Properties

Label 24.0.639...616.1
Degree $24$
Signature $[0, 12]$
Discriminant $6.399\times 10^{31}$
Root discriminant $21.15$
Ramified primes $2, 3, 71$
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 + 12*x^22 + 96*x^20 + 412*x^18 + 1260*x^16 + 2511*x^14 + 3666*x^12 + 3492*x^10 + 2322*x^8 + 736*x^6 + 165*x^4 + 15*x^2 + 1)
 
gp: K = bnfinit(x^24 + 12*x^22 + 96*x^20 + 412*x^18 + 1260*x^16 + 2511*x^14 + 3666*x^12 + 3492*x^10 + 2322*x^8 + 736*x^6 + 165*x^4 + 15*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 15, 0, 165, 0, 736, 0, 2322, 0, 3492, 0, 3666, 0, 2511, 0, 1260, 0, 412, 0, 96, 0, 12, 0, 1]);
 

\( x^{24} + 12 x^{22} + 96 x^{20} + 412 x^{18} + 1260 x^{16} + 2511 x^{14} + 3666 x^{12} + 3492 x^{10} + 2322 x^{8} + 736 x^{6} + 165 x^{4} + 15 x^{2} + 1 \)

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:  \(63990935712336613969356040175616\)\(\medspace = 2^{24}\cdot 3^{36}\cdot 71^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $21.15$
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:  $2, 3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{9} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{9} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{3} a^{10} - \frac{1}{9} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{9} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{15} + \frac{1}{3} a^{11} - \frac{1}{9} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{9} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{9} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{18} + \frac{1}{9}$, $\frac{1}{9} a^{19} + \frac{1}{9} a$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{21} + \frac{1}{9} a^{3}$, $\frac{1}{22509084141} a^{22} + \frac{299240297}{22509084141} a^{20} + \frac{1086357436}{22509084141} a^{18} + \frac{263275069}{7503028047} a^{16} + \frac{233460403}{7503028047} a^{14} + \frac{8471728}{2501009349} a^{12} + \frac{1770725249}{7503028047} a^{10} + \frac{729926189}{7503028047} a^{8} + \frac{400644671}{833669783} a^{6} + \frac{2775172954}{22509084141} a^{4} - \frac{6956139283}{22509084141} a^{2} + \frac{8821836493}{22509084141}$, $\frac{1}{22509084141} a^{23} + \frac{299240297}{22509084141} a^{21} + \frac{1086357436}{22509084141} a^{19} + \frac{263275069}{7503028047} a^{17} + \frac{233460403}{7503028047} a^{15} + \frac{8471728}{2501009349} a^{13} + \frac{1770725249}{7503028047} a^{11} + \frac{729926189}{7503028047} a^{9} + \frac{400644671}{833669783} a^{7} + \frac{2775172954}{22509084141} a^{5} - \frac{6956139283}{22509084141} a^{3} + \frac{8821836493}{22509084141} a$

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:  \( \frac{11444877047}{22509084141} a^{23} + \frac{137633056654}{22509084141} a^{21} + \frac{1101716954711}{22509084141} a^{19} + \frac{526405153627}{2501009349} a^{17} + \frac{1610660658046}{2501009349} a^{15} + \frac{3214193264483}{2501009349} a^{13} + \frac{4686196269086}{2501009349} a^{11} + \frac{4458303483595}{2501009349} a^{9} + \frac{976928675150}{833669783} a^{7} + \frac{8125534199504}{22509084141} a^{5} + \frac{1531536208654}{22509084141} a^{3} + \frac{120679604099}{22509084141} a \) (order $36$)
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:  \( 7833899.703521124 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 7833899.703521124 \cdot 2}{36\sqrt{63990935712336613969356040175616}}\approx 0.205970342829836$ (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{12})\), 6.0.465831.1, \(\Q(\zeta_{36})^+\), \(\Q(\zeta_{9})\), 6.6.29813184.1, 6.0.89439552.2, 6.6.1397493.1, 6.0.419904.1, Deg 12, Deg 12, Deg 12, 12.0.1952986685049.1, Deg 12, 12.0.888825940217856.1, \(\Q(\zeta_{36})\)

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$