Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 9*x^18 + 17*x^12 - 576*x^6 + 4096)
gp: K = bnfinit(y^24 - 9*y^18 + 17*y^12 - 576*y^6 + 4096, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 9*x^18 + 17*x^12 - 576*x^6 + 4096);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 9*x^18 + 17*x^12 - 576*x^6 + 4096)
\( x^{24} - 9x^{18} + 17x^{12} - 576x^{6} + 4096 \)
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: |
\(34854715807867200628629234134286336\)
\(\medspace = 2^{24}\cdot 3^{36}\cdot 7^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.50\) | 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: | $2\cdot 3^{3/2}7^{1/2}\approx 27.49545416973504$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| 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: | \(252=2^{2}\cdot 3^{2}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(197,·)$, $\chi_{252}(71,·)$, $\chi_{252}(139,·)$, $\chi_{252}(13,·)$, $\chi_{252}(209,·)$, $\chi_{252}(211,·)$, $\chi_{252}(85,·)$, $\chi_{252}(155,·)$, $\chi_{252}(29,·)$, $\chi_{252}(223,·)$, $\chi_{252}(97,·)$, $\chi_{252}(167,·)$, $\chi_{252}(169,·)$, $\chi_{252}(41,·)$, $\chi_{252}(43,·)$, $\chi_{252}(239,·)$, $\chi_{252}(113,·)$, $\chi_{252}(83,·)$, $\chi_{252}(181,·)$, $\chi_{252}(55,·)$, $\chi_{252}(251,·)$, $\chi_{252}(125,·)$, $\chi_{252}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{6}-\frac{1}{5}$, $\frac{1}{10}a^{13}+\frac{3}{10}a^{7}-\frac{1}{10}a$, $\frac{1}{20}a^{14}+\frac{3}{20}a^{8}+\frac{9}{20}a^{2}$, $\frac{1}{40}a^{15}-\frac{17}{40}a^{9}+\frac{9}{40}a^{3}$, $\frac{1}{80}a^{16}+\frac{23}{80}a^{10}-\frac{31}{80}a^{4}$, $\frac{1}{160}a^{17}+\frac{23}{160}a^{11}+\frac{49}{160}a^{5}$, $\frac{1}{5440}a^{18}+\frac{7}{320}a^{12}+\frac{1}{320}a^{6}-\frac{9}{85}$, $\frac{1}{10880}a^{19}+\frac{7}{640}a^{13}+\frac{1}{640}a^{7}-\frac{9}{170}a$, $\frac{1}{21760}a^{20}+\frac{7}{1280}a^{14}-\frac{639}{1280}a^{8}+\frac{161}{340}a^{2}$, $\frac{1}{43520}a^{21}+\frac{7}{2560}a^{15}-\frac{639}{2560}a^{9}+\frac{161}{680}a^{3}$, $\frac{1}{87040}a^{22}+\frac{7}{5120}a^{16}+\frac{1921}{5120}a^{10}-\frac{519}{1360}a^{4}$, $\frac{1}{174080}a^{23}+\frac{7}{10240}a^{17}-\frac{3199}{10240}a^{11}-\frac{519}{2720}a^{5}$
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
$C_{7}$, which has order $7$ (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: |
\( \frac{1}{10880} a^{19} + \frac{7}{640} a^{13} + \frac{1}{640} a^{7} - \frac{9}{170} a \)
(order $36$)
| 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: |
$\frac{99}{87040}a^{22}-\frac{11}{5120}a^{16}-\frac{93}{5120}a^{10}-\frac{44}{85}a^{4}+1$, $\frac{91}{21760}a^{20}-\frac{3}{1280}a^{14}+\frac{91}{1280}a^{8}-\frac{819}{340}a^{2}+1$, $\frac{207}{174080}a^{23}-\frac{27}{21760}a^{20}-\frac{23}{10240}a^{17}+\frac{3}{1280}a^{14}+\frac{271}{10240}a^{11}-\frac{91}{1280}a^{8}-\frac{46}{85}a^{5}+\frac{48}{85}a^{2}$, $\frac{271}{174080}a^{23}-\frac{23}{10240}a^{17}+\frac{271}{10240}a^{11}-\frac{2439}{2720}a^{5}+1$, $\frac{1}{10880}a^{19}+\frac{7}{640}a^{13}+\frac{1}{640}a^{7}-\frac{9}{170}a-1$, $\frac{31}{4352}a^{20}-\frac{3}{1280}a^{14}+\frac{91}{1280}a^{8}-\frac{553}{170}a^{2}$, $\frac{3}{1360}a^{22}+\frac{91}{21760}a^{20}+\frac{19}{5440}a^{18}-\frac{3}{1280}a^{14}+\frac{1}{64}a^{12}+\frac{91}{1280}a^{8}-\frac{9}{64}a^{6}-\frac{1541}{1360}a^{4}-\frac{819}{340}a^{2}-\frac{137}{85}$, $\frac{271}{174080}a^{23}-\frac{1}{680}a^{21}-\frac{1}{10880}a^{19}-\frac{23}{10240}a^{17}-\frac{7}{640}a^{13}+\frac{271}{10240}a^{11}-\frac{1}{640}a^{7}-\frac{2439}{2720}a^{5}+\frac{287}{680}a^{3}+\frac{179}{170}a$, $\frac{99}{87040}a^{23}-\frac{271}{87040}a^{22}+\frac{1}{680}a^{21}+\frac{9}{1088}a^{18}-\frac{11}{5120}a^{17}+\frac{23}{5120}a^{16}-\frac{1}{64}a^{12}-\frac{93}{5120}a^{11}-\frac{271}{5120}a^{10}+\frac{9}{64}a^{6}-\frac{44}{85}a^{5}+\frac{2439}{1360}a^{4}-\frac{287}{680}a^{3}-\frac{64}{17}$, $\frac{99}{87040}a^{22}+\frac{3}{680}a^{21}+\frac{1}{10880}a^{20}-\frac{63}{10880}a^{19}-\frac{11}{5120}a^{16}+\frac{7}{640}a^{14}+\frac{7}{640}a^{13}-\frac{93}{5120}a^{10}+\frac{1}{640}a^{8}+\frac{1}{640}a^{7}-\frac{44}{85}a^{4}-\frac{1541}{680}a^{3}-\frac{9}{170}a^{2}+\frac{224}{85}a$, $\frac{1}{340}a^{22}+\frac{89}{10880}a^{19}-\frac{17}{640}a^{13}+\frac{89}{640}a^{7}-\frac{627}{340}a^{4}-\frac{801}{170}a$
| 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: | \( 79907554.2564404 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 79907554.2564404 \cdot 7}{36\cdot\sqrt{34854715807867200628629234134286336}}\cr\approx \mathstrut & 0.315072881552625 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 9*x^18 + 17*x^12 - 576*x^6 + 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 - 9*x^18 + 17*x^12 - 576*x^6 + 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 - 9*x^18 + 17*x^12 - 576*x^6 + 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 - 9*x^18 + 17*x^12 - 576*x^6 + 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 $C_2^2\times C_6$ |
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 | R | ${\href{/padicField/5.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{12}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.1.0.1}{1} }^{24}$ | ${\href{/padicField/41.6.0.1}{6} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
|
\(3\)
| 3.12.18.82 | $x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$ | $6$ | $2$ | $18$ | $C_6\times C_2$ | $[2]_{2}^{2}$ |
| 3.12.18.82 | $x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$ | $6$ | $2$ | $18$ | $C_6\times C_2$ | $[2]_{2}^{2}$ | |
|
\(7\)
| 7.12.6.1 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| 7.12.6.1 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |