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Properties

Label 24.0.44455984353...4169.1
Degree $24$
Signature $[0, 12]$
Discriminant $7^{12}\cdot 13^{22}$
Root discriminant $27.78$
Ramified primes $7, 13$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, -2048, -1024, 1536, -256, -640, 448, 96, -272, 88, 92, -90, -1, -45, 23, 11, -17, 3, 7, -5, -1, 3, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - x^22 + 3*x^21 - x^20 - 5*x^19 + 7*x^18 + 3*x^17 - 17*x^16 + 11*x^15 + 23*x^14 - 45*x^13 - x^12 - 90*x^11 + 92*x^10 + 88*x^9 - 272*x^8 + 96*x^7 + 448*x^6 - 640*x^5 - 256*x^4 + 1536*x^3 - 1024*x^2 - 2048*x + 4096)
 
gp: K = bnfinit(x^24 - x^23 - x^22 + 3*x^21 - x^20 - 5*x^19 + 7*x^18 + 3*x^17 - 17*x^16 + 11*x^15 + 23*x^14 - 45*x^13 - x^12 - 90*x^11 + 92*x^10 + 88*x^9 - 272*x^8 + 96*x^7 + 448*x^6 - 640*x^5 - 256*x^4 + 1536*x^3 - 1024*x^2 - 2048*x + 4096, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - x^{22} + 3 x^{21} - x^{20} - 5 x^{19} + 7 x^{18} + 3 x^{17} - 17 x^{16} + 11 x^{15} + 23 x^{14} - 45 x^{13} - x^{12} - 90 x^{11} + 92 x^{10} + 88 x^{9} - 272 x^{8} + 96 x^{7} + 448 x^{6} - 640 x^{5} - 256 x^{4} + 1536 x^{3} - 1024 x^{2} - 2048 x + 4096 \)

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:  \(44455984353110737022824200630534169=7^{12}\cdot 13^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.78$
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:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(69,·)$, $\chi_{91}(6,·)$, $\chi_{91}(71,·)$, $\chi_{91}(8,·)$, $\chi_{91}(76,·)$, $\chi_{91}(15,·)$, $\chi_{91}(83,·)$, $\chi_{91}(20,·)$, $\chi_{91}(85,·)$, $\chi_{91}(22,·)$, $\chi_{91}(90,·)$, $\chi_{91}(27,·)$, $\chi_{91}(29,·)$, $\chi_{91}(34,·)$, $\chi_{91}(36,·)$, $\chi_{91}(41,·)$, $\chi_{91}(43,·)$, $\chi_{91}(48,·)$, $\chi_{91}(50,·)$, $\chi_{91}(55,·)$, $\chi_{91}(57,·)$, $\chi_{91}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{3}{8} a^{12} - \frac{1}{8} a^{11} + \frac{3}{8} a^{10} - \frac{1}{8} a^{9} + \frac{3}{8} a^{8} - \frac{1}{8} a^{7} + \frac{3}{8} a^{6} - \frac{1}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{16} - \frac{1}{16} a^{15} - \frac{1}{16} a^{14} + \frac{3}{16} a^{13} - \frac{1}{16} a^{12} - \frac{5}{16} a^{11} + \frac{7}{16} a^{10} + \frac{3}{16} a^{9} - \frac{1}{16} a^{8} - \frac{5}{16} a^{7} + \frac{7}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{1}{32} a^{15} + \frac{3}{32} a^{14} - \frac{1}{32} a^{13} - \frac{5}{32} a^{12} + \frac{7}{32} a^{11} + \frac{3}{32} a^{10} + \frac{15}{32} a^{9} + \frac{11}{32} a^{8} - \frac{9}{32} a^{7} - \frac{13}{32} a^{6} - \frac{1}{32} a^{5} + \frac{3}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{18} - \frac{1}{64} a^{17} - \frac{1}{64} a^{16} + \frac{3}{64} a^{15} - \frac{1}{64} a^{14} - \frac{5}{64} a^{13} + \frac{7}{64} a^{12} + \frac{3}{64} a^{11} - \frac{17}{64} a^{10} + \frac{11}{64} a^{9} + \frac{23}{64} a^{8} + \frac{19}{64} a^{7} - \frac{1}{64} a^{6} - \frac{13}{32} a^{5} + \frac{7}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{19} - \frac{1}{128} a^{18} - \frac{1}{128} a^{17} + \frac{3}{128} a^{16} - \frac{1}{128} a^{15} - \frac{5}{128} a^{14} + \frac{7}{128} a^{13} + \frac{3}{128} a^{12} - \frac{17}{128} a^{11} + \frac{11}{128} a^{10} + \frac{23}{128} a^{9} - \frac{45}{128} a^{8} - \frac{1}{128} a^{7} + \frac{19}{64} a^{6} - \frac{9}{32} a^{5} - \frac{5}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{20} - \frac{1}{256} a^{19} - \frac{1}{256} a^{18} + \frac{3}{256} a^{17} - \frac{1}{256} a^{16} - \frac{5}{256} a^{15} + \frac{7}{256} a^{14} + \frac{3}{256} a^{13} - \frac{17}{256} a^{12} + \frac{11}{256} a^{11} + \frac{23}{256} a^{10} - \frac{45}{256} a^{9} - \frac{1}{256} a^{8} - \frac{45}{128} a^{7} + \frac{23}{64} a^{6} + \frac{11}{32} a^{5} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{512} a^{21} - \frac{1}{512} a^{20} - \frac{1}{512} a^{19} + \frac{3}{512} a^{18} - \frac{1}{512} a^{17} - \frac{5}{512} a^{16} + \frac{7}{512} a^{15} + \frac{3}{512} a^{14} - \frac{17}{512} a^{13} + \frac{11}{512} a^{12} + \frac{23}{512} a^{11} - \frac{45}{512} a^{10} - \frac{1}{512} a^{9} - \frac{45}{256} a^{8} + \frac{23}{128} a^{7} + \frac{11}{64} a^{6} + \frac{15}{32} a^{5} + \frac{3}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1024} a^{22} - \frac{1}{1024} a^{21} - \frac{1}{1024} a^{20} + \frac{3}{1024} a^{19} - \frac{1}{1024} a^{18} - \frac{5}{1024} a^{17} + \frac{7}{1024} a^{16} + \frac{3}{1024} a^{15} - \frac{17}{1024} a^{14} + \frac{11}{1024} a^{13} + \frac{23}{1024} a^{12} - \frac{45}{1024} a^{11} - \frac{1}{1024} a^{10} - \frac{45}{512} a^{9} + \frac{23}{256} a^{8} + \frac{11}{128} a^{7} - \frac{17}{64} a^{6} + \frac{3}{32} a^{5} + \frac{7}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{2048} a^{23} - \frac{1}{2048} a^{22} - \frac{1}{2048} a^{21} + \frac{3}{2048} a^{20} - \frac{1}{2048} a^{19} - \frac{5}{2048} a^{18} + \frac{7}{2048} a^{17} + \frac{3}{2048} a^{16} - \frac{17}{2048} a^{15} + \frac{11}{2048} a^{14} + \frac{23}{2048} a^{13} - \frac{45}{2048} a^{12} - \frac{1}{2048} a^{11} - \frac{45}{1024} a^{10} + \frac{23}{512} a^{9} + \frac{11}{256} a^{8} - \frac{17}{128} a^{7} + \frac{3}{64} a^{6} + \frac{7}{32} a^{5} - \frac{5}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( -\frac{1}{2} a^{14} + \frac{91}{2} a \) (order $26$)
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:  \( 42757649.54957395 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{12}$ (as 24T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-91}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-7}) \), 3.3.169.1, \(\Q(\sqrt{-7}, \sqrt{13})\), 4.4.107653.1, 4.0.2197.1, 6.0.127353499.1, \(\Q(\zeta_{13})^+\), 6.0.9796423.1, 8.0.11589168409.1, 12.0.16218913707543001.1, 12.12.210845878198059013.1, \(\Q(\zeta_{13})\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{6}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
7Data not computed
13Data not computed