Properties

Label 24.0.329...561.1
Degree $24$
Signature $[0, 12]$
Discriminant $3.298\times 10^{31}$
Root discriminant $20.57$
Ramified primes $3, 7, 167$
Class number $2$ (GRH)
Class group $[2]$ (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 - 3*x^23 + 9*x^22 - 26*x^21 + 57*x^20 - 122*x^19 + 241*x^18 - 433*x^17 + 746*x^16 - 1222*x^15 + 1898*x^14 - 2838*x^13 + 4105*x^12 - 5676*x^11 + 7592*x^10 - 9776*x^9 + 11936*x^8 - 13856*x^7 + 15424*x^6 - 15616*x^5 + 14592*x^4 - 13312*x^3 + 9216*x^2 - 6144*x + 4096)
 
gp: K = bnfinit(x^24 - 3*x^23 + 9*x^22 - 26*x^21 + 57*x^20 - 122*x^19 + 241*x^18 - 433*x^17 + 746*x^16 - 1222*x^15 + 1898*x^14 - 2838*x^13 + 4105*x^12 - 5676*x^11 + 7592*x^10 - 9776*x^9 + 11936*x^8 - 13856*x^7 + 15424*x^6 - 15616*x^5 + 14592*x^4 - 13312*x^3 + 9216*x^2 - 6144*x + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, -6144, 9216, -13312, 14592, -15616, 15424, -13856, 11936, -9776, 7592, -5676, 4105, -2838, 1898, -1222, 746, -433, 241, -122, 57, -26, 9, -3, 1]);
 

\( x^{24} - 3 x^{23} + 9 x^{22} - 26 x^{21} + 57 x^{20} - 122 x^{19} + 241 x^{18} - 433 x^{17} + 746 x^{16} - 1222 x^{15} + 1898 x^{14} - 2838 x^{13} + 4105 x^{12} - 5676 x^{11} + 7592 x^{10} - 9776 x^{9} + 11936 x^{8} - 13856 x^{7} + 15424 x^{6} - 15616 x^{5} + 14592 x^{4} - 13312 x^{3} + 9216 x^{2} - 6144 x + 4096 \)

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:  \(32982361051400240694294150960561\)\(\medspace = 3^{12}\cdot 7^{20}\cdot 167^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.57$
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:  $3, 7, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{3}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{16} - \frac{1}{16} a^{15} - \frac{1}{16} a^{14} + \frac{1}{16} a^{12} + \frac{1}{16} a^{10} - \frac{3}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{1}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{1}{32} a^{15} - \frac{1}{8} a^{14} + \frac{1}{32} a^{13} - \frac{7}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{8} a^{9} + \frac{5}{16} a^{8} - \frac{1}{16} a^{7} + \frac{3}{16} a^{6} - \frac{3}{32} a^{5} + \frac{3}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{448} a^{18} - \frac{1}{64} a^{17} - \frac{1}{64} a^{16} + \frac{11}{224} a^{15} + \frac{3}{64} a^{14} + \frac{3}{32} a^{13} + \frac{29}{448} a^{12} + \frac{5}{64} a^{11} + \frac{7}{32} a^{10} + \frac{39}{224} a^{9} - \frac{13}{32} a^{8} - \frac{13}{32} a^{7} - \frac{167}{448} a^{6} + \frac{3}{8} a^{5} - \frac{7}{16} a^{4} + \frac{11}{28} a^{3} + \frac{1}{7}$, $\frac{1}{896} a^{19} - \frac{1}{896} a^{18} + \frac{1}{128} a^{17} - \frac{5}{224} a^{16} + \frac{41}{896} a^{15} - \frac{1}{8} a^{14} + \frac{1}{896} a^{13} + \frac{41}{896} a^{12} + \frac{5}{32} a^{11} - \frac{59}{448} a^{10} - \frac{109}{448} a^{9} - \frac{11}{64} a^{8} + \frac{197}{896} a^{7} - \frac{25}{448} a^{6} - \frac{13}{32} a^{5} - \frac{3}{56} a^{4} - \frac{9}{28} a^{3} + \frac{1}{14} a + \frac{3}{7}$, $\frac{1}{3584} a^{20} + \frac{1}{3584} a^{19} + \frac{1}{3584} a^{18} - \frac{17}{1792} a^{17} + \frac{85}{3584} a^{16} + \frac{81}{1792} a^{15} - \frac{83}{3584} a^{14} + \frac{267}{3584} a^{13} + \frac{53}{1792} a^{12} - \frac{353}{1792} a^{11} - \frac{339}{1792} a^{10} - \frac{3}{1792} a^{9} + \frac{1177}{3584} a^{8} + \frac{37}{112} a^{7} - \frac{425}{896} a^{6} + \frac{37}{112} a^{5} + \frac{11}{28} a^{4} - \frac{5}{112} a^{3} + \frac{1}{7} a^{2} - \frac{3}{28} a - \frac{5}{14}$, $\frac{1}{7168} a^{21} - \frac{1}{7168} a^{20} - \frac{1}{7168} a^{19} - \frac{1}{1792} a^{18} - \frac{71}{7168} a^{17} + \frac{27}{896} a^{16} - \frac{151}{7168} a^{15} - \frac{239}{7168} a^{14} - \frac{219}{1792} a^{13} + \frac{229}{3584} a^{12} - \frac{865}{3584} a^{11} + \frac{227}{3584} a^{10} - \frac{347}{7168} a^{9} - \frac{361}{3584} a^{8} - \frac{681}{1792} a^{7} + \frac{241}{896} a^{6} - \frac{43}{112} a^{5} - \frac{23}{224} a^{4} - \frac{53}{112} a^{3} - \frac{25}{56} a^{2} + \frac{3}{7} a - \frac{5}{14}$, $\frac{1}{28672} a^{22} - \frac{1}{28672} a^{21} + \frac{3}{28672} a^{20} - \frac{1}{1792} a^{19} + \frac{13}{28672} a^{18} - \frac{1}{896} a^{17} - \frac{387}{28672} a^{16} - \frac{1079}{28672} a^{15} + \frac{17}{3584} a^{14} + \frac{1427}{14336} a^{13} + \frac{1739}{14336} a^{12} - \frac{1857}{14336} a^{11} + \frac{3757}{28672} a^{10} - \frac{2405}{14336} a^{9} + \frac{251}{896} a^{8} - \frac{905}{3584} a^{7} - \frac{375}{1792} a^{6} - \frac{351}{896} a^{5} - \frac{79}{448} a^{4} - \frac{51}{112} a^{3} - \frac{11}{56} a^{2} + \frac{9}{28} a - \frac{1}{4}$, $\frac{1}{16457728} a^{23} + \frac{267}{16457728} a^{22} - \frac{225}{16457728} a^{21} - \frac{557}{4114432} a^{20} - \frac{813}{2351104} a^{19} - \frac{1889}{4114432} a^{18} + \frac{225125}{16457728} a^{17} - \frac{17197}{2351104} a^{16} + \frac{81943}{4114432} a^{15} - \frac{488377}{8228864} a^{14} + \frac{130649}{1175552} a^{13} + \frac{76491}{8228864} a^{12} + \frac{2164741}{16457728} a^{11} + \frac{82167}{1175552} a^{10} + \frac{233151}{1028608} a^{9} + \frac{728425}{2057216} a^{8} - \frac{28965}{146944} a^{7} + \frac{233885}{514304} a^{6} + \frac{118655}{257152} a^{5} - \frac{207}{9184} a^{4} - \frac{3729}{16072} a^{3} - \frac{227}{4018} a^{2} - \frac{7515}{16072} a + \frac{650}{2009}$

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:  \( \frac{21523}{16457728} a^{23} - \frac{65109}{16457728} a^{22} + \frac{117259}{16457728} a^{21} - \frac{158779}{8228864} a^{20} + \frac{69809}{2351104} a^{19} - \frac{288341}{8228864} a^{18} + \frac{752567}{16457728} a^{17} - \frac{27721}{2351104} a^{16} - \frac{591525}{8228864} a^{15} + \frac{1840337}{8228864} a^{14} - \frac{584091}{1175552} a^{13} + \frac{7888471}{8228864} a^{12} - \frac{26390885}{16457728} a^{11} + \frac{766555}{293888} a^{10} - \frac{16716023}{4114432} a^{9} + \frac{11984795}{2057216} a^{8} - \frac{303493}{36736} a^{7} + \frac{1424391}{128576} a^{6} - \frac{1696007}{128576} a^{5} + \frac{39285}{2624} a^{4} - \frac{526653}{32144} a^{3} + \frac{103367}{8036} a^{2} - \frac{170085}{16072} a + \frac{70281}{8036} \) (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:  \( 7377024.54411426 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 7377024.54411426 \cdot 2}{42\sqrt{32982361051400240694294150960561}}\approx 0.231568500385281$ (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

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\zeta_{21})^+\), 6.0.400967.1, 6.0.64827.1, 6.6.2806769.1, 6.0.75782763.1, \(\Q(\zeta_{7})\), 6.6.10826109.1, Deg 12, Deg 12, Deg 12, \(\Q(\zeta_{21})\), 12.0.7877952219361.1, 12.0.117204636079881.1, 12.12.5743027167914169.1

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 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$167$167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$