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Properties

Label 24.0.32912300299...6992.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{93}\cdot 7^{16}$
Root discriminant $53.69$
Ramified primes $2, 7$
Class number $1225$ (GRH)
Class group $[35, 35]$ (GRH)
Galois group $C_{24}$ (as 24T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 0, 384, 0, 4640, 0, 25504, 0, 75464, 0, 128416, 0, 128408, 0, 75632, 0, 26378, 0, 5440, 0, 644, 0, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 40*x^22 + 644*x^20 + 5440*x^18 + 26378*x^16 + 75632*x^14 + 128408*x^12 + 128416*x^10 + 75464*x^8 + 25504*x^6 + 4640*x^4 + 384*x^2 + 8)
 
gp: K = bnfinit(x^24 + 40*x^22 + 644*x^20 + 5440*x^18 + 26378*x^16 + 75632*x^14 + 128408*x^12 + 128416*x^10 + 75464*x^8 + 25504*x^6 + 4640*x^4 + 384*x^2 + 8, 1)
 

Normalized defining polynomial

\( x^{24} + 40 x^{22} + 644 x^{20} + 5440 x^{18} + 26378 x^{16} + 75632 x^{14} + 128408 x^{12} + 128416 x^{10} + 75464 x^{8} + 25504 x^{6} + 4640 x^{4} + 384 x^{2} + 8 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(329123002999201416128761938882499016916992=2^{93}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.69$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(224=2^{5}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{224}(1,·)$, $\chi_{224}(67,·)$, $\chi_{224}(179,·)$, $\chi_{224}(193,·)$, $\chi_{224}(9,·)$, $\chi_{224}(11,·)$, $\chi_{224}(81,·)$, $\chi_{224}(211,·)$, $\chi_{224}(121,·)$, $\chi_{224}(25,·)$, $\chi_{224}(155,·)$, $\chi_{224}(107,·)$, $\chi_{224}(163,·)$, $\chi_{224}(65,·)$, $\chi_{224}(113,·)$, $\chi_{224}(169,·)$, $\chi_{224}(43,·)$, $\chi_{224}(99,·)$, $\chi_{224}(177,·)$, $\chi_{224}(51,·)$, $\chi_{224}(219,·)$, $\chi_{224}(137,·)$, $\chi_{224}(57,·)$, $\chi_{224}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{4} a^{16}$, $\frac{1}{4} a^{17}$, $\frac{1}{4} a^{18}$, $\frac{1}{4} a^{19}$, $\frac{1}{508} a^{20} + \frac{63}{508} a^{18} - \frac{27}{254} a^{14} - \frac{5}{254} a^{12} - \frac{11}{127} a^{10} - \frac{5}{254} a^{8} + \frac{15}{127} a^{6} - \frac{4}{127} a^{4} + \frac{35}{127} a^{2} + \frac{50}{127}$, $\frac{1}{508} a^{21} + \frac{63}{508} a^{19} - \frac{27}{254} a^{15} - \frac{5}{254} a^{13} - \frac{11}{127} a^{11} - \frac{5}{254} a^{9} + \frac{15}{127} a^{7} - \frac{4}{127} a^{5} + \frac{35}{127} a^{3} + \frac{50}{127} a$, $\frac{1}{6352154428} a^{22} + \frac{1263377}{3176077214} a^{20} - \frac{352945897}{3176077214} a^{18} + \frac{414308617}{6352154428} a^{16} + \frac{578554651}{3176077214} a^{14} + \frac{168458617}{3176077214} a^{12} - \frac{481052447}{3176077214} a^{10} + \frac{35684156}{1588038607} a^{8} - \frac{649007968}{1588038607} a^{6} + \frac{213379108}{1588038607} a^{4} - \frac{718337075}{1588038607} a^{2} + \frac{651414018}{1588038607}$, $\frac{1}{6352154428} a^{23} + \frac{1263377}{3176077214} a^{21} - \frac{352945897}{3176077214} a^{19} + \frac{414308617}{6352154428} a^{17} + \frac{578554651}{3176077214} a^{15} + \frac{168458617}{3176077214} a^{13} - \frac{481052447}{3176077214} a^{11} + \frac{35684156}{1588038607} a^{9} - \frac{649007968}{1588038607} a^{7} + \frac{213379108}{1588038607} a^{5} - \frac{718337075}{1588038607} a^{3} + \frac{651414018}{1588038607} a$

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

Class group and class number

$C_{35}\times C_{35}$, which has order $1225$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 39019312.21180779 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{24}$ (as 24T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 24
The 24 conjugacy class representatives for $C_{24}$
Character table for $C_{24}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{16})^+\), 6.6.1229312.1, 8.0.2147483648.1, 12.12.49519263525896192.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $24$ $24$ R $24$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ $24$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ $24$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ $24$ $24$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$