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Properties

Label 24.0.251...000.2
Degree $24$
Signature $[0, 12]$
Discriminant $2.518\times 10^{36}$
Root discriminant \(32.86\)
Ramified primes $2,3,5$
Class number $14$ (GRH)
Class group [14] (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 + 144*x^18 + 20672*x^12 + 9216*x^6 + 4096)
 
gp: K = bnfinit(y^24 + 144*y^18 + 20672*y^12 + 9216*y^6 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 144*x^18 + 20672*x^12 + 9216*x^6 + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 144*x^18 + 20672*x^12 + 9216*x^6 + 4096)
 

\( x^{24} + 144x^{18} + 20672x^{12} + 9216x^{6} + 4096 \) 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:   \(2518170116818978404827136000000000000\) \(\medspace = 2^{36}\cdot 3^{36}\cdot 5^{12}\) 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:  \(32.86\)
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:  $2^{3/2}3^{3/2}5^{1/2}\approx 32.863353450309965$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(360=2^{3}\cdot 3^{2}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{360}(1,·)$, $\chi_{360}(161,·)$, $\chi_{360}(329,·)$, $\chi_{360}(269,·)$, $\chi_{360}(109,·)$, $\chi_{360}(209,·)$, $\chi_{360}(149,·)$, $\chi_{360}(89,·)$, $\chi_{360}(281,·)$, $\chi_{360}(29,·)$, $\chi_{360}(229,·)$, $\chi_{360}(289,·)$, $\chi_{360}(101,·)$, $\chi_{360}(241,·)$, $\chi_{360}(41,·)$, $\chi_{360}(301,·)$, $\chi_{360}(349,·)$, $\chi_{360}(49,·)$, $\chi_{360}(221,·)$, $\chi_{360}(181,·)$, $\chi_{360}(169,·)$, $\chi_{360}(121,·)$, $\chi_{360}(61,·)$, $\chi_{360}(341,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{512}a^{12}-\frac{3}{64}a^{6}+\frac{1}{8}$, $\frac{1}{512}a^{13}-\frac{3}{64}a^{7}+\frac{1}{8}a$, $\frac{1}{1024}a^{14}-\frac{3}{128}a^{8}+\frac{1}{16}a^{2}$, $\frac{1}{1024}a^{15}-\frac{3}{128}a^{9}+\frac{1}{16}a^{3}$, $\frac{1}{2048}a^{16}-\frac{3}{256}a^{10}+\frac{1}{32}a^{4}$, $\frac{1}{2048}a^{17}-\frac{3}{256}a^{11}+\frac{1}{32}a^{5}$, $\frac{1}{1323008}a^{18}+\frac{987}{2584}$, $\frac{1}{1323008}a^{19}+\frac{987}{2584}a$, $\frac{1}{2646016}a^{20}+\frac{987}{5168}a^{2}$, $\frac{1}{2646016}a^{21}+\frac{987}{5168}a^{3}$, $\frac{1}{5292032}a^{22}+\frac{987}{10336}a^{4}$, $\frac{1}{5292032}a^{23}+\frac{987}{10336}a^{5}$ 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

$C_{14}$, which has order $14$ (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:   \( \frac{987}{5292032} a^{22} + \frac{55}{2048} a^{16} + \frac{987}{256} a^{10} + \frac{8883}{5168} a^{4} \)  (order $18$) 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:   $\frac{1}{2646016}a^{22}-\frac{5473}{5168}a^{4}$, $\frac{495}{2646016}a^{22}+\frac{55}{2048}a^{16}+\frac{987}{256}a^{10}+\frac{55}{10336}a^{4}+1$, $\frac{189}{1323008}a^{20}-\frac{9}{82688}a^{18}+\frac{21}{1024}a^{14}-\frac{1}{64}a^{12}+\frac{377}{128}a^{8}-\frac{9}{4}a^{6}+\frac{21}{5168}a^{2}-\frac{324}{323}$, $\frac{495}{2646016}a^{22}+\frac{1}{1323008}a^{19}+\frac{55}{2048}a^{16}+\frac{987}{256}a^{10}+\frac{55}{10336}a^{4}-\frac{4181}{2584}a$, $\frac{1597}{5292032}a^{22}+\frac{1}{2646016}a^{20}+\frac{9}{82688}a^{18}+\frac{89}{2048}a^{16}+\frac{1}{64}a^{12}+\frac{1597}{256}a^{10}+\frac{9}{4}a^{6}+\frac{14373}{5168}a^{4}-\frac{6765}{5168}a^{2}+\frac{324}{323}$, $\frac{3}{5292032}a^{22}+\frac{1}{2646016}a^{20}-\frac{89}{1323008}a^{18}-\frac{5}{512}a^{12}-\frac{89}{64}a^{6}-\frac{17711}{10336}a^{4}-\frac{6765}{5168}a^{2}-\frac{801}{1292}$, $\frac{1597}{5292032}a^{22}-\frac{117}{661504}a^{19}+\frac{89}{2048}a^{16}-\frac{13}{512}a^{13}+\frac{1597}{256}a^{10}-\frac{233}{64}a^{7}+\frac{14373}{5168}a^{4}-\frac{13}{2584}a$, $\frac{987}{5292032}a^{22}+\frac{9}{38912}a^{21}+\frac{1}{2646016}a^{20}+\frac{55}{2048}a^{16}+\frac{17}{512}a^{15}+\frac{987}{256}a^{10}+\frac{305}{64}a^{9}+\frac{8883}{5168}a^{4}+\frac{1}{152}a^{3}-\frac{6765}{5168}a^{2}$, $\frac{987}{5292032}a^{22}+\frac{1}{1323008}a^{21}-\frac{189}{1323008}a^{20}+\frac{55}{2048}a^{16}-\frac{21}{1024}a^{14}+\frac{987}{256}a^{10}-\frac{377}{128}a^{8}+\frac{8883}{5168}a^{4}-\frac{5473}{2584}a^{3}-\frac{21}{5168}a^{2}$, $\frac{495}{2646016}a^{22}-\frac{987}{2646016}a^{21}-\frac{3}{2646016}a^{20}+\frac{189}{661504}a^{19}-\frac{9}{82688}a^{18}+\frac{55}{2048}a^{16}-\frac{55}{1024}a^{15}+\frac{21}{512}a^{13}-\frac{1}{64}a^{12}+\frac{987}{256}a^{10}-\frac{987}{128}a^{9}+\frac{377}{64}a^{7}-\frac{9}{4}a^{6}+\frac{55}{10336}a^{4}-\frac{8883}{2584}a^{3}+\frac{17711}{5168}a^{2}+\frac{21}{2584}a-\frac{324}{323}$, $\frac{495}{2646016}a^{23}-\frac{1}{5292032}a^{22}-\frac{377}{2646016}a^{21}+\frac{189}{1323008}a^{20}+\frac{55}{2048}a^{17}-\frac{21}{1024}a^{15}+\frac{21}{1024}a^{14}+\frac{987}{256}a^{11}-\frac{377}{128}a^{9}+\frac{377}{128}a^{8}+\frac{55}{10336}a^{5}+\frac{6765}{10336}a^{4}-\frac{3393}{2584}a^{3}+\frac{21}{5168}a^{2}+1$ 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:  \( 81723202.08229 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 81723202.08229 \cdot 14}{18\cdot\sqrt{2518170116818978404827136000000000000}}\cr\approx \mathstrut & 0.1516410534442 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 + 144*x^18 + 20672*x^12 + 9216*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 + 144*x^18 + 20672*x^12 + 9216*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 + 144*x^18 + 20672*x^12 + 9216*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 + 144*x^18 + 20672*x^12 + 9216*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 $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\zeta_{9})\), 6.0.10077696.1, 6.6.3359232.1, 6.6.419904000.1, 6.0.1259712000.1, 6.0.2460375.1, 6.6.820125.1, 8.0.207360000.1, 12.0.101559956668416.2, 12.0.1586874322944000000.2, 12.0.6053445140625.1, 12.0.1586874322944000000.6, 12.0.1586874322944000000.4, 12.12.176319369216000000.1, 12.0.1586874322944000000.3

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 R ${\href{/padicField/7.6.0.1}{6} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{12}$ ${\href{/padicField/19.2.0.1}{2} }^{12}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{12}$ ${\href{/padicField/41.6.0.1}{6} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.12.18.23$x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.23$x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
\(3\) Copy content Toggle raw display 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}$
\(5\) 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}$