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Properties

Label 24.0.13489901283...5625.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 5^{18}\cdot 13^{16}$
Root discriminant $32.02$
Ramified primes $3, 5, 13$
Class number $16$ (GRH)
Class group $[4, 4]$ (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, 4, 1, -50, -196, -698, -162, 8891, 34835, 8739, -11444, -11446, -4594, -2771, 2796, 3344, 670, -809, 163, 62, -16, 15, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 4*x^22 + 15*x^21 - 16*x^20 + 62*x^19 + 163*x^18 - 809*x^17 + 670*x^16 + 3344*x^15 + 2796*x^14 - 2771*x^13 - 4594*x^12 - 11446*x^11 - 11444*x^10 + 8739*x^9 + 34835*x^8 + 8891*x^7 - 162*x^6 - 698*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*x + 1)
 
gp: K = bnfinit(x^24 - x^23 - 4*x^22 + 15*x^21 - 16*x^20 + 62*x^19 + 163*x^18 - 809*x^17 + 670*x^16 + 3344*x^15 + 2796*x^14 - 2771*x^13 - 4594*x^12 - 11446*x^11 - 11444*x^10 + 8739*x^9 + 34835*x^8 + 8891*x^7 - 162*x^6 - 698*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 4 x^{22} + 15 x^{21} - 16 x^{20} + 62 x^{19} + 163 x^{18} - 809 x^{17} + 670 x^{16} + 3344 x^{15} + 2796 x^{14} - 2771 x^{13} - 4594 x^{12} - 11446 x^{11} - 11444 x^{10} + 8739 x^{9} + 34835 x^{8} + 8891 x^{7} - 162 x^{6} - 698 x^{5} - 196 x^{4} - 50 x^{3} + x^{2} + 4 x + 1 \)

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:  \(1348990128329918967746280670166015625=3^{12}\cdot 5^{18}\cdot 13^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.02$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(1,·)$, $\chi_{195}(131,·)$, $\chi_{195}(68,·)$, $\chi_{195}(133,·)$, $\chi_{195}(74,·)$, $\chi_{195}(139,·)$, $\chi_{195}(14,·)$, $\chi_{195}(79,·)$, $\chi_{195}(16,·)$, $\chi_{195}(146,·)$, $\chi_{195}(22,·)$, $\chi_{195}(152,·)$, $\chi_{195}(92,·)$, $\chi_{195}(29,·)$, $\chi_{195}(94,·)$, $\chi_{195}(107,·)$, $\chi_{195}(172,·)$, $\chi_{195}(157,·)$, $\chi_{195}(113,·)$, $\chi_{195}(178,·)$, $\chi_{195}(53,·)$, $\chi_{195}(118,·)$, $\chi_{195}(61,·)$, $\chi_{195}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{6781} a^{18} + \frac{675}{6781} a^{17} - \frac{1401}{6781} a^{15} - \frac{3116}{6781} a^{14} - \frac{532}{6781} a^{13} + \frac{3385}{6781} a^{12} - \frac{1448}{6781} a^{11} - \frac{578}{6781} a^{10} - \frac{2466}{6781} a^{9} + \frac{2556}{6781} a^{8} + \frac{3162}{6781} a^{7} + \frac{3337}{6781} a^{6} - \frac{588}{6781} a^{5} - \frac{1969}{6781} a^{4} + \frac{1}{6781} a^{3} - \frac{1298}{6781} a - \frac{1401}{6781}$, $\frac{1}{6781} a^{19} - \frac{1298}{6781} a^{17} - \frac{1401}{6781} a^{16} + \frac{658}{6781} a^{14} + \frac{3092}{6781} a^{13} - \frac{1126}{6781} a^{12} + \frac{358}{6781} a^{11} + \frac{1167}{6781} a^{10} - \frac{1020}{6781} a^{9} + \frac{236}{6781} a^{8} - \frac{1779}{6781} a^{7} - \frac{1771}{6781} a^{6} + \frac{1633}{6781} a^{5} - \frac{675}{6781} a^{3} - \frac{1298}{6781} a^{2} + \frac{3116}{6781}$, $\frac{1}{535699} a^{20} - \frac{5}{535699} a^{19} + \frac{243825}{535699} a^{17} + \frac{115501}{535699} a^{16} - \frac{136152}{535699} a^{15} - \frac{240625}{535699} a^{14} + \frac{99817}{535699} a^{13} + \frac{5630}{535699} a^{12} - \frac{164534}{535699} a^{11} + \frac{110869}{535699} a^{10} - \frac{137301}{535699} a^{9} + \frac{222593}{535699} a^{8} - \frac{4667}{535699} a^{7} + \frac{185161}{535699} a^{6} + \frac{83017}{535699} a^{5} - \frac{33}{79} a^{4} - \frac{125464}{535699} a^{3} + \frac{683}{6781} a^{2} - \frac{30}{79} a - \frac{247324}{535699}$, $\frac{1}{47184545236369} a^{21} - \frac{17080124}{47184545236369} a^{20} + \frac{1500080850}{47184545236369} a^{19} - \frac{1414680224}{47184545236369} a^{18} - \frac{1877983318866}{47184545236369} a^{17} - \frac{7007454641359}{47184545236369} a^{16} - \frac{3379018699049}{47184545236369} a^{15} + \frac{7040323998634}{47184545236369} a^{14} - \frac{4027367370299}{47184545236369} a^{13} + \frac{3887941698954}{47184545236369} a^{12} + \frac{13286602800625}{47184545236369} a^{11} - \frac{23009996697831}{47184545236369} a^{10} - \frac{22749799957235}{47184545236369} a^{9} - \frac{5503154423107}{47184545236369} a^{8} + \frac{8677897803958}{47184545236369} a^{7} - \frac{19535957558520}{47184545236369} a^{6} - \frac{1983064517430}{47184545236369} a^{5} + \frac{10595512088738}{47184545236369} a^{4} - \frac{11404493465426}{47184545236369} a^{3} - \frac{10846588671388}{47184545236369} a^{2} - \frac{11873330592484}{47184545236369} a - \frac{2387836934179}{47184545236369}$, $\frac{1}{47184545236369} a^{22} - \frac{33076046}{47184545236369} a^{20} + \frac{372334935}{47184545236369} a^{19} - \frac{2619680}{597272724511} a^{18} + \frac{11004207107753}{47184545236369} a^{17} + \frac{7332756595023}{47184545236369} a^{16} - \frac{22941253632919}{47184545236369} a^{15} + \frac{760943789928}{47184545236369} a^{14} - \frac{7122442825305}{47184545236369} a^{13} - \frac{8561061457225}{47184545236369} a^{12} + \frac{7808099354305}{47184545236369} a^{11} - \frac{11281529368793}{47184545236369} a^{10} - \frac{12651888130908}{47184545236369} a^{9} - \frac{20849388313915}{47184545236369} a^{8} + \frac{3928612349415}{47184545236369} a^{7} + \frac{15741425842458}{47184545236369} a^{6} - \frac{5418135874254}{47184545236369} a^{5} - \frac{1681538238348}{47184545236369} a^{4} - \frac{6454077701611}{47184545236369} a^{3} - \frac{270554649143}{597272724511} a^{2} + \frac{19384762548293}{47184545236369} a + \frac{7060940013033}{47184545236369}$, $\frac{1}{47184545236369} a^{23} - \frac{19998312}{47184545236369} a^{20} - \frac{22417316}{597272724511} a^{19} - \frac{172364210}{47184545236369} a^{18} + \frac{23018114605162}{47184545236369} a^{17} + \frac{174855771762}{597272724511} a^{16} + \frac{21648219442167}{47184545236369} a^{15} + \frac{22172224036193}{47184545236369} a^{14} - \frac{6893582052213}{47184545236369} a^{13} - \frac{7596374724848}{47184545236369} a^{12} - \frac{4217535432354}{47184545236369} a^{11} + \frac{19943358953226}{47184545236369} a^{10} - \frac{20112121712830}{47184545236369} a^{9} + \frac{7114695148}{597272724511} a^{8} + \frac{14206707948609}{47184545236369} a^{7} + \frac{19539348120385}{47184545236369} a^{6} + \frac{154812702830}{597272724511} a^{5} + \frac{4932422499134}{47184545236369} a^{4} - \frac{280990663123}{597272724511} a^{3} + \frac{103429228327}{597272724511} a^{2} + \frac{17904962031941}{47184545236369} a + \frac{30858710583}{597272724511}$

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

Class group and class number

$C_{4}\times C_{4}$, 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:  \( -\frac{766315375}{1522082104399} a^{23} - \frac{75711959050}{47184545236369} a^{22} + \frac{313852124985}{47184545236369} a^{21} - \frac{376230196510}{47184545236369} a^{20} - \frac{1211416582770}{47184545236369} a^{19} + \frac{3003189954625}{47184545236369} a^{18} - \frac{16723423039085}{47184545236369} a^{17} + \frac{16453220237860}{47184545236369} a^{16} + \frac{63936327861140}{47184545236369} a^{15} - \frac{286037666260465}{47184545236369} a^{14} - \frac{65923812765125}{47184545236369} a^{13} - \frac{80688717621400}{47184545236369} a^{12} - \frac{209958458230150}{47184545236369} a^{11} - \frac{183454674670400}{47184545236369} a^{10} + \frac{1844357238745210}{47184545236369} a^{9} + \frac{659239690934615}{47184545236369} a^{8} + \frac{42332763293135}{47184545236369} a^{7} - \frac{37561837031460}{47184545236369} a^{6} - \frac{13260014722850}{47184545236369} a^{5} - \frac{8388487554590514}{47184545236369} a^{4} - \frac{237680376710}{47184545236369} a^{3} + \frac{213005021635}{47184545236369} a^{2} + \frac{75711959050}{47184545236369} a + \frac{4444629175}{47184545236369} \) (order $30$)
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:  \( 183169998.253836 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), 3.3.169.1, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), 6.0.771147.1, 6.6.3570125.1, 6.0.96393375.1, \(\Q(\zeta_{15})\), 12.0.9291682743890625.1, 12.12.1161460342986328125.1, 12.0.1593224064453125.1

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 R ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
3Data not computed
$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$$[\ ]_{4}^{2}$
$13$13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$