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Properties

Label 24.0.228...000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2.284\times 10^{36}$
Root discriminant \(32.73\)
Ramified primes $2,5,7$
Class number $27$ (GRH)
Class group [3, 3, 3] (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 91*x^20 + 1391*x^16 + 2688*x^12 + 1287*x^8 + 182*x^4 + 1)
 
gp: K = bnfinit(y^24 + 91*y^20 + 1391*y^16 + 2688*y^12 + 1287*y^8 + 182*y^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 91*x^20 + 1391*x^16 + 2688*x^12 + 1287*x^8 + 182*x^4 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 91*x^20 + 1391*x^16 + 2688*x^12 + 1287*x^8 + 182*x^4 + 1)
 

\( x^{24} + 91x^{20} + 1391x^{16} + 2688x^{12} + 1287x^{8} + 182x^{4} + 1 \) 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:   \(2283749599146799148302336000000000000\) \(\medspace = 2^{48}\cdot 5^{12}\cdot 7^{16}\) 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:  \(32.73\)
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:  $2^{2}5^{1/2}7^{2/3}\approx 32.72982527225799$
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:  \(280=2^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(261,·)$, $\chi_{280}(51,·)$, $\chi_{280}(71,·)$, $\chi_{280}(9,·)$, $\chi_{280}(11,·)$, $\chi_{280}(239,·)$, $\chi_{280}(141,·)$, $\chi_{280}(79,·)$, $\chi_{280}(81,·)$, $\chi_{280}(211,·)$, $\chi_{280}(149,·)$, $\chi_{280}(151,·)$, $\chi_{280}(219,·)$, $\chi_{280}(29,·)$, $\chi_{280}(99,·)$, $\chi_{280}(39,·)$, $\chi_{280}(169,·)$, $\chi_{280}(109,·)$, $\chi_{280}(221,·)$, $\chi_{280}(179,·)$, $\chi_{280}(121,·)$, $\chi_{280}(249,·)$, $\chi_{280}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{8}-\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{9}-\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{10}-\frac{1}{3}a^{6}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{11}-\frac{1}{3}a^{7}-\frac{1}{3}a^{3}$, $\frac{1}{87}a^{16}-\frac{4}{29}a^{12}+\frac{10}{87}a^{8}+\frac{31}{87}a^{4}-\frac{4}{87}$, $\frac{1}{87}a^{17}-\frac{4}{29}a^{13}+\frac{10}{87}a^{9}+\frac{31}{87}a^{5}-\frac{4}{87}a$, $\frac{1}{87}a^{18}-\frac{4}{29}a^{14}+\frac{10}{87}a^{10}+\frac{31}{87}a^{6}-\frac{4}{87}a^{2}$, $\frac{1}{87}a^{19}-\frac{4}{29}a^{15}+\frac{10}{87}a^{11}+\frac{31}{87}a^{7}-\frac{4}{87}a^{3}$, $\frac{1}{11988687}a^{20}-\frac{44207}{11988687}a^{16}+\frac{544270}{11988687}a^{12}+\frac{2211407}{11988687}a^{8}+\frac{929836}{3996229}a^{4}-\frac{1401487}{11988687}$, $\frac{1}{11988687}a^{21}-\frac{44207}{11988687}a^{17}+\frac{544270}{11988687}a^{13}+\frac{2211407}{11988687}a^{9}+\frac{929836}{3996229}a^{5}-\frac{1401487}{11988687}a$, $\frac{1}{11988687}a^{22}-\frac{44207}{11988687}a^{18}+\frac{544270}{11988687}a^{14}+\frac{2211407}{11988687}a^{10}+\frac{929836}{3996229}a^{6}-\frac{1401487}{11988687}a^{2}$, $\frac{1}{11988687}a^{23}-\frac{44207}{11988687}a^{19}+\frac{544270}{11988687}a^{15}+\frac{2211407}{11988687}a^{11}+\frac{929836}{3996229}a^{7}-\frac{1401487}{11988687}a^{3}$ 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

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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:   \( -\frac{375523}{11988687} a^{21} - \frac{34178056}{11988687} a^{17} - \frac{522828208}{11988687} a^{13} - \frac{1015070498}{11988687} a^{9} - \frac{156565458}{3996229} a^{5} - \frac{46242242}{11988687} a \)  (order $8$) 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:   $\frac{1300859}{3996229}a^{23}+\frac{118548367}{3996229}a^{19}+\frac{1824920195}{3996229}a^{15}+\frac{3727776101}{3996229}a^{11}+\frac{2046894315}{3996229}a^{7}+\frac{322334222}{3996229}a^{3}$, $\frac{665846}{11988687}a^{22}+\frac{60836525}{11988687}a^{18}+\frac{316130941}{3996229}a^{14}+\frac{708403187}{3996229}a^{10}+\frac{1441531697}{11988687}a^{6}+\frac{97910943}{3996229}a^{2}$, $\frac{199640}{3996229}a^{22}+\frac{53956687}{11988687}a^{18}+\frac{783671744}{11988687}a^{14}+\frac{867353768}{11988687}a^{10}-\frac{460006726}{11988687}a^{6}-\frac{66756352}{3996229}a^{2}$, $\frac{30212}{137801}a^{22}+\frac{238228634}{11988687}a^{18}+\frac{3569113265}{11988687}a^{14}+\frac{1918743886}{3996229}a^{10}+\frac{398237995}{3996229}a^{6}-\frac{93018658}{11988687}a^{2}$, $\frac{1505218}{11988687}a^{22}+\frac{136873346}{11988687}a^{18}+\frac{2084519656}{11988687}a^{14}+\frac{1301545993}{3996229}a^{10}+\frac{1664184067}{11988687}a^{6}+\frac{177862147}{11988687}a^{2}$, $\frac{1253679}{3996229}a^{23}-\frac{6020}{97469}a^{22}+\frac{114069059}{3996229}a^{19}-\frac{1643771}{292407}a^{18}+\frac{5227289627}{11988687}a^{15}-\frac{8383274}{97469}a^{14}+\frac{10042358653}{11988687}a^{11}-\frac{48983498}{292407}a^{10}+\frac{4689687892}{11988687}a^{7}-\frac{24170447}{292407}a^{6}+\frac{602658088}{11988687}a^{3}-\frac{4001842}{292407}a^{2}+1$, $\frac{1253679}{3996229}a^{23}-\frac{375523}{11988687}a^{21}-\frac{129745}{3996229}a^{20}+\frac{114069059}{3996229}a^{19}-\frac{34178056}{11988687}a^{17}-\frac{11766893}{3996229}a^{16}+\frac{5227289627}{11988687}a^{15}-\frac{522828208}{11988687}a^{13}-\frac{176849582}{3996229}a^{12}+\frac{10042358653}{11988687}a^{11}-\frac{1015070498}{11988687}a^{9}-\frac{293748021}{3996229}a^{8}+\frac{4689687892}{11988687}a^{7}-\frac{156565458}{3996229}a^{5}-\frac{67833122}{3996229}a^{4}+\frac{602658088}{11988687}a^{3}-\frac{46242242}{11988687}a+\frac{6822310}{3996229}$, $\frac{6020}{97469}a^{22}-\frac{1654001}{11988687}a^{21}+\frac{1643771}{292407}a^{18}-\frac{50031297}{3996229}a^{17}+\frac{8383274}{97469}a^{14}-\frac{2262613313}{11988687}a^{13}+\frac{48983498}{292407}a^{10}-\frac{3873807392}{11988687}a^{9}+\frac{24170447}{292407}a^{6}-\frac{1185024337}{11988687}a^{5}+\frac{4001842}{292407}a^{2}-\frac{20336052}{3996229}a+1$, $\frac{1206499}{3996229}a^{23}+\frac{676508}{3996229}a^{22}-\frac{28106}{413403}a^{20}+\frac{109589751}{3996229}a^{19}+\frac{184271947}{11988687}a^{18}-\frac{850031}{137801}a^{16}+\frac{4979818669}{11988687}a^{15}+\frac{928480507}{3996229}a^{14}-\frac{12803363}{137801}a^{12}+\frac{8901389003}{11988687}a^{11}+\frac{4888877890}{11988687}a^{10}-\frac{65221975}{413403}a^{8}+\frac{3238692839}{11988687}a^{7}+\frac{1654720711}{11988687}a^{6}-\frac{6212011}{137801}a^{4}+\frac{238313510}{11988687}a^{3}+\frac{107250398}{11988687}a^{2}-\frac{360029}{413403}$, $\frac{3385514}{11988687}a^{22}-\frac{1115983}{11988687}a^{21}+\frac{328333}{11988687}a^{20}+\frac{308029121}{11988687}a^{18}-\frac{101572667}{11988687}a^{17}+\frac{9955072}{3996229}a^{16}+\frac{4704461419}{11988687}a^{14}-\frac{1553970910}{11988687}a^{13}+\frac{455519405}{11988687}a^{12}+\frac{9027288155}{11988687}a^{10}-\frac{1007797972}{3996229}a^{9}+\frac{288101257}{3996229}a^{8}+\frac{4219991518}{11988687}a^{6}-\frac{1460684701}{11988687}a^{5}+\frac{387845728}{11988687}a^{4}+\frac{556415846}{11988687}a^{2}-\frac{198329077}{11988687}a+\frac{42481205}{11988687}$, $\frac{7420264}{11988687}a^{23}-\frac{1654001}{11988687}a^{21}-\frac{284213}{3996229}a^{20}+\frac{674488891}{11988687}a^{19}-\frac{50031297}{3996229}a^{17}-\frac{77263817}{11988687}a^{16}+\frac{3417703403}{3996229}a^{15}-\frac{2262613313}{11988687}a^{13}-\frac{1156441270}{11988687}a^{12}+\frac{6305684264}{3996229}a^{11}-\frac{3873807392}{11988687}a^{9}-\frac{1848543682}{11988687}a^{8}+\frac{7847225527}{11988687}a^{7}-\frac{1185024337}{11988687}a^{5}-\frac{13047668}{413403}a^{4}+\frac{307068513}{3996229}a^{3}-\frac{24332281}{3996229}a+\frac{1384390}{3996229}$ 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:  \( 27409659.83250929 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 27409659.83250929 \cdot 27}{8\cdot\sqrt{2283749599146799148302336000000000000}}\cr\approx \mathstrut & 0.231745655457395 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 + 91*x^20 + 1391*x^16 + 2688*x^12 + 1287*x^8 + 182*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 + 91*x^20 + 1391*x^16 + 2688*x^12 + 1287*x^8 + 182*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 + 91*x^20 + 1391*x^16 + 2688*x^12 + 1287*x^8 + 182*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 + 91*x^20 + 1391*x^16 + 2688*x^12 + 1287*x^8 + 182*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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), 6.0.153664.1, 6.0.1229312.1, 6.6.1229312.1, 6.6.300125.1, 6.0.19208000.1, 6.0.153664000.1, 6.6.153664000.1, 8.0.40960000.1, 12.0.96717311574016.1, 12.0.368947264000000.1, 12.0.1511207993344000000.3, 12.0.23612624896000000.1, 12.0.1511207993344000000.1, 12.12.23612624896000000.1, 12.0.1511207993344000000.2

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 ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{12}$ ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{4}$ ${\href{/padicField/41.1.0.1}{1} }^{24}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $24$$4$$6$$48$
\(5\) Copy content Toggle raw display 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}$
\(7\) Copy content Toggle raw display 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}$