↑

Properties

Label 24.0.16176178662...5625.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 5^{18}\cdot 7^{20}$
Root discriminant $29.31$
Ramified primes $3, 5, 7$
Class number $26$ (GRH)
Class group $[26]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 56, -378, 2534, -3548, 6187, -8224, 10534, -4802, 6504, -2395, 3557, -1394, 1618, -588, 629, -146, 153, -34, 35, -7, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 7*x^22 - 7*x^21 + 35*x^20 - 34*x^19 + 153*x^18 - 146*x^17 + 629*x^16 - 588*x^15 + 1618*x^14 - 1394*x^13 + 3557*x^12 - 2395*x^11 + 6504*x^10 - 4802*x^9 + 10534*x^8 - 8224*x^7 + 6187*x^6 - 3548*x^5 + 2534*x^4 - 378*x^3 + 56*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^24 - x^23 + 7*x^22 - 7*x^21 + 35*x^20 - 34*x^19 + 153*x^18 - 146*x^17 + 629*x^16 - 588*x^15 + 1618*x^14 - 1394*x^13 + 3557*x^12 - 2395*x^11 + 6504*x^10 - 4802*x^9 + 10534*x^8 - 8224*x^7 + 6187*x^6 - 3548*x^5 + 2534*x^4 - 378*x^3 + 56*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 7 x^{22} - 7 x^{21} + 35 x^{20} - 34 x^{19} + 153 x^{18} - 146 x^{17} + 629 x^{16} - 588 x^{15} + 1618 x^{14} - 1394 x^{13} + 3557 x^{12} - 2395 x^{11} + 6504 x^{10} - 4802 x^{9} + 10534 x^{8} - 8224 x^{7} + 6187 x^{6} - 3548 x^{5} + 2534 x^{4} - 378 x^{3} + 56 x^{2} - 8 x + 1 \)

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:  \(161761786626698377317203521728515625=3^{12}\cdot 5^{18}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.31$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(67,·)$, $\chi_{105}(4,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(17,·)$, $\chi_{105}(83,·)$, $\chi_{105}(22,·)$, $\chi_{105}(89,·)$, $\chi_{105}(88,·)$, $\chi_{105}(68,·)$, $\chi_{105}(26,·)$, $\chi_{105}(101,·)$, $\chi_{105}(37,·)$, $\chi_{105}(38,·)$, $\chi_{105}(104,·)$, $\chi_{105}(41,·)$, $\chi_{105}(43,·)$, $\chi_{105}(46,·)$, $\chi_{105}(47,·)$, $\chi_{105}(58,·)$, $\chi_{105}(59,·)$, $\chi_{105}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{586382566462601} a^{21} - \frac{171538548989333}{586382566462601} a^{20} - \frac{70125785132858}{586382566462601} a^{19} + \frac{213659624122068}{586382566462601} a^{18} + \frac{95502070532581}{586382566462601} a^{17} - \frac{192023246337468}{586382566462601} a^{16} - \frac{90437479961259}{586382566462601} a^{15} - \frac{27070111217202}{586382566462601} a^{14} + \frac{33815946558713}{586382566462601} a^{13} - \frac{180351711312451}{586382566462601} a^{12} + \frac{82019629063228}{586382566462601} a^{11} + \frac{214886966053931}{586382566462601} a^{10} - \frac{109698136901594}{586382566462601} a^{9} + \frac{158581104538649}{586382566462601} a^{8} + \frac{52627448885242}{586382566462601} a^{7} - \frac{205693131590540}{586382566462601} a^{6} - \frac{5661939767723}{586382566462601} a^{5} - \frac{285708907636718}{586382566462601} a^{4} - \frac{216861545151990}{586382566462601} a^{3} + \frac{120963384291680}{586382566462601} a^{2} + \frac{206237200571440}{586382566462601} a - \frac{186258710837106}{586382566462601}$, $\frac{1}{586382566462601} a^{22} - \frac{186215045205283}{586382566462601} a^{20} - \frac{15661384932351}{586382566462601} a^{19} + \frac{71136246625847}{586382566462601} a^{18} - \frac{94132030330856}{586382566462601} a^{17} + \frac{173051526941893}{586382566462601} a^{16} - \frac{276103737934819}{586382566462601} a^{15} + \frac{105342405320544}{586382566462601} a^{14} + \frac{62354549756924}{586382566462601} a^{13} + \frac{12272366125966}{586382566462601} a^{12} + \frac{195850814920183}{586382566462601} a^{11} + \frac{102355701839496}{586382566462601} a^{10} - \frac{279782116565472}{586382566462601} a^{9} + \frac{193898351814680}{586382566462601} a^{8} - \frac{64494144274257}{586382566462601} a^{7} + \frac{72569749588502}{586382566462601} a^{6} + \frac{92788436701071}{586382566462601} a^{5} - \frac{107780608046877}{586382566462601} a^{4} - \frac{241904384436198}{586382566462601} a^{3} - \frac{161289357740575}{586382566462601} a^{2} + \frac{268365935802107}{586382566462601} a - \frac{139808032505916}{586382566462601}$, $\frac{1}{586382566462601} a^{23} - \frac{179373992853381}{586382566462601} a^{20} - \frac{161398474765477}{586382566462601} a^{19} + \frac{273417314765986}{586382566462601} a^{18} + \frac{42975809566870}{586382566462601} a^{17} + \frac{211995488701823}{586382566462601} a^{16} + \frac{219490812382700}{586382566462601} a^{15} - \frac{264594525294754}{586382566462601} a^{14} - \frac{60077322863200}{586382566462601} a^{13} - \frac{135662957979370}{586382566462601} a^{12} - \frac{285027841217279}{586382566462601} a^{11} - \frac{23733207765291}{586382566462601} a^{10} - \frac{269918783075074}{586382566462601} a^{9} + \frac{237259122924358}{586382566462601} a^{8} + \frac{243057377254032}{586382566462601} a^{7} - \frac{115133424425142}{586382566462601} a^{6} - \frac{257645977686546}{586382566462601} a^{5} - \frac{34917752774359}{586382566462601} a^{4} + \frac{34716266604898}{586382566462601} a^{3} - \frac{173574082386098}{586382566462601} a^{2} - \frac{278327275127020}{586382566462601} a - \frac{57165082815132}{586382566462601}$

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

Class group and class number

$C_{26}$, which has order $26$ (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:  \( -\frac{87543327880547}{586382566462601} a^{23} + \frac{87683025935477}{586382566462601} a^{22} - \frac{612803295163829}{586382566462601} a^{21} + \frac{613455219420169}{586382566462601} a^{20} - \frac{3064007668486569}{586382566462601} a^{19} + \frac{2979243826028043}{586382566462601} a^{18} - \frac{13393966184659606}{586382566462601} a^{17} + \frac{12792571563981727}{586382566462601} a^{16} - \frac{55063798633488708}{586382566462601} a^{15} + \frac{51520059395520031}{586382566462601} a^{14} - \frac{141639889116674326}{586382566462601} a^{13} + \frac{122085760214284783}{586382566462601} a^{12} - \frac{311364562414467569}{586382566462601} a^{11} + \frac{209761940158527960}{586382566462601} a^{10} - \frac{569255340353525742}{586382566462601} a^{9} + \frac{420516518690863154}{586382566462601} a^{8} - \frac{922127632142534048}{586382566462601} a^{7} + \frac{719908900999969793}{586382566462601} a^{6} - \frac{541569125735784244}{586382566462601} a^{5} + \frac{308960363330819046}{586382566462601} a^{4} - \frac{221784594681567918}{586382566462601} a^{3} + \frac{33083880809898856}{586382566462601} a^{2} - \frac{4901308776871192}{586382566462601} a + \frac{700183641980291}{586382566462601} \) (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:  \( 9216624.607005743 \) (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

\(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{21})\), 4.0.55125.1, \(\Q(\zeta_{5})\), 6.6.56723625.1, 6.6.300125.1, \(\Q(\zeta_{21})^+\), 8.0.3038765625.3, 12.12.3217569633140625.1, 12.0.402196204142578125.1, 12.0.11259376953125.1

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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
7Data not computed