Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1)
gp: K = bnfinit(y^24 - 13*y^20 + 143*y^16 - 336*y^12 + 663*y^8 - 26*y^4 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1)
\( x^{24} - 13x^{20} + 143x^{16} - 336x^{12} + 663x^{8} - 26x^{4} + 1 \)
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: |
\(4971225787269833056964369392336896\)
\(\medspace = 2^{48}\cdot 3^{12}\cdot 7^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.35\) | 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))
| |
| 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: | \(168=2^{3}\cdot 3\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{168}(1,·)$, $\chi_{168}(67,·)$, $\chi_{168}(65,·)$, $\chi_{168}(137,·)$, $\chi_{168}(11,·)$, $\chi_{168}(79,·)$, $\chi_{168}(107,·)$, $\chi_{168}(149,·)$, $\chi_{168}(23,·)$, $\chi_{168}(25,·)$, $\chi_{168}(71,·)$, $\chi_{168}(155,·)$, $\chi_{168}(29,·)$, $\chi_{168}(95,·)$, $\chi_{168}(163,·)$, $\chi_{168}(37,·)$, $\chi_{168}(43,·)$, $\chi_{168}(109,·)$, $\chi_{168}(113,·)$, $\chi_{168}(53,·)$, $\chi_{168}(121,·)$, $\chi_{168}(127,·)$, $\chi_{168}(151,·)$, $\chi_{168}(85,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}+\frac{1}{13}$, $\frac{1}{13}a^{13}+\frac{1}{13}a$, $\frac{1}{13}a^{14}+\frac{1}{13}a^{2}$, $\frac{1}{13}a^{15}+\frac{1}{13}a^{3}$, $\frac{1}{377}a^{16}+\frac{10}{377}a^{12}-\frac{12}{29}a^{8}+\frac{66}{377}a^{4}-\frac{120}{377}$, $\frac{1}{377}a^{17}+\frac{10}{377}a^{13}-\frac{12}{29}a^{9}+\frac{66}{377}a^{5}-\frac{120}{377}a$, $\frac{1}{377}a^{18}+\frac{10}{377}a^{14}-\frac{12}{29}a^{10}+\frac{66}{377}a^{6}-\frac{120}{377}a^{2}$, $\frac{1}{377}a^{19}+\frac{10}{377}a^{15}-\frac{12}{29}a^{11}+\frac{66}{377}a^{7}-\frac{120}{377}a^{3}$, $\frac{1}{16211}a^{20}-\frac{11}{16211}a^{16}-\frac{395}{16211}a^{12}-\frac{51}{16211}a^{8}+\frac{2}{16211}a^{4}-\frac{1656}{16211}$, $\frac{1}{16211}a^{21}-\frac{11}{16211}a^{17}-\frac{395}{16211}a^{13}-\frac{51}{16211}a^{9}+\frac{2}{16211}a^{5}-\frac{1656}{16211}a$, $\frac{1}{16211}a^{22}-\frac{11}{16211}a^{18}-\frac{395}{16211}a^{14}-\frac{51}{16211}a^{10}+\frac{2}{16211}a^{6}-\frac{1656}{16211}a^{2}$, $\frac{1}{16211}a^{23}-\frac{11}{16211}a^{19}-\frac{395}{16211}a^{15}-\frac{51}{16211}a^{11}+\frac{2}{16211}a^{7}-\frac{1656}{16211}a^{3}$
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
$C_{3}$, which has order $3$ (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: |
\( -\frac{7213}{16211} a^{23} + \frac{93791}{16211} a^{19} - \frac{1031701}{16211} a^{15} + \frac{2426101}{16211} a^{11} - \frac{4783341}{16211} a^{7} + \frac{187582}{16211} a^{3} \)
(order $24$)
| 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: |
$\frac{1164}{16211}a^{20}-\frac{15083}{16211}a^{16}+\frac{165870}{16211}a^{12}-\frac{384702}{16211}a^{8}+\frac{759730}{16211}a^{4}-\frac{582}{16211}$, $\frac{280}{16211}a^{20}-\frac{3682}{16211}a^{16}+\frac{40502}{16211}a^{12}-\frac{98689}{16211}a^{8}+\frac{187782}{16211}a^{4}-\frac{7364}{16211}$, $\frac{76}{377}a^{22}-\frac{22}{559}a^{20}-\frac{985}{377}a^{18}+\frac{285}{559}a^{16}+\frac{10830}{377}a^{14}-\frac{3135}{559}a^{12}-\frac{25118}{377}a^{10}+\frac{7271}{559}a^{8}+\frac{49507}{377}a^{6}-\frac{14535}{559}a^{4}-\frac{38}{377}a^{2}+\frac{11}{559}$, $\frac{7364}{16211}a^{23}-\frac{95452}{16211}a^{19}+\frac{1049370}{16211}a^{15}-\frac{2433802}{16211}a^{11}+\frac{4783643}{16211}a^{7}-\frac{3682}{16211}a^{3}+1$, $\frac{76}{377}a^{22}+\frac{1466}{16211}a^{21}-\frac{985}{377}a^{18}-\frac{19007}{16211}a^{17}+\frac{10830}{377}a^{14}+\frac{208905}{16211}a^{13}-\frac{25118}{377}a^{10}-\frac{484513}{16211}a^{9}+\frac{49507}{377}a^{6}+\frac{947556}{16211}a^{5}-\frac{38}{377}a^{2}-\frac{733}{16211}a$, $\frac{13262}{16211}a^{23}+\frac{56}{16211}a^{22}-\frac{171897}{16211}a^{19}-\frac{616}{16211}a^{18}+\frac{1889835}{16211}a^{15}+\frac{6561}{16211}a^{14}-\frac{4383091}{16211}a^{11}-\frac{2856}{16211}a^{10}+\frac{8619730}{16211}a^{7}+\frac{112}{16211}a^{6}-\frac{6631}{16211}a^{3}+\frac{81844}{16211}a^{2}-1$, $\frac{56}{16211}a^{22}+\frac{1785}{16211}a^{21}-\frac{22}{559}a^{20}-\frac{616}{16211}a^{18}-\frac{23161}{16211}a^{17}+\frac{285}{559}a^{16}+\frac{6561}{16211}a^{14}+\frac{254771}{16211}a^{13}-\frac{3135}{559}a^{12}-\frac{2856}{16211}a^{10}-\frac{594694}{16211}a^{9}+\frac{7271}{559}a^{8}+\frac{112}{16211}a^{6}+\frac{1181211}{16211}a^{5}-\frac{14535}{559}a^{4}+\frac{81844}{16211}a^{2}-\frac{46322}{16211}a+\frac{11}{559}$, $\frac{7364}{16211}a^{23}-\frac{1466}{16211}a^{21}+\frac{22}{16211}a^{20}-\frac{95452}{16211}a^{19}+\frac{19007}{16211}a^{17}-\frac{242}{16211}a^{16}+\frac{1049370}{16211}a^{15}-\frac{208905}{16211}a^{13}+\frac{2533}{16211}a^{12}-\frac{2433802}{16211}a^{11}+\frac{484513}{16211}a^{9}-\frac{1122}{16211}a^{8}+\frac{4783643}{16211}a^{7}-\frac{947556}{16211}a^{5}+\frac{44}{16211}a^{4}-\frac{3682}{16211}a^{3}+\frac{733}{16211}a+\frac{7213}{16211}$, $\frac{13262}{16211}a^{23}-\frac{56}{16211}a^{22}-\frac{171897}{16211}a^{19}+\frac{616}{16211}a^{18}+\frac{1889835}{16211}a^{15}-\frac{6561}{16211}a^{14}-\frac{4383091}{16211}a^{11}+\frac{2856}{16211}a^{10}+\frac{8619730}{16211}a^{7}-\frac{112}{16211}a^{6}-\frac{6631}{16211}a^{3}-\frac{81844}{16211}a^{2}+1$, $\frac{7213}{16211}a^{23}-\frac{95}{16211}a^{22}-\frac{1466}{16211}a^{21}-\frac{93791}{16211}a^{19}+\frac{1045}{16211}a^{18}+\frac{19007}{16211}a^{17}+\frac{1031701}{16211}a^{15}-\frac{11108}{16211}a^{14}-\frac{208905}{16211}a^{13}-\frac{2426101}{16211}a^{11}+\frac{4845}{16211}a^{10}+\frac{484513}{16211}a^{9}+\frac{4783341}{16211}a^{7}-\frac{190}{16211}a^{6}-\frac{947556}{16211}a^{5}-\frac{187582}{16211}a^{3}-\frac{102056}{16211}a^{2}+\frac{733}{16211}a$, $\frac{7364}{16211}a^{23}+\frac{5786}{16211}a^{22}-\frac{3}{1247}a^{21}-\frac{95452}{16211}a^{19}-\frac{75213}{16211}a^{18}+\frac{33}{1247}a^{17}+\frac{1049370}{16211}a^{15}+\frac{827343}{16211}a^{14}-\frac{4547}{16211}a^{13}-\frac{2433802}{16211}a^{11}-\frac{1943577}{16211}a^{10}+\frac{153}{1247}a^{9}+\frac{4783643}{16211}a^{7}+\frac{3835863}{16211}a^{6}-\frac{6}{1247}a^{5}-\frac{3682}{16211}a^{3}-\frac{150426}{16211}a^{2}-\frac{36423}{16211}a$
| 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: | \( 24013423.00725467 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 24013423.00725467 \cdot 3}{24\cdot\sqrt{4971225787269833056964369392336896}}\cr\approx \mathstrut & 0.161172432351781 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1)
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 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1, 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 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1);
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 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1);
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 | R | ${\href{/padicField/5.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $24$ | $4$ | $6$ | $48$ | |||
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |