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Properties

Label 24.24.2296127442...0417.1
Degree $24$
Signature $[24, 0]$
Discriminant $7^{16}\cdot 17^{21}$
Root discriminant $43.66$
Ramified primes $7, 17$
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, 8, -76, -214, 1370, 2302, -9753, -11047, 34507, 26654, -67445, -34634, 75800, 25292, -49508, -10883, 18986, 2783, -4276, -411, 550, 32, -37, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 37*x^22 + 32*x^21 + 550*x^20 - 411*x^19 - 4276*x^18 + 2783*x^17 + 18986*x^16 - 10883*x^15 - 49508*x^14 + 25292*x^13 + 75800*x^12 - 34634*x^11 - 67445*x^10 + 26654*x^9 + 34507*x^8 - 11047*x^7 - 9753*x^6 + 2302*x^5 + 1370*x^4 - 214*x^3 - 76*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^24 - x^23 - 37*x^22 + 32*x^21 + 550*x^20 - 411*x^19 - 4276*x^18 + 2783*x^17 + 18986*x^16 - 10883*x^15 - 49508*x^14 + 25292*x^13 + 75800*x^12 - 34634*x^11 - 67445*x^10 + 26654*x^9 + 34507*x^8 - 11047*x^7 - 9753*x^6 + 2302*x^5 + 1370*x^4 - 214*x^3 - 76*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 37 x^{22} + 32 x^{21} + 550 x^{20} - 411 x^{19} - 4276 x^{18} + 2783 x^{17} + 18986 x^{16} - 10883 x^{15} - 49508 x^{14} + 25292 x^{13} + 75800 x^{12} - 34634 x^{11} - 67445 x^{10} + 26654 x^{9} + 34507 x^{8} - 11047 x^{7} - 9753 x^{6} + 2302 x^{5} + 1370 x^{4} - 214 x^{3} - 76 x^{2} + 8 x + 1 \)

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:  \(2296127442650479958000916502307873630417=7^{16}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.66$
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:  $7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(119=7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{119}(64,·)$, $\chi_{119}(1,·)$, $\chi_{119}(2,·)$, $\chi_{119}(67,·)$, $\chi_{119}(4,·)$, $\chi_{119}(100,·)$, $\chi_{119}(8,·)$, $\chi_{119}(9,·)$, $\chi_{119}(15,·)$, $\chi_{119}(16,·)$, $\chi_{119}(81,·)$, $\chi_{119}(18,·)$, $\chi_{119}(86,·)$, $\chi_{119}(25,·)$, $\chi_{119}(93,·)$, $\chi_{119}(30,·)$, $\chi_{119}(32,·)$, $\chi_{119}(36,·)$, $\chi_{119}(106,·)$, $\chi_{119}(43,·)$, $\chi_{119}(72,·)$, $\chi_{119}(50,·)$, $\chi_{119}(53,·)$, $\chi_{119}(60,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{124443115529047578349036543249} a^{23} - \frac{20319689947768478839077300118}{124443115529047578349036543249} a^{22} - \frac{37828199293480920255676478719}{124443115529047578349036543249} a^{21} + \frac{17866155357051436238243468201}{124443115529047578349036543249} a^{20} + \frac{31712804218282491071928431476}{124443115529047578349036543249} a^{19} - \frac{31699999820067286091170648211}{124443115529047578349036543249} a^{18} - \frac{62069235766515930856952936747}{124443115529047578349036543249} a^{17} - \frac{4090132866511296148974581950}{124443115529047578349036543249} a^{16} - \frac{27416116458636780838789168422}{124443115529047578349036543249} a^{15} - \frac{41552152405288752687989498604}{124443115529047578349036543249} a^{14} - \frac{49498745926423890137414539699}{124443115529047578349036543249} a^{13} + \frac{55391603907048174063873888666}{124443115529047578349036543249} a^{12} - \frac{10808182309188718112766972200}{124443115529047578349036543249} a^{11} - \frac{52443071696719988466585642685}{124443115529047578349036543249} a^{10} - \frac{47130390839157268238008261703}{124443115529047578349036543249} a^{9} + \frac{32964444807806539419286268518}{124443115529047578349036543249} a^{8} + \frac{36387085809786144338291080306}{124443115529047578349036543249} a^{7} + \frac{11025080937397248722558418643}{124443115529047578349036543249} a^{6} - \frac{30983760426174364837278468808}{124443115529047578349036543249} a^{5} - \frac{37516466679554263825751339394}{124443115529047578349036543249} a^{4} - \frac{17094957812877852287728542141}{124443115529047578349036543249} a^{3} - \frac{3232662452616566710628292234}{124443115529047578349036543249} a^{2} + \frac{30976940305332140579470968625}{124443115529047578349036543249} a - \frac{46938636406964374786251312547}{124443115529047578349036543249}$

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:  \( 657704246476.2357 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.4913.1, 6.6.11796113.1, \(\Q(\zeta_{17})^+\), 12.12.683635509017782097.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ $24$ $24$ R $24$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ R ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
17Data not computed