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Properties

Label 24.0.212...000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2.127\times 10^{33}$
Root discriminant \(24.47\)
Ramified primes $2,5,7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 5*x^22 + 19*x^20 - 66*x^18 + 221*x^16 - 358*x^14 + 530*x^12 - 723*x^10 + 793*x^8 - 157*x^6 + 31*x^4 - 6*x^2 + 1)
 
gp: K = bnfinit(y^24 - 5*y^22 + 19*y^20 - 66*y^18 + 221*y^16 - 358*y^14 + 530*y^12 - 723*y^10 + 793*y^8 - 157*y^6 + 31*y^4 - 6*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 5*x^22 + 19*x^20 - 66*x^18 + 221*x^16 - 358*x^14 + 530*x^12 - 723*x^10 + 793*x^8 - 157*x^6 + 31*x^4 - 6*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 5*x^22 + 19*x^20 - 66*x^18 + 221*x^16 - 358*x^14 + 530*x^12 - 723*x^10 + 793*x^8 - 157*x^6 + 31*x^4 - 6*x^2 + 1)
 

\( x^{24} - 5 x^{22} + 19 x^{20} - 66 x^{18} + 221 x^{16} - 358 x^{14} + 530 x^{12} - 723 x^{10} + 793 x^{8} - 157 x^{6} + 31 x^{4} - 6 x^{2} + 1 \) Copy content Toggle raw display

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:   \(2126907556454464000000000000000000\) \(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{16}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(24.47\)
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:   \(2\), \(5\), \(7\) Copy content Toggle raw display
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:  \(140=2^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{140}(1,·)$, $\chi_{140}(67,·)$, $\chi_{140}(71,·)$, $\chi_{140}(9,·)$, $\chi_{140}(11,·)$, $\chi_{140}(79,·)$, $\chi_{140}(81,·)$, $\chi_{140}(107,·)$, $\chi_{140}(121,·)$, $\chi_{140}(23,·)$, $\chi_{140}(29,·)$, $\chi_{140}(99,·)$, $\chi_{140}(37,·)$, $\chi_{140}(39,·)$, $\chi_{140}(43,·)$, $\chi_{140}(109,·)$, $\chi_{140}(93,·)$, $\chi_{140}(113,·)$, $\chi_{140}(51,·)$, $\chi_{140}(53,·)$, $\chi_{140}(137,·)$, $\chi_{140}(57,·)$, $\chi_{140}(123,·)$, $\chi_{140}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{164891}a^{18}+\frac{2495}{164891}a^{16}-\frac{40833}{164891}a^{14}+\frac{24303}{164891}a^{12}-\frac{43903}{164891}a^{10}+\frac{57450}{164891}a^{8}+\frac{47471}{164891}a^{6}+\frac{48407}{164891}a^{4}+\frac{75253}{164891}a^{2}-\frac{54614}{164891}$, $\frac{1}{164891}a^{19}+\frac{2495}{164891}a^{17}-\frac{40833}{164891}a^{15}+\frac{24303}{164891}a^{13}-\frac{43903}{164891}a^{11}+\frac{57450}{164891}a^{9}+\frac{47471}{164891}a^{7}+\frac{48407}{164891}a^{5}+\frac{75253}{164891}a^{3}-\frac{54614}{164891}a$, $\frac{1}{164891}a^{20}-\frac{57080}{164891}a^{10}+\frac{61964}{164891}$, $\frac{1}{164891}a^{21}-\frac{57080}{164891}a^{11}+\frac{61964}{164891}a$, $\frac{1}{164891}a^{22}-\frac{57080}{164891}a^{12}+\frac{61964}{164891}a^{2}$, $\frac{1}{164891}a^{23}-\frac{57080}{164891}a^{13}+\frac{61964}{164891}a^{3}$ Copy content Toggle raw display

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:   \( -\frac{73393}{164891} a^{23} + \frac{366965}{164891} a^{21} - \frac{1394378}{164891} a^{19} + \frac{4843938}{164891} a^{17} - \frac{16219853}{164891} a^{15} + \frac{26274694}{164891} a^{13} - \frac{38898290}{164891} a^{11} + \frac{53094640}{164891} a^{9} - \frac{58200649}{164891} a^{7} + \frac{11522701}{164891} a^{5} - \frac{2275183}{164891} a^{3} + \frac{440358}{164891} a \)  (order $20$) Copy content Toggle raw display
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:   $\frac{6079}{164891}a^{22}-\frac{23940}{164891}a^{20}+\frac{83160}{164891}a^{18}-\frac{278460}{164891}a^{16}+\frac{917429}{164891}a^{14}-\frac{749727}{164891}a^{12}+\frac{910980}{164891}a^{10}-\frac{999180}{164891}a^{8}+\frac{197820}{164891}a^{6}+\frac{4164412}{164891}a^{4}-\frac{824986}{164891}a^{2}+\frac{163631}{164891}$, $\frac{3315}{164891}a^{22}-\frac{12597}{164891}a^{20}+\frac{43758}{164891}a^{18}-\frac{146523}{164891}a^{16}+\frac{483135}{164891}a^{14}-\frac{351390}{164891}a^{12}+\frac{479349}{164891}a^{10}-\frac{525759}{164891}a^{8}+\frac{104091}{164891}a^{6}+\frac{2312194}{164891}a^{4}+\frac{3978}{164891}a^{2}-\frac{663}{164891}$, $\frac{597}{164891}a^{22}+\frac{220568}{164891}a^{12}+\frac{1870725}{164891}a^{2}$, $\frac{73393}{164891}a^{23}-\frac{366965}{164891}a^{21}+\frac{1394378}{164891}a^{19}-\frac{4843938}{164891}a^{17}+\frac{16219853}{164891}a^{15}-\frac{26274694}{164891}a^{13}+\frac{38898290}{164891}a^{11}-\frac{53094640}{164891}a^{9}+\frac{58200649}{164891}a^{7}-\frac{11522701}{164891}a^{5}+\frac{2275183}{164891}a^{3}-\frac{440358}{164891}a-1$, $\frac{73393}{164891}a^{23}-\frac{6079}{164891}a^{22}-\frac{366810}{164891}a^{21}+\frac{23940}{164891}a^{20}+\frac{1394378}{164891}a^{19}-\frac{83160}{164891}a^{18}-\frac{4843938}{164891}a^{17}+\frac{278460}{164891}a^{16}+\frac{16219853}{164891}a^{15}-\frac{917429}{164891}a^{14}-\frac{26274694}{164891}a^{13}+\frac{749727}{164891}a^{12}+\frac{38955004}{164891}a^{11}-\frac{910980}{164891}a^{10}-\frac{53094640}{164891}a^{9}+\frac{999180}{164891}a^{8}+\frac{58200649}{164891}a^{7}-\frac{197820}{164891}a^{6}-\frac{11522701}{164891}a^{5}-\frac{4164412}{164891}a^{4}+\frac{2275183}{164891}a^{3}+\frac{824986}{164891}a^{2}-\frac{69834}{164891}a+\frac{1260}{164891}$, $\frac{6521}{164891}a^{22}-\frac{23940}{164891}a^{20}+\frac{83160}{164891}a^{18}-\frac{278460}{164891}a^{16}+\frac{917429}{164891}a^{14}-\frac{585873}{164891}a^{12}+\frac{910980}{164891}a^{10}-\frac{999180}{164891}a^{8}+\frac{197820}{164891}a^{6}+\frac{4164412}{164891}a^{4}+\frac{675215}{164891}a^{2}+\frac{163631}{164891}$, $\frac{6145}{164891}a^{22}-\frac{23940}{164891}a^{20}+\frac{83160}{164891}a^{18}-\frac{278460}{164891}a^{16}+\frac{917429}{164891}a^{14}-\frac{724514}{164891}a^{12}+\frac{910980}{164891}a^{10}-\frac{999180}{164891}a^{8}+\frac{197820}{164891}a^{6}+\frac{4164412}{164891}a^{4}-\frac{362964}{164891}a^{2}+\frac{163631}{164891}$, $\frac{38730}{164891}a^{23}-\frac{71381}{164891}a^{22}-\frac{187195}{164891}a^{21}+\frac{350450}{164891}a^{20}+\frac{703595}{164891}a^{19}-\frac{1324030}{164891}a^{18}-\frac{2433426}{164891}a^{17}+\frac{4588392}{164891}a^{16}+\frac{8133300}{164891}a^{15}-\frac{15349171}{164891}a^{14}-\frac{12438785}{164891}a^{13}+\frac{133303}{911}a^{12}+\frac{18216010}{164891}a^{11}-\frac{35521040}{164891}a^{10}-\frac{24580640}{164891}a^{9}+\frac{48162100}{164891}a^{8}+\frac{26090063}{164891}a^{7}-\frac{51982306}{164891}a^{6}-\frac{961795}{164891}a^{5}+\frac{6088002}{164891}a^{4}+\frac{187195}{164891}a^{3}-\frac{1199376}{164891}a^{2}-\frac{32275}{164891}a+\frac{228181}{164891}$, $\frac{80}{911}a^{23}-\frac{58847}{164891}a^{22}-\frac{55179}{164891}a^{21}+\frac{294235}{164891}a^{20}+\frac{1056}{911}a^{19}-\frac{1118116}{164891}a^{18}-\frac{3536}{911}a^{17}+\frac{3883902}{164891}a^{16}+\frac{11647}{911}a^{15}-\frac{13005187}{164891}a^{14}-\frac{8480}{911}a^{13}+\frac{21067226}{164891}a^{12}+\frac{2037094}{164891}a^{11}-\frac{31188910}{164891}a^{10}-\frac{12688}{911}a^{9}+\frac{42540093}{164891}a^{8}+\frac{2512}{911}a^{7}-\frac{46665671}{164891}a^{6}+\frac{51687}{911}a^{5}+\frac{9238979}{164891}a^{4}+\frac{96}{911}a^{3}-\frac{1824257}{164891}a^{2}-\frac{373420}{164891}a+\frac{353082}{164891}$, $\frac{73703}{164891}a^{23}+\frac{287}{164891}a^{22}-\frac{389869}{164891}a^{21}+\frac{1503657}{164891}a^{19}-\frac{5252329}{164891}a^{17}+\frac{17629816}{164891}a^{15}-\frac{30870014}{164891}a^{13}+\frac{107140}{164891}a^{12}+\frac{45920783}{164891}a^{11}-\frac{63299266}{164891}a^{9}+\frac{71999151}{164891}a^{7}-\frac{26032727}{164891}a^{5}+\frac{2815084}{164891}a^{3}+\frac{1294568}{164891}a^{2}-\frac{545429}{164891}a$, $\frac{160669}{164891}a^{23}+\frac{40742}{164891}a^{22}-\frac{788799}{164891}a^{21}-\frac{203710}{164891}a^{20}+\frac{2979892}{164891}a^{19}+\frac{773943}{164891}a^{18}-\frac{10327892}{164891}a^{17}-\frac{2688972}{164891}a^{16}+\frac{34547813}{164891}a^{15}+\frac{9003982}{164891}a^{14}-\frac{54304836}{164891}a^{13}-\frac{14585636}{164891}a^{12}+\frac{79947102}{164891}a^{11}+\frac{21593260}{164891}a^{10}-\frac{108485808}{164891}a^{9}-\frac{29513180}{164891}a^{8}+\frac{116855970}{164891}a^{7}+\frac{32308406}{164891}a^{6}-\frac{13690055}{164891}a^{5}-\frac{6396494}{164891}a^{4}+\frac{2697017}{164891}a^{3}+\frac{1263002}{164891}a^{2}-\frac{513088}{164891}a-\frac{244452}{164891}$ Copy content Toggle raw display (assuming GRH)
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:  \( 18292450.943147723 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 18292450.943147723 \cdot 3}{20\cdot\sqrt{2126907556454464000000000000000000}}\cr\approx \mathstrut & 0.225240908955274 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 5*x^22 + 19*x^20 - 66*x^18 + 221*x^16 - 358*x^14 + 530*x^12 - 723*x^10 + 793*x^8 - 157*x^6 + 31*x^4 - 6*x^2 + 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 - 5*x^22 + 19*x^20 - 66*x^18 + 221*x^16 - 358*x^14 + 530*x^12 - 723*x^10 + 793*x^8 - 157*x^6 + 31*x^4 - 6*x^2 + 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 - 5*x^22 + 19*x^20 - 66*x^18 + 221*x^16 - 358*x^14 + 530*x^12 - 723*x^10 + 793*x^8 - 157*x^6 + 31*x^4 - 6*x^2 + 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 - 5*x^22 + 19*x^20 - 66*x^18 + 221*x^16 - 358*x^14 + 530*x^12 - 723*x^10 + 793*x^8 - 157*x^6 + 31*x^4 - 6*x^2 + 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

$C_2\times C_{12}$ (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{7})^+\), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 6.0.153664.1, 6.6.300125.1, 6.0.19208000.1, \(\Q(\zeta_{20})\), 12.0.368947264000000.1, 12.0.11259376953125.1, 12.12.46118408000000000.1

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 ${\href{/padicField/3.12.0.1}{12} }^{2}$ R R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{6}$ ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.1.0.1}{1} }^{24}$ ${\href{/padicField/43.4.0.1}{4} }^{6}$ ${\href{/padicField/47.12.0.1}{12} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $24$$2$$12$$24$
\(5\) Copy content Toggle raw display 5.12.9.2$x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.2$x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(7\) Copy content Toggle raw display 7.12.8.1$x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
7.12.8.1$x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$