↑

Properties

Label 24.0.42013892329...0736.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 7^{20}\cdot 11^{12}$
Root discriminant $33.57$
Ramified primes $2, 7, 11$
Class number $14$ (GRH)
Class group $[14]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![531441, 0, 295245, 0, 104976, 0, 25515, 0, 2511, 0, -1440, 0, -1079, 0, -160, 0, 31, 0, 35, 0, 16, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 5*x^22 + 16*x^20 + 35*x^18 + 31*x^16 - 160*x^14 - 1079*x^12 - 1440*x^10 + 2511*x^8 + 25515*x^6 + 104976*x^4 + 295245*x^2 + 531441)
 
gp: K = bnfinit(x^24 + 5*x^22 + 16*x^20 + 35*x^18 + 31*x^16 - 160*x^14 - 1079*x^12 - 1440*x^10 + 2511*x^8 + 25515*x^6 + 104976*x^4 + 295245*x^2 + 531441, 1)
 

Normalized defining polynomial

\( x^{24} + 5 x^{22} + 16 x^{20} + 35 x^{18} + 31 x^{16} - 160 x^{14} - 1079 x^{12} - 1440 x^{10} + 2511 x^{8} + 25515 x^{6} + 104976 x^{4} + 295245 x^{2} + 531441 \)

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:  \(4201389232919273300173938291676020736=2^{24}\cdot 7^{20}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.57$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(308=2^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(131,·)$, $\chi_{308}(197,·)$, $\chi_{308}(263,·)$, $\chi_{308}(265,·)$, $\chi_{308}(87,·)$, $\chi_{308}(109,·)$, $\chi_{308}(67,·)$, $\chi_{308}(23,·)$, $\chi_{308}(153,·)$, $\chi_{308}(155,·)$, $\chi_{308}(285,·)$, $\chi_{308}(199,·)$, $\chi_{308}(219,·)$, $\chi_{308}(243,·)$, $\chi_{308}(65,·)$, $\chi_{308}(241,·)$, $\chi_{308}(43,·)$, $\chi_{308}(45,·)$, $\chi_{308}(221,·)$, $\chi_{308}(177,·)$, $\chi_{308}(307,·)$, $\chi_{308}(89,·)$, $\chi_{308}(111,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9711} a^{14} + \frac{4}{9} a^{12} + \frac{2}{9} a^{10} + \frac{1}{9} a^{8} - \frac{4}{9} a^{6} - \frac{2}{9} a^{4} - \frac{1}{9} a^{2} - \frac{160}{1079}$, $\frac{1}{29133} a^{15} + \frac{4}{27} a^{13} + \frac{2}{27} a^{11} + \frac{1}{27} a^{9} - \frac{13}{27} a^{7} + \frac{7}{27} a^{5} - \frac{10}{27} a^{3} + \frac{919}{3237} a$, $\frac{1}{87399} a^{16} - \frac{4}{87399} a^{14} + \frac{2}{81} a^{12} - \frac{26}{81} a^{10} + \frac{14}{81} a^{8} - \frac{20}{81} a^{6} + \frac{17}{81} a^{4} + \frac{919}{9711} a^{2} + \frac{191}{1079}$, $\frac{1}{262197} a^{17} - \frac{4}{262197} a^{15} + \frac{2}{243} a^{13} - \frac{26}{243} a^{11} + \frac{95}{243} a^{9} - \frac{20}{243} a^{7} + \frac{17}{243} a^{5} + \frac{919}{29133} a^{3} + \frac{191}{3237} a$, $\frac{1}{786591} a^{18} - \frac{4}{786591} a^{16} - \frac{29}{786591} a^{14} - \frac{269}{729} a^{12} + \frac{95}{729} a^{10} - \frac{20}{729} a^{8} - \frac{226}{729} a^{6} + \frac{919}{87399} a^{4} + \frac{191}{9711} a^{2} + \frac{4}{1079}$, $\frac{1}{2359773} a^{19} - \frac{4}{2359773} a^{17} - \frac{29}{2359773} a^{15} - \frac{269}{2187} a^{13} + \frac{824}{2187} a^{11} - \frac{20}{2187} a^{9} - \frac{955}{2187} a^{7} - \frac{86480}{262197} a^{5} + \frac{9902}{29133} a^{3} - \frac{1075}{3237} a$, $\frac{1}{7079319} a^{20} - \frac{4}{7079319} a^{18} - \frac{29}{7079319} a^{16} - \frac{109}{7079319} a^{14} - \frac{2092}{6561} a^{12} - \frac{1478}{6561} a^{10} - \frac{1684}{6561} a^{8} + \frac{919}{786591} a^{6} + \frac{191}{87399} a^{4} + \frac{4}{9711} a^{2} - \frac{19}{1079}$, $\frac{1}{21237957} a^{21} - \frac{4}{21237957} a^{19} - \frac{29}{21237957} a^{17} - \frac{109}{21237957} a^{15} - \frac{2092}{19683} a^{13} + \frac{5083}{19683} a^{11} + \frac{4877}{19683} a^{9} + \frac{787510}{2359773} a^{7} - \frac{87208}{262197} a^{5} + \frac{9715}{29133} a^{3} - \frac{366}{1079} a$, $\frac{1}{63713871} a^{22} - \frac{4}{63713871} a^{20} - \frac{29}{63713871} a^{18} - \frac{109}{63713871} a^{16} - \frac{284}{63713871} a^{14} - \frac{1478}{59049} a^{12} + \frac{11438}{59049} a^{10} + \frac{919}{7079319} a^{8} + \frac{191}{786591} a^{6} + \frac{4}{87399} a^{4} - \frac{19}{9711} a^{2} - \frac{11}{1079}$, $\frac{1}{191141613} a^{23} - \frac{4}{191141613} a^{21} - \frac{29}{191141613} a^{19} - \frac{109}{191141613} a^{17} - \frac{284}{191141613} a^{15} - \frac{1478}{177147} a^{13} - \frac{47611}{177147} a^{11} + \frac{919}{21237957} a^{9} + \frac{191}{2359773} a^{7} + \frac{4}{262197} a^{5} - \frac{19}{29133} a^{3} - \frac{11}{3237} a$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( -\frac{1}{262197} a^{19} + \frac{15467}{262197} a^{5} \) (order $28$)
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:  \( 151397348.565616 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_6$ (as 24T3):

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 $C_2^2\times C_6$
Character table for $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-77}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(i, \sqrt{77})\), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{7}, \sqrt{-11})\), \(\Q(\sqrt{-7}, \sqrt{11})\), \(\Q(\sqrt{-7}, \sqrt{-11})\), \(\Q(\sqrt{7}, \sqrt{11})\), 6.0.153664.1, 6.0.1431687488.1, 6.6.22370117.1, 6.0.3195731.1, 6.6.204526784.1, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{7})\), 8.0.8999178496.1, 12.0.2049729063295750144.2, 12.0.41831205373382656.1, \(\Q(\zeta_{28})\), 12.0.2049729063295750144.3, 12.0.2049729063295750144.1, 12.0.500422134593689.1, 12.12.2049729063295750144.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$11$11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$