Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^22 + 14*x^20 - 48*x^18 + 164*x^16 - 560*x^14 + 1912*x^12 - 1120*x^10 + 656*x^8 - 384*x^6 + 224*x^4 - 128*x^2 + 64)
gp: K = bnfinit(y^24 - 4*y^22 + 14*y^20 - 48*y^18 + 164*y^16 - 560*y^14 + 1912*y^12 - 1120*y^10 + 656*y^8 - 384*y^6 + 224*y^4 - 128*y^2 + 64, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 4*x^22 + 14*x^20 - 48*x^18 + 164*x^16 - 560*x^14 + 1912*x^12 - 1120*x^10 + 656*x^8 - 384*x^6 + 224*x^4 - 128*x^2 + 64);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 4*x^22 + 14*x^20 - 48*x^18 + 164*x^16 - 560*x^14 + 1912*x^12 - 1120*x^10 + 656*x^8 - 384*x^6 + 224*x^4 - 128*x^2 + 64)
\( x^{24} - 4 x^{22} + 14 x^{20} - 48 x^{18} + 164 x^{16} - 560 x^{14} + 1912 x^{12} - 1120 x^{10} + \cdots + 64 \)
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: |
\(5887630061813314259984772021002174464\)
\(\medspace = 2^{66}\cdot 7^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.05\) | 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: | $2^{11/4}7^{5/6}\approx 34.04715710793443$ | ||
| Ramified primes: |
\(2\), \(7\)
| 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: | \(112=2^{4}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{112}(1,·)$, $\chi_{112}(83,·)$, $\chi_{112}(3,·)$, $\chi_{112}(97,·)$, $\chi_{112}(65,·)$, $\chi_{112}(9,·)$, $\chi_{112}(11,·)$, $\chi_{112}(17,·)$, $\chi_{112}(75,·)$, $\chi_{112}(19,·)$, $\chi_{112}(89,·)$, $\chi_{112}(25,·)$, $\chi_{112}(27,·)$, $\chi_{112}(107,·)$, $\chi_{112}(33,·)$, $\chi_{112}(67,·)$, $\chi_{112}(99,·)$, $\chi_{112}(81,·)$, $\chi_{112}(41,·)$, $\chi_{112}(43,·)$, $\chi_{112}(51,·)$, $\chi_{112}(73,·)$, $\chi_{112}(57,·)$, $\chi_{112}(59,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{1912}a^{14}+\frac{99}{239}$, $\frac{1}{1912}a^{15}+\frac{99}{239}a$, $\frac{1}{3824}a^{16}-\frac{70}{239}a^{2}$, $\frac{1}{3824}a^{17}-\frac{70}{239}a^{3}$, $\frac{1}{3824}a^{18}+\frac{99}{478}a^{4}$, $\frac{1}{3824}a^{19}+\frac{99}{478}a^{5}$, $\frac{1}{7648}a^{20}-\frac{35}{239}a^{6}$, $\frac{1}{7648}a^{21}-\frac{35}{239}a^{7}$, $\frac{1}{7648}a^{22}+\frac{99}{956}a^{8}$, $\frac{1}{7648}a^{23}+\frac{99}{956}a^{9}$
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
$C_{3}\times C_{6}$, which has order $18$ (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: |
\( \frac{41}{3824} a^{22} - \frac{287}{7648} a^{20} + \frac{123}{956} a^{18} - \frac{1681}{3824} a^{16} + \frac{1435}{956} a^{14} - \frac{41}{8} a^{12} + \frac{35}{2} a^{10} - \frac{1681}{956} a^{8} + \frac{246}{239} a^{6} - \frac{287}{478} a^{4} + \frac{82}{239} a^{2} - \frac{41}{239} \)
(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: |
$\frac{1}{3824}a^{16}+\frac{408}{239}a^{2}-1$, $\frac{3}{1912}a^{22}-\frac{1}{3824}a^{16}+\frac{8119}{956}a^{8}-\frac{408}{239}a^{2}$, $\frac{1}{1912}a^{14}+\frac{99}{239}$, $\frac{35}{3824}a^{22}-\frac{145}{3824}a^{20}+\frac{245}{1912}a^{18}-\frac{105}{239}a^{16}+\frac{1435}{956}a^{14}-\frac{41}{8}a^{12}+\frac{35}{2}a^{10}-\frac{2450}{239}a^{8}-\frac{493}{478}a^{6}-\frac{840}{239}a^{4}+\frac{490}{239}a^{2}-\frac{41}{239}$, $\frac{17}{7648}a^{22}-\frac{109}{7648}a^{20}+\frac{201}{3824}a^{18}-\frac{695}{3824}a^{16}+\frac{1189}{1912}a^{14}-\frac{17}{8}a^{12}+\frac{29}{4}a^{10}-\frac{12179}{956}a^{8}+\frac{3567}{478}a^{6}-\frac{2089}{478}a^{4}+\frac{611}{239}a^{2}-\frac{116}{239}$, $\frac{29}{7648}a^{23}+\frac{35}{3824}a^{22}-\frac{273}{7648}a^{20}+\frac{245}{1912}a^{18}-\frac{105}{239}a^{16}+\frac{1435}{956}a^{14}-\frac{41}{8}a^{12}+\frac{35}{2}a^{10}+\frac{19601}{956}a^{9}-\frac{2450}{239}a^{8}+\frac{2624}{239}a^{6}-\frac{840}{239}a^{4}+\frac{490}{239}a^{2}-\frac{280}{239}$, $\frac{29}{7648}a^{23}+\frac{1}{1912}a^{18}+\frac{19601}{956}a^{9}+\frac{1393}{478}a^{4}-1$, $\frac{41}{3824}a^{23}-\frac{3}{1912}a^{22}-\frac{287}{7648}a^{21}+\frac{123}{956}a^{19}-\frac{1681}{3824}a^{17}+\frac{1435}{956}a^{15}-\frac{41}{8}a^{13}+\frac{35}{2}a^{11}-\frac{1681}{956}a^{9}-\frac{8119}{956}a^{8}+\frac{246}{239}a^{7}-\frac{287}{478}a^{5}+\frac{82}{239}a^{3}-\frac{41}{239}a+1$, $\frac{1}{1912}a^{18}-\frac{3}{3824}a^{17}+\frac{1}{3824}a^{16}+\frac{1393}{478}a^{4}-\frac{985}{239}a^{3}+\frac{408}{239}a^{2}$, $\frac{35}{3824}a^{23}+\frac{3}{1912}a^{22}-\frac{35}{956}a^{21}+\frac{245}{1912}a^{19}-\frac{1}{1912}a^{18}-\frac{105}{239}a^{17}+\frac{1435}{956}a^{15}-\frac{41}{8}a^{13}+\frac{35}{2}a^{11}-\frac{2450}{239}a^{9}+\frac{8119}{956}a^{8}+\frac{1435}{239}a^{7}-\frac{840}{239}a^{5}-\frac{1393}{478}a^{4}+\frac{490}{239}a^{3}-\frac{280}{239}a$, $\frac{3}{1912}a^{23}-\frac{47}{3824}a^{22}+\frac{287}{7648}a^{20}-\frac{123}{956}a^{18}+\frac{1681}{3824}a^{16}-\frac{1435}{956}a^{14}+\frac{41}{8}a^{12}-\frac{35}{2}a^{10}+\frac{8119}{956}a^{9}-\frac{3219}{478}a^{8}-\frac{246}{239}a^{6}+\frac{287}{478}a^{4}-\frac{82}{239}a^{2}+\frac{41}{239}$
| 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: | \( 215040822.68029645 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 215040822.68029645 \cdot 18}{14\cdot\sqrt{5887630061813314259984772021002174464}}\cr\approx \mathstrut & 0.431373582817108 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^22 + 14*x^20 - 48*x^18 + 164*x^16 - 560*x^14 + 1912*x^12 - 1120*x^10 + 656*x^8 - 384*x^6 + 224*x^4 - 128*x^2 + 64)
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 - 4*x^22 + 14*x^20 - 48*x^18 + 164*x^16 - 560*x^14 + 1912*x^12 - 1120*x^10 + 656*x^8 - 384*x^6 + 224*x^4 - 128*x^2 + 64, 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 - 4*x^22 + 14*x^20 - 48*x^18 + 164*x^16 - 560*x^14 + 1912*x^12 - 1120*x^10 + 656*x^8 - 384*x^6 + 224*x^4 - 128*x^2 + 64);
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 - 4*x^22 + 14*x^20 - 48*x^18 + 164*x^16 - 560*x^14 + 1912*x^12 - 1120*x^10 + 656*x^8 - 384*x^6 + 224*x^4 - 128*x^2 + 64);
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
$C_2\times C_{12}$ (as 24T2):
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 $C_2\times C_{12}$ |
| Character table for $C_2\times C_{12}$ 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 | ${\href{/padicField/3.12.0.1}{12} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }^{2}$ | R | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.33.344 | $x^{12} + 56 x^{10} - 392 x^{9} + 226 x^{8} - 640 x^{7} - 2480 x^{6} + 3968 x^{5} + 1276 x^{4} - 384 x^{3} + 9280 x^{2} + 24224 x + 31544$ | $4$ | $3$ | $33$ | $C_{12}$ | $[3, 4]^{3}$ |
| 2.12.33.344 | $x^{12} + 56 x^{10} - 392 x^{9} + 226 x^{8} - 640 x^{7} - 2480 x^{6} + 3968 x^{5} + 1276 x^{4} - 384 x^{3} + 9280 x^{2} + 24224 x + 31544$ | $4$ | $3$ | $33$ | $C_{12}$ | $[3, 4]^{3}$ | |
|
\(7\)
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |