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Properties

Label 24.0.224...696.1
Degree $24$
Signature $[0, 12]$
Discriminant $2.244\times 10^{33}$
Root discriminant $24.53$
Ramified primes $2, 3, 23, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3\times S_4$ (as 24T400)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 8*x^22 - 12*x^21 + 24*x^20 - 56*x^19 + 104*x^18 - 152*x^17 + 224*x^16 - 376*x^15 + 608*x^14 - 848*x^13 + 1156*x^12 - 1696*x^11 + 2432*x^10 - 3008*x^9 + 3584*x^8 - 4864*x^7 + 6656*x^6 - 7168*x^5 + 6144*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096)
 
gp: K = bnfinit(x^24 - 4*x^23 + 8*x^22 - 12*x^21 + 24*x^20 - 56*x^19 + 104*x^18 - 152*x^17 + 224*x^16 - 376*x^15 + 608*x^14 - 848*x^13 + 1156*x^12 - 1696*x^11 + 2432*x^10 - 3008*x^9 + 3584*x^8 - 4864*x^7 + 6656*x^6 - 7168*x^5 + 6144*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, -8192, 8192, -6144, 6144, -7168, 6656, -4864, 3584, -3008, 2432, -1696, 1156, -848, 608, -376, 224, -152, 104, -56, 24, -12, 8, -4, 1]);
 

\( x^{24} - 4 x^{23} + 8 x^{22} - 12 x^{21} + 24 x^{20} - 56 x^{19} + 104 x^{18} - 152 x^{17} + 224 x^{16} - 376 x^{15} + 608 x^{14} - 848 x^{13} + 1156 x^{12} - 1696 x^{11} + 2432 x^{10} - 3008 x^{9} + 3584 x^{8} - 4864 x^{7} + 6656 x^{6} - 7168 x^{5} + 6144 x^{4} - 6144 x^{3} + 8192 x^{2} - 8192 x + 4096 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2243572222946525052726785077149696\)\(\medspace = 2^{32}\cdot 3^{12}\cdot 23^{4}\cdot 37^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $24.53$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 23, 37$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{7} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{14} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5}$, $\frac{1}{16} a^{15} + \frac{1}{4} a^{3}$, $\frac{1}{32} a^{16} + \frac{1}{8} a^{4}$, $\frac{1}{64} a^{17} + \frac{1}{16} a^{5}$, $\frac{1}{256} a^{18} - \frac{1}{64} a^{15} - \frac{1}{32} a^{14} - \frac{1}{32} a^{13} + \frac{1}{32} a^{12} + \frac{1}{32} a^{11} + \frac{1}{32} a^{9} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} + \frac{1}{64} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2}$, $\frac{1}{512} a^{19} - \frac{1}{128} a^{16} + \frac{1}{64} a^{15} + \frac{1}{64} a^{14} + \frac{1}{64} a^{13} + \frac{1}{64} a^{12} + \frac{1}{64} a^{10} - \frac{1}{32} a^{8} - \frac{31}{128} a^{7} + \frac{7}{32} a^{6} - \frac{3}{16} a^{5} + \frac{3}{16} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{1024} a^{20} - \frac{1}{256} a^{17} + \frac{1}{128} a^{16} + \frac{1}{128} a^{15} + \frac{1}{128} a^{14} + \frac{1}{128} a^{13} + \frac{1}{128} a^{11} - \frac{1}{16} a^{10} - \frac{1}{64} a^{9} - \frac{31}{256} a^{8} + \frac{15}{64} a^{7} - \frac{3}{32} a^{6} + \frac{3}{32} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{2048} a^{21} - \frac{1}{512} a^{18} + \frac{1}{256} a^{17} + \frac{1}{256} a^{16} + \frac{1}{256} a^{15} + \frac{1}{256} a^{14} + \frac{1}{256} a^{12} - \frac{1}{32} a^{11} + \frac{7}{128} a^{10} - \frac{31}{512} a^{9} + \frac{15}{128} a^{8} + \frac{5}{64} a^{7} + \frac{3}{64} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{7}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4096} a^{22} - \frac{1}{1024} a^{19} - \frac{1}{512} a^{18} + \frac{1}{512} a^{17} + \frac{1}{512} a^{16} + \frac{9}{512} a^{15} - \frac{1}{32} a^{14} - \frac{15}{512} a^{13} - \frac{3}{64} a^{12} - \frac{1}{256} a^{11} - \frac{31}{1024} a^{10} + \frac{7}{256} a^{9} + \frac{5}{128} a^{8} - \frac{21}{128} a^{7} - \frac{9}{64} a^{6} - \frac{1}{8} a^{5} - \frac{11}{32} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1564672} a^{23} - \frac{69}{782336} a^{22} + \frac{41}{391168} a^{21} - \frac{149}{391168} a^{20} + \frac{27}{97792} a^{19} - \frac{367}{195584} a^{18} + \frac{1059}{195584} a^{17} + \frac{1707}{195584} a^{16} + \frac{1773}{97792} a^{15} - \frac{4587}{195584} a^{14} - \frac{2817}{97792} a^{13} - \frac{1519}{97792} a^{12} - \frac{22871}{391168} a^{11} - \frac{8079}{195584} a^{10} + \frac{5117}{97792} a^{9} - \frac{5627}{48896} a^{8} + \frac{1559}{24448} a^{7} + \frac{283}{3056} a^{6} - \frac{5139}{12224} a^{5} + \frac{1249}{6112} a^{4} - \frac{777}{3056} a^{3} + \frac{73}{382} a^{2} - \frac{39}{382} a + \frac{67}{382}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{2121}{1564672} a^{23} - \frac{2027}{391168} a^{22} + \frac{4067}{391168} a^{21} - \frac{5081}{391168} a^{20} + \frac{3945}{195584} a^{19} - \frac{7913}{195584} a^{18} + \frac{15641}{195584} a^{17} - \frac{21823}{195584} a^{16} + \frac{1663}{12224} a^{15} - \frac{36923}{195584} a^{14} + \frac{14423}{48896} a^{13} - \frac{38211}{97792} a^{12} + \frac{195609}{391168} a^{11} - \frac{8905}{12224} a^{10} + \frac{101385}{97792} a^{9} - \frac{49319}{48896} a^{8} + \frac{16473}{24448} a^{7} - \frac{8111}{12224} a^{6} + \frac{18505}{12224} a^{5} - \frac{4509}{3056} a^{4} - \frac{451}{3056} a^{3} + \frac{887}{382} a^{2} - \frac{1751}{764} a + \frac{97}{191} \) (order $12$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 25169728.518406376 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 25169728.518406376 \cdot 2}{12\sqrt{2243572222946525052726785077149696}}\approx 0.335286196293265$ (assuming GRH)

Galois group

$C_2^3\times S_4$ (as 24T400):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 192
The 40 conjugacy class representatives for $C_2^3\times S_4$
Character table for $C_2^3\times S_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), 3.3.148.1, \(\Q(\zeta_{12})\), 6.6.9462528.1, 6.0.503792.1, 6.0.591408.1, 6.6.8060672.1, 6.0.217638144.1, 6.0.350464.1, 6.6.13602384.1, Deg 12, Deg 12, Deg 12, 12.0.89539436150784.1, 12.0.64974433091584.2, Deg 12, 12.0.185024850483456.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 24 siblings: data not computed
Degree 32 siblings: data not computed

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$37$37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$