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Properties

Label 24.0.72498183345...5625.1
Degree $24$
Signature $[0, 12]$
Discriminant $5^{18}\cdot 13^{20}$
Root discriminant $28.35$
Ramified primes $5, 13$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (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![1, 3, 15, 59, 250, -193, 614, -593, 1501, -237, 1318, -86, 1114, -353, 643, -205, 317, -58, 94, -19, 27, -7, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 6*x^22 - 7*x^21 + 27*x^20 - 19*x^19 + 94*x^18 - 58*x^17 + 317*x^16 - 205*x^15 + 643*x^14 - 353*x^13 + 1114*x^12 - 86*x^11 + 1318*x^10 - 237*x^9 + 1501*x^8 - 593*x^7 + 614*x^6 - 193*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^24 - x^23 + 6*x^22 - 7*x^21 + 27*x^20 - 19*x^19 + 94*x^18 - 58*x^17 + 317*x^16 - 205*x^15 + 643*x^14 - 353*x^13 + 1114*x^12 - 86*x^11 + 1318*x^10 - 237*x^9 + 1501*x^8 - 593*x^7 + 614*x^6 - 193*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 6 x^{22} - 7 x^{21} + 27 x^{20} - 19 x^{19} + 94 x^{18} - 58 x^{17} + 317 x^{16} - 205 x^{15} + 643 x^{14} - 353 x^{13} + 1114 x^{12} - 86 x^{11} + 1318 x^{10} - 237 x^{9} + 1501 x^{8} - 593 x^{7} + 614 x^{6} - 193 x^{5} + 250 x^{4} + 59 x^{3} + 15 x^{2} + 3 x + 1 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(72498183345339963679508209228515625=5^{18}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.35$
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:  $5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(65=5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{65}(64,·)$, $\chi_{65}(1,·)$, $\chi_{65}(3,·)$, $\chi_{65}(4,·)$, $\chi_{65}(9,·)$, $\chi_{65}(12,·)$, $\chi_{65}(14,·)$, $\chi_{65}(16,·)$, $\chi_{65}(17,·)$, $\chi_{65}(22,·)$, $\chi_{65}(23,·)$, $\chi_{65}(27,·)$, $\chi_{65}(29,·)$, $\chi_{65}(36,·)$, $\chi_{65}(38,·)$, $\chi_{65}(42,·)$, $\chi_{65}(43,·)$, $\chi_{65}(48,·)$, $\chi_{65}(49,·)$, $\chi_{65}(51,·)$, $\chi_{65}(53,·)$, $\chi_{65}(56,·)$, $\chi_{65}(61,·)$, $\chi_{65}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{1445643759919} a^{21} - \frac{545276790517}{1445643759919} a^{20} + \frac{615546033350}{1445643759919} a^{19} - \frac{450642466093}{1445643759919} a^{18} - \frac{98378289150}{1445643759919} a^{17} + \frac{31198496828}{1445643759919} a^{16} + \frac{529838603505}{1445643759919} a^{15} + \frac{113352996699}{1445643759919} a^{14} - \frac{564197865009}{1445643759919} a^{13} + \frac{16331163075}{1445643759919} a^{12} - \frac{15567765193}{1445643759919} a^{11} - \frac{141867390675}{1445643759919} a^{10} - \frac{300826242972}{1445643759919} a^{9} - \frac{208465917818}{1445643759919} a^{8} + \frac{508367897020}{1445643759919} a^{7} - \frac{636551380589}{1445643759919} a^{6} - \frac{107303772139}{1445643759919} a^{5} - \frac{700355649864}{1445643759919} a^{4} + \frac{674570434851}{1445643759919} a^{3} + \frac{371462421767}{1445643759919} a^{2} - \frac{685643609770}{1445643759919} a - \frac{195689327083}{1445643759919}$, $\frac{1}{1445643759919} a^{22} - \frac{11518196924}{1445643759919} a^{20} - \frac{467276115929}{1445643759919} a^{19} + \frac{409685131303}{1445643759919} a^{18} - \frac{557375797327}{1445643759919} a^{17} + \frac{149281353153}{1445643759919} a^{16} + \frac{386299897432}{1445643759919} a^{15} + \frac{514143645886}{1445643759919} a^{14} - \frac{674526751747}{1445643759919} a^{13} + \frac{150765032714}{1445643759919} a^{12} - \frac{496141072407}{1445643759919} a^{11} + \frac{123635468964}{1445643759919} a^{10} - \frac{484124230646}{1445643759919} a^{9} - \frac{694772769238}{1445643759919} a^{8} + \frac{376286591203}{1445643759919} a^{7} + \frac{650871630636}{1445643759919} a^{6} + \frac{443994073545}{1445643759919} a^{5} - \frac{4893280191}{1445643759919} a^{4} - \frac{166106579447}{1445643759919} a^{3} - \frac{530482488819}{1445643759919} a^{2} + \frac{166490853936}{1445643759919} a - \frac{290199457394}{1445643759919}$, $\frac{1}{1445643759919} a^{23} - \frac{2921529482}{1445643759919} a^{20} + \frac{624776915251}{1445643759919} a^{19} + \frac{149734378737}{1445643759919} a^{18} - \frac{585348258765}{1445643759919} a^{17} - \frac{94937716659}{1445643759919} a^{16} - \frac{298676141967}{1445643759919} a^{15} - \frac{61454505577}{1445643759919} a^{14} + \frac{387113597936}{1445643759919} a^{13} + \frac{543085858416}{1445643759919} a^{12} - \frac{36957153463}{1445643759919} a^{11} + \frac{351554201768}{1445643759919} a^{10} + \frac{528222622044}{1445643759919} a^{9} + \frac{694909117606}{1445643759919} a^{8} + \frac{643944498019}{1445643759919} a^{7} + \frac{363349832697}{1445643759919} a^{6} - \frac{138138769758}{1445643759919} a^{5} - \frac{669101080326}{1445643759919} a^{4} - \frac{707601791392}{1445643759919} a^{3} - \frac{435431176893}{1445643759919} a^{2} - \frac{115414909970}{1445643759919} a + \frac{577837003617}{1445643759919}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  \( \frac{341728824381}{1445643759919} a^{23} - \frac{336966501786}{1445643759919} a^{22} + \frac{2049420481767}{1445643759919} a^{21} - \frac{2373052480287}{1445643759919} a^{20} + \frac{9219314197064}{1445643759919} a^{19} - \frac{6421412824314}{1445643759919} a^{18} + \frac{32156798214498}{1445643759919} a^{17} - \frac{149416667967}{11035448549} a^{16} + \frac{108434713354905}{1445643759919} a^{15} - \frac{69218274924038}{1445643759919} a^{14} + \frac{220007848787493}{1445643759919} a^{13} - \frac{119905449507534}{1445643759919} a^{12} + \frac{381665043885966}{1445643759919} a^{11} - \frac{28215242609358}{1445643759919} a^{10} + \frac{454112955994722}{1445643759919} a^{9} - \frac{79785816226281}{1445643759919} a^{8} + \frac{513799803179133}{1445643759919} a^{7} - \frac{202625191103034}{1445643759919} a^{6} + \frac{210222485732433}{1445643759919} a^{5} - \frac{73055463437345}{1445643759919} a^{4} + \frac{85726517631621}{1445643759919} a^{3} + \frac{20232483012885}{1445643759919} a^{2} + \frac{5143076727057}{1445643759919} a + \frac{1028996331219}{1445643759919} \) (order $10$)
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:  \( 7346081.887826216 \) (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

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\zeta_{5})\), 4.0.21125.1, 6.6.46411625.1, 6.6.3570125.1, \(\Q(\zeta_{13})^+\), 8.0.446265625.1, 12.12.2154038935140625.1, 12.0.1593224064453125.1, 12.0.269254866892578125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13Data not computed