Properties

Label 24.0.394...336.1
Degree $24$
Signature $[0, 12]$
Discriminant $3.949\times 10^{33}$
Root discriminant $25.11$
Ramified primes $2, 3, 199$
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 + 6*x^22 + 9*x^20 - 21*x^18 - 72*x^16 + 33*x^14 + 341*x^12 + 132*x^10 - 1152*x^8 - 1344*x^6 + 2304*x^4 + 6144*x^2 + 4096)
 
gp: K = bnfinit(x^24 + 6*x^22 + 9*x^20 - 21*x^18 - 72*x^16 + 33*x^14 + 341*x^12 + 132*x^10 - 1152*x^8 - 1344*x^6 + 2304*x^4 + 6144*x^2 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 0, 6144, 0, 2304, 0, -1344, 0, -1152, 0, 132, 0, 341, 0, 33, 0, -72, 0, -21, 0, 9, 0, 6, 0, 1]);
 

\( x^{24} + 6 x^{22} + 9 x^{20} - 21 x^{18} - 72 x^{16} + 33 x^{14} + 341 x^{12} + 132 x^{10} - 1152 x^{8} - 1344 x^{6} + 2304 x^{4} + 6144 x^{2} + 4096 \)

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:  \(3949093091982271355222362303758336\)\(\medspace = 2^{24}\cdot 3^{36}\cdot 199^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $25.11$
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:  $2, 3, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{13} + \frac{1}{8} a^{11} + \frac{3}{8} a^{9} + \frac{1}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{14} - \frac{7}{16} a^{12} + \frac{3}{16} a^{10} + \frac{1}{16} a^{6} - \frac{3}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{17} - \frac{1}{16} a^{15} - \frac{7}{32} a^{13} + \frac{3}{32} a^{11} + \frac{1}{32} a^{7} + \frac{13}{32} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{1216} a^{18} + \frac{13}{608} a^{16} + \frac{33}{1216} a^{14} - \frac{545}{1216} a^{12} - \frac{1}{16} a^{10} - \frac{5}{64} a^{8} - \frac{599}{1216} a^{6} + \frac{1}{152} a^{4} - \frac{37}{76} a^{2} - \frac{8}{19}$, $\frac{1}{2432} a^{19} + \frac{13}{1216} a^{17} + \frac{33}{2432} a^{15} - \frac{545}{2432} a^{13} - \frac{1}{32} a^{11} + \frac{59}{128} a^{9} + \frac{617}{2432} a^{7} - \frac{151}{304} a^{5} + \frac{39}{152} a^{3} + \frac{11}{38} a$, $\frac{1}{4864} a^{20} - \frac{1}{2432} a^{18} - \frac{87}{4864} a^{16} - \frac{253}{4864} a^{14} + \frac{75}{304} a^{12} + \frac{11}{256} a^{10} + \frac{845}{4864} a^{8} - \frac{213}{1216} a^{6} + \frac{63}{304} a^{4} - \frac{17}{38} a^{2} - \frac{1}{19}$, $\frac{1}{9728} a^{21} - \frac{1}{4864} a^{19} - \frac{87}{9728} a^{17} - \frac{253}{9728} a^{15} + \frac{75}{608} a^{13} + \frac{11}{512} a^{11} - \frac{4019}{9728} a^{9} - \frac{213}{2432} a^{7} + \frac{63}{608} a^{5} - \frac{17}{76} a^{3} + \frac{9}{19} a$, $\frac{1}{19456} a^{22} - \frac{1}{9728} a^{20} - \frac{7}{19456} a^{18} - \frac{605}{19456} a^{16} - \frac{1}{19} a^{14} - \frac{6911}{19456} a^{12} + \frac{2061}{19456} a^{10} + \frac{319}{4864} a^{8} + \frac{141}{304} a^{6} - \frac{31}{152} a^{4} + \frac{23}{76} a^{2} - \frac{2}{19}$, $\frac{1}{38912} a^{23} - \frac{1}{19456} a^{21} - \frac{7}{38912} a^{19} - \frac{605}{38912} a^{17} - \frac{1}{38} a^{15} - \frac{6911}{38912} a^{13} - \frac{17395}{38912} a^{11} + \frac{319}{9728} a^{9} + \frac{141}{608} a^{7} - \frac{31}{304} a^{5} - \frac{53}{152} a^{3} - \frac{1}{19} a$

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:  \( \frac{21}{1216} a^{23} + \frac{403}{4864} a^{21} + \frac{37}{1216} a^{19} - \frac{2269}{4864} a^{17} - \frac{2825}{4864} a^{15} + \frac{4259}{2432} a^{13} + \frac{17647}{4864} a^{11} - \frac{21019}{4864} a^{9} - \frac{37199}{2432} a^{7} + \frac{1703}{608} a^{5} + \frac{6523}{152} a^{3} + \frac{1421}{38} a \) (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:  \( 43460476.10350948 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 43460476.10350948 \cdot 3}{36\sqrt{3949093091982271355222362303758336}}\approx 0.218184034503331$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{12})\), 6.6.83560896.1, \(\Q(\zeta_{9})\), 6.0.1305639.1, \(\Q(\zeta_{36})^+\), 6.0.250682688.4, 6.0.419904.1, 6.6.3916917.1, Deg 12, Deg 12, Deg 12, 12.0.6982423340322816.1, \(\Q(\zeta_{36})\), Deg 12, 12.0.15342238784889.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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}$
$199$199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$