 880 x^{16} - 4608 x^{14} + 24128 x^{12} - 18432 x^{10} + \cdots + 4096 \)
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: |
\(1338692086804556824969216000000000000\)
\(\medspace = 2^{36}\cdot 5^{12}\cdot 7^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.01\) | 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: | $2^{3/2}5^{1/2}7^{5/6}\approx 32.00946108272415$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $24$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(280=2^{3}\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(131,·)$, $\chi_{280}(179,·)$, $\chi_{280}(211,·)$, $\chi_{280}(129,·)$, $\chi_{280}(9,·)$, $\chi_{280}(11,·)$, $\chi_{280}(209,·)$, $\chi_{280}(19,·)$, $\chi_{280}(139,·)$, $\chi_{280}(121,·)$, $\chi_{280}(89,·)$, $\chi_{280}(219,·)$, $\chi_{280}(99,·)$, $\chi_{280}(251,·)$, $\chi_{280}(81,·)$, $\chi_{280}(41,·)$, $\chi_{280}(171,·)$, $\chi_{280}(241,·)$, $\chi_{280}(51,·)$, $\chi_{280}(201,·)$, $\chi_{280}(249,·)$, $\chi_{280}(59,·)$, $\chi_{280}(169,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{48256}a^{14}-\frac{144}{377}$, $\frac{1}{48256}a^{15}-\frac{144}{377}a$, $\frac{1}{96512}a^{16}-\frac{72}{377}a^{2}$, $\frac{1}{96512}a^{17}-\frac{72}{377}a^{3}$, $\frac{1}{193024}a^{18}-\frac{36}{377}a^{4}$, $\frac{1}{193024}a^{19}-\frac{36}{377}a^{5}$, $\frac{1}{386048}a^{20}-\frac{18}{377}a^{6}$, $\frac{1}{386048}a^{21}-\frac{18}{377}a^{7}$, $\frac{1}{772096}a^{22}-\frac{9}{377}a^{8}$, $\frac{1}{772096}a^{23}-\frac{9}{377}a^{9}$
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
$C_{3}\times C_{3}$, which has order $9$ (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: |
\( -\frac{9}{48256} a^{22} + \frac{27}{24128} a^{20} - \frac{9}{1508} a^{18} + \frac{189}{6032} a^{16} - \frac{495}{3016} a^{14} + \frac{55}{64} a^{12} - \frac{9}{2} a^{10} + \frac{1296}{377} a^{8} - \frac{990}{377} a^{6} + \frac{756}{377} a^{4} - \frac{576}{377} a^{2} + \frac{432}{377} \)
(order $14$)
| 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: |
$\frac{51}{386048}a^{22}-\frac{17}{24128}a^{20}+\frac{357}{96512}a^{18}-\frac{935}{48256}a^{16}+\frac{153}{1508}a^{14}-\frac{17}{32}a^{12}+\frac{89}{32}a^{10}-\frac{935}{3016}a^{8}+\frac{357}{1508}a^{6}-\frac{68}{377}a^{4}+\frac{51}{377}a^{2}-\frac{34}{377}$, $\frac{3}{193024}a^{18}+\frac{646}{377}a^{4}+1$, $\frac{21}{772096}a^{22}-\frac{1}{96512}a^{16}+\frac{17711}{6032}a^{8}-\frac{987}{754}a^{2}$, $\frac{9}{48256}a^{22}-\frac{55}{48256}a^{20}+\frac{1153}{193024}a^{18}-\frac{189}{6032}a^{16}+\frac{7919}{48256}a^{14}-\frac{55}{64}a^{12}+\frac{9}{2}a^{10}-\frac{1296}{377}a^{8}+\frac{1155}{3016}a^{6}-\frac{2037}{1508}a^{4}+\frac{775}{754}a^{2}-\frac{288}{377}$, $\frac{1}{48256}a^{20}-\frac{5}{193024}a^{18}+\frac{1}{96512}a^{16}+\frac{6765}{3016}a^{6}-\frac{4181}{1508}a^{4}+\frac{987}{754}a^{2}$, $\frac{17}{386048}a^{23}-\frac{93}{386048}a^{22}+\frac{55}{48256}a^{20}-\frac{1155}{193024}a^{18}+\frac{3025}{96512}a^{16}-\frac{495}{3016}a^{14}+\frac{55}{64}a^{12}-\frac{9}{2}a^{10}+\frac{28657}{6032}a^{9}-\frac{7343}{3016}a^{8}-\frac{1155}{3016}a^{6}+\frac{110}{377}a^{4}-\frac{165}{754}a^{2}+\frac{55}{377}$, $\frac{9}{48256}a^{22}-\frac{27}{24128}a^{20}+\frac{1149}{193024}a^{18}-\frac{189}{6032}a^{16}-\frac{1}{48256}a^{15}+\frac{495}{3016}a^{14}-\frac{55}{64}a^{12}+\frac{9}{2}a^{10}-\frac{1296}{377}a^{8}+\frac{990}{377}a^{6}-\frac{1402}{377}a^{4}+\frac{576}{377}a^{2}-\frac{610}{377}a-\frac{432}{377}$, $\frac{17}{386048}a^{23}+\frac{3}{193024}a^{18}+\frac{28657}{6032}a^{9}+\frac{646}{377}a^{4}-1$, $\frac{17}{386048}a^{23}-\frac{53}{193024}a^{22}-\frac{17}{193024}a^{21}+\frac{89}{48256}a^{20}+\frac{1}{14848}a^{19}-\frac{1869}{193024}a^{18}-\frac{5}{96512}a^{17}+\frac{2449}{48256}a^{16}+\frac{1}{48256}a^{15}-\frac{12817}{48256}a^{14}+\frac{89}{64}a^{12}-\frac{233}{32}a^{10}+\frac{28657}{6032}a^{9}+\frac{51263}{6032}a^{8}-\frac{28657}{3016}a^{7}-\frac{1869}{3016}a^{6}+\frac{421}{58}a^{5}+\frac{178}{377}a^{4}-\frac{4181}{754}a^{3}+\frac{2317}{754}a^{2}+\frac{610}{377}a-\frac{521}{377}$, $\frac{1}{2048}a^{23}+\frac{199}{772096}a^{22}-\frac{9}{3016}a^{21}-\frac{89}{48256}a^{20}+\frac{3011}{193024}a^{19}+\frac{1}{104}a^{18}-\frac{3961}{48256}a^{17}-\frac{153}{3016}a^{16}+\frac{10367}{24128}a^{15}+\frac{6407}{24128}a^{14}-\frac{9}{4}a^{13}-\frac{89}{64}a^{12}+\frac{377}{32}a^{11}+\frac{233}{32}a^{10}-9a^{9}-\frac{62209}{6032}a^{8}+\frac{378}{377}a^{7}+\frac{1869}{3016}a^{6}-\frac{6049}{754}a^{5}-\frac{201}{26}a^{4}-\frac{1165}{754}a^{3}-\frac{360}{377}a^{2}-\frac{1741}{377}a-\frac{1309}{377}$, $\frac{21}{772096}a^{22}+\frac{21}{386048}a^{21}+\frac{21}{386048}a^{20}+\frac{1}{24128}a^{19}+\frac{1}{24128}a^{18}+\frac{3}{96512}a^{17}+\frac{1}{96512}a^{16}+\frac{17711}{6032}a^{8}+\frac{17711}{3016}a^{7}+\frac{17711}{3016}a^{6}+\frac{6765}{1508}a^{5}+\frac{6765}{1508}a^{4}+\frac{1292}{377}a^{3}+\frac{987}{754}a^{2}$
| 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: | \( 69523271.83090398 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 69523271.83090398 \cdot 9}{14\cdot\sqrt{1338692086804556824969216000000000000}}\cr\approx \mathstrut & 0.146238844949408 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^22 + 32*x^20 - 168*x^18 + 880*x^16 - 4608*x^14 + 24128*x^12 - 18432*x^10 + 14080*x^8 - 10752*x^6 + 8192*x^4 - 6144*x^2 + 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 - 6*x^22 + 32*x^20 - 168*x^18 + 880*x^16 - 4608*x^14 + 24128*x^12 - 18432*x^10 + 14080*x^8 - 10752*x^6 + 8192*x^4 - 6144*x^2 + 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 - 6*x^22 + 32*x^20 - 168*x^18 + 880*x^16 - 4608*x^14 + 24128*x^12 - 18432*x^10 + 14080*x^8 - 10752*x^6 + 8192*x^4 - 6144*x^2 + 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 - 6*x^22 + 32*x^20 - 168*x^18 + 880*x^16 - 4608*x^14 + 24128*x^12 - 18432*x^10 + 14080*x^8 - 10752*x^6 + 8192*x^4 - 6144*x^2 + 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
$C_2^2\times C_6$ (as 24T3):
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 24 |
| The 24 conjugacy class representatives for $C_2^2\times C_6$ |
| Character table for $C_2^2\times C_6$ 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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{4}$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
| 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(5\)
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
| 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |