Properties

Label 24.0.138...681.1
Degree $24$
Signature $[0, 12]$
Discriminant $1.384\times 10^{32}$
Root discriminant $21.84$
Ramified primes $3, 7, 239$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2^3\times A_4$ (as 24T135)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 + 2*x^22 + 10*x^21 - 20*x^20 + 13*x^19 + 57*x^18 - 98*x^17 + 19*x^16 + 228*x^15 - 267*x^14 - 159*x^13 + 711*x^12 - 318*x^11 - 1068*x^10 + 1824*x^9 + 304*x^8 - 3136*x^7 + 3648*x^6 + 1664*x^5 - 5120*x^4 + 5120*x^3 + 2048*x^2 - 4096*x + 4096)
 
gp: K = bnfinit(x^24 - 2*x^23 + 2*x^22 + 10*x^21 - 20*x^20 + 13*x^19 + 57*x^18 - 98*x^17 + 19*x^16 + 228*x^15 - 267*x^14 - 159*x^13 + 711*x^12 - 318*x^11 - 1068*x^10 + 1824*x^9 + 304*x^8 - 3136*x^7 + 3648*x^6 + 1664*x^5 - 5120*x^4 + 5120*x^3 + 2048*x^2 - 4096*x + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, -4096, 2048, 5120, -5120, 1664, 3648, -3136, 304, 1824, -1068, -318, 711, -159, -267, 228, 19, -98, 57, 13, -20, 10, 2, -2, 1]);
 

\( x^{24} - 2 x^{23} + 2 x^{22} + 10 x^{21} - 20 x^{20} + 13 x^{19} + 57 x^{18} - 98 x^{17} + 19 x^{16} + 228 x^{15} - 267 x^{14} - 159 x^{13} + 711 x^{12} - 318 x^{11} - 1068 x^{10} + 1824 x^{9} + 304 x^{8} - 3136 x^{7} + 3648 x^{6} + 1664 x^{5} - 5120 x^{4} + 5120 x^{3} + 2048 x^{2} - 4096 x + 4096 \)

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:  \(138359014736314946502328332753681\)\(\medspace = 3^{12}\cdot 7^{20}\cdot 239^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $21.84$
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:  $3, 7, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{3}{16} a^{11} + \frac{3}{16} a^{10} - \frac{1}{16} a^{8} + \frac{1}{8} a^{7} + \frac{5}{16} a^{6} - \frac{5}{16} a^{5} - \frac{7}{16} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{32} a^{17} - \frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{3}{32} a^{12} - \frac{5}{32} a^{11} + \frac{7}{32} a^{9} - \frac{7}{16} a^{8} - \frac{11}{32} a^{7} + \frac{3}{32} a^{6} - \frac{7}{32} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{64} a^{18} - \frac{1}{32} a^{16} - \frac{1}{32} a^{15} + \frac{5}{64} a^{13} + \frac{3}{64} a^{12} - \frac{1}{64} a^{10} + \frac{5}{32} a^{9} - \frac{19}{64} a^{8} - \frac{21}{64} a^{7} + \frac{25}{64} a^{6} - \frac{3}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{128} a^{19} - \frac{1}{64} a^{17} - \frac{1}{64} a^{16} + \frac{5}{128} a^{14} - \frac{13}{128} a^{13} - \frac{1}{8} a^{12} + \frac{31}{128} a^{11} + \frac{13}{64} a^{10} + \frac{29}{128} a^{9} + \frac{59}{128} a^{8} + \frac{9}{128} a^{7} + \frac{1}{4} a^{6} - \frac{3}{16} a^{5} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{512} a^{20} - \frac{1}{256} a^{19} - \frac{1}{256} a^{18} + \frac{1}{256} a^{17} + \frac{1}{128} a^{16} + \frac{5}{512} a^{15} + \frac{41}{512} a^{14} + \frac{21}{256} a^{13} - \frac{33}{512} a^{12} + \frac{23}{128} a^{11} - \frac{119}{512} a^{10} + \frac{33}{512} a^{9} - \frac{141}{512} a^{8} - \frac{41}{256} a^{7} - \frac{3}{64} a^{6} + \frac{29}{64} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1024} a^{21} + \frac{1}{512} a^{19} - \frac{1}{512} a^{18} - \frac{1}{128} a^{17} - \frac{3}{1024} a^{16} + \frac{51}{1024} a^{15} + \frac{9}{256} a^{14} + \frac{75}{1024} a^{13} - \frac{51}{512} a^{12} + \frac{185}{1024} a^{11} - \frac{125}{1024} a^{10} - \frac{227}{1024} a^{9} + \frac{123}{256} a^{8} - \frac{29}{256} a^{7} - \frac{41}{128} a^{6} + \frac{29}{64} a^{5} + \frac{7}{16} a^{4} - \frac{5}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{176128} a^{22} + \frac{19}{44032} a^{21} + \frac{83}{88064} a^{20} + \frac{39}{88064} a^{19} - \frac{165}{22016} a^{18} + \frac{613}{176128} a^{17} - \frac{2465}{176128} a^{16} + \frac{127}{44032} a^{15} + \frac{4927}{176128} a^{14} - \frac{10685}{88064} a^{13} - \frac{11763}{176128} a^{12} - \frac{41249}{176128} a^{11} + \frac{12293}{176128} a^{10} + \frac{8171}{44032} a^{9} - \frac{319}{22016} a^{8} + \frac{343}{1376} a^{7} - \frac{5403}{11008} a^{6} + \frac{2301}{5504} a^{5} + \frac{155}{2752} a^{4} - \frac{575}{1376} a^{3} - \frac{3}{172} a^{2} + \frac{109}{344} a + \frac{45}{172}$, $\frac{1}{44736512} a^{23} + \frac{7}{22368256} a^{22} - \frac{6745}{22368256} a^{21} - \frac{17491}{22368256} a^{20} - \frac{10139}{11184128} a^{19} + \frac{28789}{44736512} a^{18} - \frac{656919}{44736512} a^{17} + \frac{7599}{520192} a^{16} - \frac{2742105}{44736512} a^{15} + \frac{789297}{11184128} a^{14} - \frac{39205}{1040384} a^{13} - \frac{2106599}{44736512} a^{12} + \frac{67475}{44736512} a^{11} - \frac{2905869}{22368256} a^{10} - \frac{305561}{2796032} a^{9} - \frac{1375407}{2796032} a^{8} + \frac{714569}{2796032} a^{7} - \frac{303565}{699008} a^{6} + \frac{15273}{174752} a^{5} + \frac{741}{5461} a^{4} - \frac{42903}{174752} a^{3} + \frac{33849}{87376} a^{2} - \frac{7859}{21844} a + \frac{2905}{21844}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{233629}{22368256} a^{23} + \frac{356283}{22368256} a^{22} - \frac{244523}{11184128} a^{21} + \frac{59507}{699008} a^{20} + \frac{1973597}{11184128} a^{19} - \frac{5059799}{22368256} a^{18} + \frac{2173295}{11184128} a^{17} + \frac{22093205}{22368256} a^{16} - \frac{502931}{520192} a^{15} - \frac{13893057}{22368256} a^{14} + \frac{77191467}{22368256} a^{13} - \frac{16518885}{11184128} a^{12} - \frac{59569811}{11184128} a^{11} + \frac{181994879}{22368256} a^{10} + \frac{23259973}{5592064} a^{9} - \frac{43889855}{2796032} a^{8} + \frac{18372467}{1398016} a^{7} + \frac{35452205}{1398016} a^{6} - \frac{16496667}{699008} a^{5} + \frac{4517191}{349504} a^{4} + \frac{9277671}{174752} a^{3} - \frac{653149}{43688} a^{2} + \frac{263005}{43688} a + \frac{1000995}{21844} \) (order $42$)
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:  \( 9137650.497592166 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 9137650.497592166 \cdot 3}{42\sqrt{138359014736314946502328332753681}}\approx 0.210068627107942$ (assuming GRH)

Galois group

$C_2^3\times A_4$ (as 24T135):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 96
The 32 conjugacy class representatives for $C_2^3\times A_4$
Character table for $C_2^3\times A_4$ is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.0.64827.1, 6.6.4016873.1, 6.0.573839.1, \(\Q(\zeta_{21})^+\), 6.0.108455571.1, 6.6.15493653.1, \(\Q(\zeta_{7})\), Deg 12, Deg 12, Deg 12, 12.0.240053283284409.1, Deg 12, 12.0.16135268698129.1, \(\Q(\zeta_{21})\)

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 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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

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
$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}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
239Data not computed