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Properties

Label 24.0.31188962191...4896.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 7^{20}\cdot 13^{12}$
Root discriminant $36.50$
Ramified primes $2, 7, 13$
Class number $48$ (GRH)
Class group $[4, 12]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![531441, 0, -413343, 0, 262440, 0, -158193, 0, 93879, 0, -55440, 0, 32689, 0, -6160, 0, 1159, 0, -217, 0, 40, 0, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 7*x^22 + 40*x^20 - 217*x^18 + 1159*x^16 - 6160*x^14 + 32689*x^12 - 55440*x^10 + 93879*x^8 - 158193*x^6 + 262440*x^4 - 413343*x^2 + 531441)
 
gp: K = bnfinit(x^24 - 7*x^22 + 40*x^20 - 217*x^18 + 1159*x^16 - 6160*x^14 + 32689*x^12 - 55440*x^10 + 93879*x^8 - 158193*x^6 + 262440*x^4 - 413343*x^2 + 531441, 1)
 

Normalized defining polynomial

\( x^{24} - 7 x^{22} + 40 x^{20} - 217 x^{18} + 1159 x^{16} - 6160 x^{14} + 32689 x^{12} - 55440 x^{10} + 93879 x^{8} - 158193 x^{6} + 262440 x^{4} - 413343 x^{2} + 531441 \)

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:  \(31188962191164288779371841230414544896=2^{24}\cdot 7^{20}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.50$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(364=2^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(131,·)$, $\chi_{364}(261,·)$, $\chi_{364}(129,·)$, $\chi_{364}(103,·)$, $\chi_{364}(183,·)$, $\chi_{364}(207,·)$, $\chi_{364}(79,·)$, $\chi_{364}(209,·)$, $\chi_{364}(339,·)$, $\chi_{364}(25,·)$, $\chi_{364}(27,·)$, $\chi_{364}(285,·)$, $\chi_{364}(155,·)$, $\chi_{364}(363,·)$, $\chi_{364}(337,·)$, $\chi_{364}(233,·)$, $\chi_{364}(235,·)$, $\chi_{364}(157,·)$, $\chi_{364}(51,·)$, $\chi_{364}(53,·)$, $\chi_{364}(311,·)$, $\chi_{364}(313,·)$, $\chi_{364}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{294201} a^{14} + \frac{2}{9} a^{12} + \frac{4}{9} a^{10} - \frac{1}{9} a^{8} - \frac{2}{9} a^{6} - \frac{4}{9} a^{4} + \frac{1}{9} a^{2} - \frac{6160}{32689}$, $\frac{1}{882603} a^{15} + \frac{2}{27} a^{13} + \frac{4}{27} a^{11} + \frac{8}{27} a^{9} - \frac{11}{27} a^{7} + \frac{5}{27} a^{5} + \frac{10}{27} a^{3} - \frac{38849}{98067} a$, $\frac{1}{2647809} a^{16} + \frac{2}{2647809} a^{14} + \frac{40}{81} a^{12} + \frac{26}{81} a^{10} + \frac{25}{81} a^{8} - \frac{4}{81} a^{6} - \frac{35}{81} a^{4} + \frac{8843}{98067} a^{2} - \frac{5001}{32689}$, $\frac{1}{7943427} a^{17} + \frac{2}{7943427} a^{15} + \frac{40}{243} a^{13} + \frac{26}{243} a^{11} - \frac{56}{243} a^{9} - \frac{85}{243} a^{7} - \frac{116}{243} a^{5} + \frac{8843}{294201} a^{3} - \frac{1667}{32689} a$, $\frac{1}{23830281} a^{18} + \frac{2}{23830281} a^{16} - \frac{23}{23830281} a^{14} + \frac{269}{729} a^{12} - \frac{56}{729} a^{10} + \frac{158}{729} a^{8} + \frac{127}{729} a^{6} + \frac{8843}{882603} a^{4} - \frac{1667}{98067} a^{2} + \frac{942}{32689}$, $\frac{1}{71490843} a^{19} + \frac{2}{71490843} a^{17} - \frac{23}{71490843} a^{15} + \frac{269}{2187} a^{13} - \frac{56}{2187} a^{11} + \frac{158}{2187} a^{9} - \frac{602}{2187} a^{7} - \frac{873760}{2647809} a^{5} + \frac{96400}{294201} a^{3} - \frac{31747}{98067} a$, $\frac{1}{214472529} a^{20} + \frac{2}{214472529} a^{18} - \frac{23}{214472529} a^{16} + \frac{143}{214472529} a^{14} - \frac{785}{6561} a^{12} + \frac{3074}{6561} a^{10} - \frac{1331}{6561} a^{8} + \frac{8843}{7943427} a^{6} - \frac{1667}{882603} a^{4} + \frac{314}{98067} a^{2} - \frac{177}{32689}$, $\frac{1}{643417587} a^{21} + \frac{2}{643417587} a^{19} - \frac{23}{643417587} a^{17} + \frac{143}{643417587} a^{15} - \frac{785}{19683} a^{13} - \frac{3487}{19683} a^{11} - \frac{7892}{19683} a^{9} - \frac{7934584}{23830281} a^{7} + \frac{880936}{2647809} a^{5} - \frac{97753}{294201} a^{3} + \frac{32512}{98067} a$, $\frac{1}{1930252761} a^{22} + \frac{2}{1930252761} a^{20} - \frac{23}{1930252761} a^{18} + \frac{143}{1930252761} a^{16} - \frac{794}{1930252761} a^{14} + \frac{22757}{59049} a^{12} + \frac{24913}{59049} a^{10} + \frac{8843}{71490843} a^{8} - \frac{1667}{7943427} a^{6} + \frac{314}{882603} a^{4} - \frac{59}{98067} a^{2} + \frac{33}{32689}$, $\frac{1}{5790758283} a^{23} + \frac{2}{5790758283} a^{21} - \frac{23}{5790758283} a^{19} + \frac{143}{5790758283} a^{17} - \frac{794}{5790758283} a^{15} + \frac{22757}{177147} a^{13} + \frac{83962}{177147} a^{11} + \frac{71499686}{214472529} a^{9} - \frac{7945094}{23830281} a^{7} + \frac{882917}{2647809} a^{5} - \frac{98126}{294201} a^{3} + \frac{32722}{98067} a$

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

Class group and class number

$C_{4}\times C_{12}$, which has order $48$ (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:  \( \frac{18781}{5790758283} a^{23} - \frac{107320}{5790758283} a^{21} + \frac{582211}{5790758283} a^{19} - \frac{3109597}{5790758283} a^{17} + \frac{16527280}{5790758283} a^{15} - \frac{2683}{177147} a^{13} + \frac{14209}{177147} a^{11} - \frac{3109597}{71490843} a^{9} + \frac{582211}{7943427} a^{7} - \frac{107320}{882603} a^{5} + \frac{18781}{98067} a^{3} - \frac{8049}{32689} a \) (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:  \( 268691479.40790963 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{91}) \), \(\Q(\sqrt{-91}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\zeta_{7})^+\), \(\Q(i, \sqrt{91})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{13})\), \(\Q(\sqrt{7}, \sqrt{13})\), \(\Q(\sqrt{-7}, \sqrt{-13})\), \(\Q(\sqrt{7}, \sqrt{-13})\), \(\Q(\sqrt{-7}, \sqrt{13})\), 6.0.153664.1, 6.6.2363198656.1, 6.0.36924979.1, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{7})\), 6.6.5274997.1, 6.0.337599808.1, 8.0.17555190016.1, 12.0.5584707887720206336.1, \(\Q(\zeta_{28})\), 12.0.113973630361636864.1, 12.12.5584707887720206336.1, 12.0.5584707887720206336.2, 12.0.5584707887720206336.3, 12.0.1363454074150441.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$