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Properties

Label 24.24.1280287484...6497.1
Degree $24$
Signature $[24, 0]$
Discriminant $3^{32}\cdot 17^{21}$
Root discriminant $51.62$
Ramified primes $3, 17$
Class number $1$ (GRH)
Class group Trivial (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![1, -12, -90, 424, 1995, -4968, -18724, 24507, 83349, -56140, -185646, 62388, 211167, -36012, -126369, 10764, 41631, -1692, -7783, 132, 819, -4, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 45*x^22 - 4*x^21 + 819*x^20 + 132*x^19 - 7783*x^18 - 1692*x^17 + 41631*x^16 + 10764*x^15 - 126369*x^14 - 36012*x^13 + 211167*x^12 + 62388*x^11 - 185646*x^10 - 56140*x^9 + 83349*x^8 + 24507*x^7 - 18724*x^6 - 4968*x^5 + 1995*x^4 + 424*x^3 - 90*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^24 - 45*x^22 - 4*x^21 + 819*x^20 + 132*x^19 - 7783*x^18 - 1692*x^17 + 41631*x^16 + 10764*x^15 - 126369*x^14 - 36012*x^13 + 211167*x^12 + 62388*x^11 - 185646*x^10 - 56140*x^9 + 83349*x^8 + 24507*x^7 - 18724*x^6 - 4968*x^5 + 1995*x^4 + 424*x^3 - 90*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{24} - 45 x^{22} - 4 x^{21} + 819 x^{20} + 132 x^{19} - 7783 x^{18} - 1692 x^{17} + 41631 x^{16} + 10764 x^{15} - 126369 x^{14} - 36012 x^{13} + 211167 x^{12} + 62388 x^{11} - 185646 x^{10} - 56140 x^{9} + 83349 x^{8} + 24507 x^{7} - 18724 x^{6} - 4968 x^{5} + 1995 x^{4} + 424 x^{3} - 90 x^{2} - 12 x + 1 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[24, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(128028748427622359924863503266793533356497=3^{32}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.62$
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:  $3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(153=3^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{153}(64,·)$, $\chi_{153}(1,·)$, $\chi_{153}(67,·)$, $\chi_{153}(4,·)$, $\chi_{153}(70,·)$, $\chi_{153}(76,·)$, $\chi_{153}(13,·)$, $\chi_{153}(16,·)$, $\chi_{153}(145,·)$, $\chi_{153}(19,·)$, $\chi_{153}(151,·)$, $\chi_{153}(25,·)$, $\chi_{153}(94,·)$, $\chi_{153}(100,·)$, $\chi_{153}(103,·)$, $\chi_{153}(106,·)$, $\chi_{153}(43,·)$, $\chi_{153}(49,·)$, $\chi_{153}(115,·)$, $\chi_{153}(52,·)$, $\chi_{153}(118,·)$, $\chi_{153}(55,·)$, $\chi_{153}(121,·)$, $\chi_{153}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{378306675502303071398085545224554737} a^{23} + \frac{166696513998291653232651827014502456}{378306675502303071398085545224554737} a^{22} - \frac{80258515865554001512031890615097837}{378306675502303071398085545224554737} a^{21} + \frac{74910079712383370945038735518607944}{378306675502303071398085545224554737} a^{20} - \frac{176213530082876468325386999067783950}{378306675502303071398085545224554737} a^{19} - \frac{176054618356105719164835052654808979}{378306675502303071398085545224554737} a^{18} - \frac{131519271064972140549773597584593562}{378306675502303071398085545224554737} a^{17} + \frac{128694555704573727601348519329091371}{378306675502303071398085545224554737} a^{16} + \frac{86141807643069871785546297157632617}{378306675502303071398085545224554737} a^{15} - \frac{74748607728494152530005875691016516}{378306675502303071398085545224554737} a^{14} + \frac{183281007092872619322361834157588581}{378306675502303071398085545224554737} a^{13} + \frac{99794821359111223626895926642470692}{378306675502303071398085545224554737} a^{12} + \frac{44838904540027529681179217323209227}{378306675502303071398085545224554737} a^{11} + \frac{152727856552119109204886472444986071}{378306675502303071398085545224554737} a^{10} - \frac{21784265066905589500603881626358346}{378306675502303071398085545224554737} a^{9} - \frac{52332756136993304531731965049701023}{378306675502303071398085545224554737} a^{8} - \frac{32437445009661604640947698104630561}{378306675502303071398085545224554737} a^{7} + \frac{59346276957710248751956628210976146}{378306675502303071398085545224554737} a^{6} + \frac{134991017087438194638247530076563456}{378306675502303071398085545224554737} a^{5} + \frac{50801498813711671179860484141721165}{378306675502303071398085545224554737} a^{4} + \frac{98784380562228105740545136287890435}{378306675502303071398085545224554737} a^{3} + \frac{72924855862895952593036166208162096}{378306675502303071398085545224554737} a^{2} - \frac{48074842754807292037106218578595405}{378306675502303071398085545224554737} a + \frac{131613847938353394322834448743552639}{378306675502303071398085545224554737}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $23$
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:  \( 5894416573994.629 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{9})^+\), 4.4.4913.1, 6.6.32234193.1, \(\Q(\zeta_{17})^+\), 12.12.5104819233548816337.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ R $24$ $24$ $24$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ $24$ $24$ $24$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{3}$ $24$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
3Data not computed
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$