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Properties

Label 24.0.10080126276...7049.1
Degree $24$
Signature $[0, 12]$
Discriminant $11^{12}\cdot 13^{22}$
Root discriminant $34.82$
Ramified primes $11, 13$
Class number $35$ (GRH)
Class group $[35]$ (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![531441, -177147, -118098, 98415, 6561, -34992, 9477, 8505, -5994, -837, 2277, -480, -599, -160, 253, -31, -74, 35, 13, -16, 1, 5, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 2*x^22 + 5*x^21 + x^20 - 16*x^19 + 13*x^18 + 35*x^17 - 74*x^16 - 31*x^15 + 253*x^14 - 160*x^13 - 599*x^12 - 480*x^11 + 2277*x^10 - 837*x^9 - 5994*x^8 + 8505*x^7 + 9477*x^6 - 34992*x^5 + 6561*x^4 + 98415*x^3 - 118098*x^2 - 177147*x + 531441)
 
gp: K = bnfinit(x^24 - x^23 - 2*x^22 + 5*x^21 + x^20 - 16*x^19 + 13*x^18 + 35*x^17 - 74*x^16 - 31*x^15 + 253*x^14 - 160*x^13 - 599*x^12 - 480*x^11 + 2277*x^10 - 837*x^9 - 5994*x^8 + 8505*x^7 + 9477*x^6 - 34992*x^5 + 6561*x^4 + 98415*x^3 - 118098*x^2 - 177147*x + 531441, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 2 x^{22} + 5 x^{21} + x^{20} - 16 x^{19} + 13 x^{18} + 35 x^{17} - 74 x^{16} - 31 x^{15} + 253 x^{14} - 160 x^{13} - 599 x^{12} - 480 x^{11} + 2277 x^{10} - 837 x^{9} - 5994 x^{8} + 8505 x^{7} + 9477 x^{6} - 34992 x^{5} + 6561 x^{4} + 98415 x^{3} - 118098 x^{2} - 177147 x + 531441 \)

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:  \(10080126276029253939999801345921207049=11^{12}\cdot 13^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.82$
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:  $11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(143=11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{143}(1,·)$, $\chi_{143}(131,·)$, $\chi_{143}(133,·)$, $\chi_{143}(76,·)$, $\chi_{143}(10,·)$, $\chi_{143}(87,·)$, $\chi_{143}(12,·)$, $\chi_{143}(98,·)$, $\chi_{143}(142,·)$, $\chi_{143}(109,·)$, $\chi_{143}(120,·)$, $\chi_{143}(67,·)$, $\chi_{143}(21,·)$, $\chi_{143}(23,·)$, $\chi_{143}(89,·)$, $\chi_{143}(32,·)$, $\chi_{143}(34,·)$, $\chi_{143}(100,·)$, $\chi_{143}(43,·)$, $\chi_{143}(45,·)$, $\chi_{143}(111,·)$, $\chi_{143}(54,·)$, $\chi_{143}(56,·)$, $\chi_{143}(122,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{1797} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{160}{599}$, $\frac{1}{5391} a^{14} - \frac{1}{5391} a^{13} - \frac{4}{9} a^{12} + \frac{1}{9} a^{11} + \frac{2}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{9} a^{8} - \frac{2}{9} a^{7} - \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2} - \frac{160}{1797} a + \frac{253}{599}$, $\frac{1}{16173} a^{15} - \frac{1}{16173} a^{14} - \frac{2}{16173} a^{13} + \frac{1}{27} a^{12} + \frac{11}{27} a^{11} + \frac{13}{27} a^{10} + \frac{8}{27} a^{9} + \frac{7}{27} a^{8} - \frac{4}{27} a^{7} + \frac{10}{27} a^{6} + \frac{2}{27} a^{5} - \frac{5}{27} a^{4} - \frac{1}{27} a^{3} - \frac{160}{5391} a^{2} + \frac{253}{1797} a - \frac{31}{599}$, $\frac{1}{48519} a^{16} - \frac{1}{48519} a^{15} - \frac{2}{48519} a^{14} + \frac{5}{48519} a^{13} + \frac{38}{81} a^{12} + \frac{40}{81} a^{11} + \frac{8}{81} a^{10} + \frac{34}{81} a^{9} + \frac{23}{81} a^{8} + \frac{37}{81} a^{7} - \frac{25}{81} a^{6} - \frac{5}{81} a^{5} - \frac{1}{81} a^{4} - \frac{160}{16173} a^{3} + \frac{253}{5391} a^{2} - \frac{31}{1797} a - \frac{74}{599}$, $\frac{1}{145557} a^{17} - \frac{1}{145557} a^{16} - \frac{2}{145557} a^{15} + \frac{5}{145557} a^{14} + \frac{1}{145557} a^{13} - \frac{41}{243} a^{12} - \frac{73}{243} a^{11} - \frac{47}{243} a^{10} + \frac{23}{243} a^{9} + \frac{118}{243} a^{8} + \frac{56}{243} a^{7} + \frac{76}{243} a^{6} - \frac{1}{243} a^{5} - \frac{160}{48519} a^{4} + \frac{253}{16173} a^{3} - \frac{31}{5391} a^{2} - \frac{74}{1797} a + \frac{35}{599}$, $\frac{1}{436671} a^{18} - \frac{1}{436671} a^{17} - \frac{2}{436671} a^{16} + \frac{5}{436671} a^{15} + \frac{1}{436671} a^{14} - \frac{16}{436671} a^{13} - \frac{73}{729} a^{12} + \frac{196}{729} a^{11} + \frac{23}{729} a^{10} + \frac{118}{729} a^{9} - \frac{187}{729} a^{8} - \frac{167}{729} a^{7} - \frac{1}{729} a^{6} - \frac{160}{145557} a^{5} + \frac{253}{48519} a^{4} - \frac{31}{16173} a^{3} - \frac{74}{5391} a^{2} + \frac{35}{1797} a + \frac{13}{599}$, $\frac{1}{1310013} a^{19} - \frac{1}{1310013} a^{18} - \frac{2}{1310013} a^{17} + \frac{5}{1310013} a^{16} + \frac{1}{1310013} a^{15} - \frac{16}{1310013} a^{14} + \frac{13}{1310013} a^{13} - \frac{533}{2187} a^{12} + \frac{752}{2187} a^{11} + \frac{847}{2187} a^{10} - \frac{916}{2187} a^{9} + \frac{562}{2187} a^{8} - \frac{1}{2187} a^{7} - \frac{160}{436671} a^{6} + \frac{253}{145557} a^{5} - \frac{31}{48519} a^{4} - \frac{74}{16173} a^{3} + \frac{35}{5391} a^{2} + \frac{13}{1797} a - \frac{16}{599}$, $\frac{1}{3930039} a^{20} - \frac{1}{3930039} a^{19} - \frac{2}{3930039} a^{18} + \frac{5}{3930039} a^{17} + \frac{1}{3930039} a^{16} - \frac{16}{3930039} a^{15} + \frac{13}{3930039} a^{14} + \frac{35}{3930039} a^{13} - \frac{1435}{6561} a^{12} + \frac{3034}{6561} a^{11} + \frac{1271}{6561} a^{10} + \frac{2749}{6561} a^{9} - \frac{1}{6561} a^{8} - \frac{160}{1310013} a^{7} + \frac{253}{436671} a^{6} - \frac{31}{145557} a^{5} - \frac{74}{48519} a^{4} + \frac{35}{16173} a^{3} + \frac{13}{5391} a^{2} - \frac{16}{1797} a + \frac{1}{599}$, $\frac{1}{11790117} a^{21} - \frac{1}{11790117} a^{20} - \frac{2}{11790117} a^{19} + \frac{5}{11790117} a^{18} + \frac{1}{11790117} a^{17} - \frac{16}{11790117} a^{16} + \frac{13}{11790117} a^{15} + \frac{35}{11790117} a^{14} - \frac{74}{11790117} a^{13} + \frac{9595}{19683} a^{12} - \frac{5290}{19683} a^{11} - \frac{3812}{19683} a^{10} - \frac{1}{19683} a^{9} - \frac{160}{3930039} a^{8} + \frac{253}{1310013} a^{7} - \frac{31}{436671} a^{6} - \frac{74}{145557} a^{5} + \frac{35}{48519} a^{4} + \frac{13}{16173} a^{3} - \frac{16}{5391} a^{2} + \frac{1}{1797} a + \frac{5}{599}$, $\frac{1}{35370351} a^{22} - \frac{1}{35370351} a^{21} - \frac{2}{35370351} a^{20} + \frac{5}{35370351} a^{19} + \frac{1}{35370351} a^{18} - \frac{16}{35370351} a^{17} + \frac{13}{35370351} a^{16} + \frac{35}{35370351} a^{15} - \frac{74}{35370351} a^{14} - \frac{31}{35370351} a^{13} + \frac{14393}{59049} a^{12} + \frac{15871}{59049} a^{11} - \frac{1}{59049} a^{10} - \frac{160}{11790117} a^{9} + \frac{253}{3930039} a^{8} - \frac{31}{1310013} a^{7} - \frac{74}{436671} a^{6} + \frac{35}{145557} a^{5} + \frac{13}{48519} a^{4} - \frac{16}{16173} a^{3} + \frac{1}{5391} a^{2} + \frac{5}{1797} a - \frac{2}{599}$, $\frac{1}{106111053} a^{23} - \frac{1}{106111053} a^{22} - \frac{2}{106111053} a^{21} + \frac{5}{106111053} a^{20} + \frac{1}{106111053} a^{19} - \frac{16}{106111053} a^{18} + \frac{13}{106111053} a^{17} + \frac{35}{106111053} a^{16} - \frac{74}{106111053} a^{15} - \frac{31}{106111053} a^{14} + \frac{253}{106111053} a^{13} - \frac{43178}{177147} a^{12} - \frac{1}{177147} a^{11} - \frac{160}{35370351} a^{10} + \frac{253}{11790117} a^{9} - \frac{31}{3930039} a^{8} - \frac{74}{1310013} a^{7} + \frac{35}{436671} a^{6} + \frac{13}{145557} a^{5} - \frac{16}{48519} a^{4} + \frac{1}{16173} a^{3} + \frac{5}{5391} a^{2} - \frac{2}{1797} a - \frac{1}{599}$

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

Class group and class number

$C_{35}$, which has order $35$ (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:  \( -\frac{5}{48519} a^{17} + \frac{1801}{48519} a^{4} \) (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:  \( 166366823.90883145 \) (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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-143}) \), 3.3.169.1, \(\Q(\sqrt{-11}, \sqrt{13})\), 4.4.265837.1, 4.0.2197.1, 6.0.38014691.1, \(\Q(\zeta_{13})^+\), 6.0.494190983.1, 8.0.70669310569.1, 12.0.244224727678506289.1, 12.12.3174921459820581757.1, \(\Q(\zeta_{13})\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
11Data not computed
13Data not computed