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Properties

Label 24.0.17369039582...0000.2
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{12}\cdot 7^{20}$
Root discriminant $39.20$
Ramified primes $2, 3, 5, 7$
Class number $104$ (GRH)
Class group $[2, 52]$ (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![707281, 0, 666072, 0, 126028, 0, -35932, 0, 38660, 0, 46879, 0, 26174, 0, 10696, 0, 3338, 0, 776, 0, 133, 0, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 15*x^22 + 133*x^20 + 776*x^18 + 3338*x^16 + 10696*x^14 + 26174*x^12 + 46879*x^10 + 38660*x^8 - 35932*x^6 + 126028*x^4 + 666072*x^2 + 707281)
 
gp: K = bnfinit(x^24 + 15*x^22 + 133*x^20 + 776*x^18 + 3338*x^16 + 10696*x^14 + 26174*x^12 + 46879*x^10 + 38660*x^8 - 35932*x^6 + 126028*x^4 + 666072*x^2 + 707281, 1)
 

Normalized defining polynomial

\( x^{24} + 15 x^{22} + 133 x^{20} + 776 x^{18} + 3338 x^{16} + 10696 x^{14} + 26174 x^{12} + 46879 x^{10} + 38660 x^{8} - 35932 x^{6} + 126028 x^{4} + 666072 x^{2} + 707281 \)

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:  \(173690395826049922758414336000000000000=2^{24}\cdot 3^{12}\cdot 5^{12}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.20$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(11,·)$, $\chi_{420}(71,·)$, $\chi_{420}(139,·)$, $\chi_{420}(269,·)$, $\chi_{420}(401,·)$, $\chi_{420}(19,·)$, $\chi_{420}(151,·)$, $\chi_{420}(409,·)$, $\chi_{420}(331,·)$, $\chi_{420}(221,·)$, $\chi_{420}(199,·)$, $\chi_{420}(419,·)$, $\chi_{420}(229,·)$, $\chi_{420}(209,·)$, $\chi_{420}(361,·)$, $\chi_{420}(299,·)$, $\chi_{420}(349,·)$, $\chi_{420}(211,·)$, $\chi_{420}(89,·)$, $\chi_{420}(121,·)$, $\chi_{420}(59,·)$, $\chi_{420}(281,·)$, $\chi_{420}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} - \frac{4}{13} a^{8} + \frac{3}{13} a^{6} + \frac{1}{13} a^{4} - \frac{4}{13} a^{2} + \frac{3}{13}$, $\frac{1}{13} a^{11} - \frac{4}{13} a^{9} + \frac{3}{13} a^{7} + \frac{1}{13} a^{5} - \frac{4}{13} a^{3} + \frac{3}{13} a$, $\frac{1}{13} a^{12} - \frac{1}{13}$, $\frac{1}{377} a^{13} + \frac{7}{377} a^{11} + \frac{180}{377} a^{9} - \frac{148}{377} a^{7} + \frac{176}{377} a^{5} + \frac{11}{377} a^{3} + \frac{20}{377} a$, $\frac{1}{377} a^{14} + \frac{7}{377} a^{12} + \frac{6}{377} a^{10} + \frac{171}{377} a^{8} + \frac{31}{377} a^{6} - \frac{163}{377} a^{4} - \frac{38}{377} a^{2} - \frac{5}{13}$, $\frac{1}{377} a^{15} - \frac{14}{377} a^{11} - \frac{74}{377} a^{9} + \frac{23}{377} a^{7} + \frac{142}{377} a^{5} + \frac{146}{377} a^{3} + \frac{179}{377} a$, $\frac{1}{377} a^{16} - \frac{14}{377} a^{12} + \frac{1}{29} a^{10} + \frac{4}{29} a^{8} + \frac{2}{29} a^{6} - \frac{144}{377} a^{4} - \frac{13}{29} a^{2} - \frac{4}{13}$, $\frac{1}{377} a^{17} - \frac{5}{377} a^{11} + \frac{20}{377} a^{9} - \frac{132}{377} a^{7} - \frac{2}{13} a^{5} + \frac{72}{377} a^{3} - \frac{184}{377} a$, $\frac{1}{377} a^{18} - \frac{5}{377} a^{12} - \frac{9}{377} a^{10} - \frac{16}{377} a^{8} - \frac{5}{13} a^{6} + \frac{43}{377} a^{4} - \frac{68}{377} a^{2} - \frac{3}{13}$, $\frac{1}{377} a^{19} - \frac{3}{377} a^{11} - \frac{131}{377} a^{9} + \frac{159}{377} a^{7} + \frac{140}{377} a^{5} + \frac{103}{377} a^{3} - \frac{74}{377} a$, $\frac{1}{4901} a^{20} + \frac{5}{4901} a^{18} - \frac{4}{4901} a^{16} + \frac{4}{4901} a^{14} - \frac{176}{4901} a^{12} + \frac{57}{4901} a^{10} - \frac{1620}{4901} a^{8} + \frac{1726}{4901} a^{6} - \frac{2136}{4901} a^{4} + \frac{574}{4901} a^{2} - \frac{10}{169}$, $\frac{1}{4901} a^{21} + \frac{5}{4901} a^{19} - \frac{4}{4901} a^{17} + \frac{4}{4901} a^{15} + \frac{6}{4901} a^{13} - \frac{177}{4901} a^{11} - \frac{2036}{4901} a^{9} - \frac{328}{4901} a^{7} - \frac{1018}{4901} a^{5} - \frac{1194}{4901} a^{3} - \frac{1174}{4901} a$, $\frac{1}{20423139363731702429} a^{22} - \frac{1550166015890593}{20423139363731702429} a^{20} - \frac{11377673098399102}{20423139363731702429} a^{18} + \frac{24071398270583312}{20423139363731702429} a^{16} - \frac{22296408850646244}{20423139363731702429} a^{14} - \frac{29469502588022361}{20423139363731702429} a^{12} - \frac{590151798345555139}{20423139363731702429} a^{10} + \frac{262267545808498847}{1571010720287054033} a^{8} - \frac{4403228701799317193}{20423139363731702429} a^{6} - \frac{738551272140147508}{20423139363731702429} a^{4} + \frac{1531672289193118740}{20423139363731702429} a^{2} - \frac{9014218316092810}{24284351205388469}$, $\frac{1}{592271041548219370441} a^{23} + \frac{15118382740470723}{592271041548219370441} a^{21} - \frac{307244413523812461}{592271041548219370441} a^{19} - \frac{259293930587559060}{592271041548219370441} a^{17} - \frac{226486131116072365}{592271041548219370441} a^{15} + \frac{70541789950145535}{592271041548219370441} a^{13} - \frac{14808423887521757687}{592271041548219370441} a^{11} + \frac{185675891609132385142}{592271041548219370441} a^{9} - \frac{120057954247812308259}{592271041548219370441} a^{7} - \frac{13480122169422058824}{45559310888324566957} a^{5} - \frac{206837688576890602247}{592271041548219370441} a^{3} - \frac{95087155428682709}{704246184956265601} a$

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

Class group and class number

$C_{2}\times C_{52}$, which has order $104$ (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:  \( \frac{2511707008300}{704246184956265601} a^{23} + \frac{32458926990364}{704246184956265601} a^{21} + \frac{267108195416055}{704246184956265601} a^{19} + \frac{1402721106847039}{704246184956265601} a^{17} + \frac{5568709201308533}{704246184956265601} a^{15} + \frac{15998685007301094}{704246184956265601} a^{13} + \frac{36574428351788362}{704246184956265601} a^{11} + \frac{58668200384014340}{704246184956265601} a^{9} + \frac{27936455098991370}{704246184956265601} a^{7} - \frac{3198296674575037}{54172783458174277} a^{5} + \frac{17295319642698244}{24284351205388469} a^{3} + \frac{512894168565599205}{704246184956265601} a \) (order $12$)
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:  \( 366249358.8925901 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{35}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{105})\), \(\Q(i, \sqrt{35})\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{35})\), \(\Q(\sqrt{-3}, \sqrt{35})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), 6.0.153664.1, 6.6.4148928.1, 6.0.64827.1, 6.0.3630312000.2, 6.6.56723625.1, 6.0.2100875.1, 6.6.134456000.1, 8.0.31116960000.10, 12.0.17213603549184.1, 12.0.13179165217344000000.2, 12.0.18078415936000000.2, 12.0.13179165217344000000.9, 12.12.13179165217344000000.2, 12.0.13179165217344000000.6, 12.0.3217569633140625.2

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 R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{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}$